Direct Path-Integral Scheme for Fatigue Simulation of Reinforced Concrete in Shear

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1 Journal of Advanced Concrete Technology Vol. 4, No., 59-77, February 6 / Copyright 6 Japan Concrete Institute 59 Scientific paper Direct Path-Integral Scheme for Fatigue Simulation of Reinforced Concrete in Shear Koichi Maekawa, Kukrit Toongoenthong, Esayas Gebreyouhannes 3 and Toshiharu Kishi 4 Received September 5, accepted 3 December 5 Abstract Path-dependent fatigue constitutive models for concrete tension, compression and rough crack shear are proposed and directly integrated with respect to time and deformational paths actualized in structural concrete. This approach is experimentally verified to be consistent with the fatigue life of materials and structural members under high repetition of forces. The mechanistic background of the extended truss model for fatigue design is also investigated. The coupling of fatigue loads with initial defects is simulated and its applicability is discussed as a versatile tool of performance assessment.. Introduction The truss analogy of reinforced concrete (RC) with web reinforcement in shear was first proposed by Ritter and Morch in the early th century and has been worldwide used in practice. Afterwards, this load-carrying mechanistic analogy was upgraded to take into account the remaining force carried by cracked web concrete (denoted by Vc), and the simple summation law of Vc with Vs carried by web steel was generally recognized. This modified truss model became the basis of the subsequent compression field theory (Collins and Vecchio 98) and strut-tie models (Schlaich and Weischede 98, Marti 985, etc). This macro-behavioral modeling of RC members is the reflection of primary load-carrying mechanism. This truss analogy was further extended to highly repeated shear (e.g., Hawkins 974, Okamura et al. 98, Ueda and Okamura 98), and at present, it serves some fatigue design codes for estimating stress amplitudes of web reinforcement. Figure shows the typical relation of the repeated shear force and mean strain of web reinforcement. Followings are experimentally obtained characteristics by Ueda and Okamura (98). () The increase in the mean strain of stirrups caused by fatigue loading is not highly dependent on the level of maximum shear force if the minimum shear is nearly zero. It means that the shear capacity by concrete decreases according to the numbers of cycles but the reduction rate is almost independent on the magnitude of shear forces. () The shear capacity remains about 6% after million cycles and 7-75% after thousands cycles and Professor, University of Tokyo, Japan. maekawa@concrete.t.u-tokyo.ac.jp Engineer, Taisei Corporation, Japan. 3 Research Fellow, University of Tokyo, Japan. 4 Associate Professor, University of Tokyo, Japan. this S-N relation of Vc is not much dependent on the amount of web reinforcement. (3) Unloading/reloading curves have a unique focal point on the shear force versus stirrup strain diagram before yielding. The steel strain does not reach null even when the shear force is completely removed. This is the great difference from the flexural cracking and longitudinal reinforcement. (4) When the minimum shear force is large, the shear fatigue life is prolonged exponentially. This behavioral knowledge is practical for assessment of stress amplitude of web steel with reasonable accuracy. But, its microscopic mechanism is still under discussion. Ueda et al. (999) proposed a new FE fatigue analysis method. Here, tailored constitutive models were applied in consideration of fatigue effects by reducing stiffness and strength of material models. In this case, the following hypothesis is assumed; (a) initial states of materials are non-damaged, and (b) from the beginning to the end of loading, stress states do not vary much during the damaging process. This original study brought success in simulating highly cyclic responses although cyclic deterioration process of materials is not strictly traced. The authors understand that the above hypothesis holds for initially non-damaged cases and works well for fatigue design. However, the above stated hypothesis may not hold when environmental actions cause corrosion of steel, drying shrinkage and cracking. Stress amplitudes and paths may drastically change and proportional stress history can no longer be assumed. Here, it is required to track the exact damaging process similar to the nonlinear seismic analysis as shown in Fig.. This study aims to investigate the fatigue mechanism by tracing the exact transient process of gradual damaging under repeated shear in use of direct path-integral scheme. It means that the computerized RC members are virtually created, and high cyclic actions are reproduced by integrating the constitutive modeling of materials in regard to damage accumulation. The analysis is extended to the fatigue life

2 6 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, 6 N= 6 shear force N= shear force Truss model at N= (initial) cm Vc.6Vc Vc.6 Vc Truss model at N=million D6@cm f y = 345 MPa 4cm fc =35MPa, ft=mpa 75cm time mean strain of web stirrup D9@4 f y =69 MPa 5cm 35cm section A D6@5cm f y = 345 MPa 5cm double reinforced in web and axial directions section B -Vc focal point Fig. Extended truss model for RC members in highly repeated shear. Seismic-dynamic problems low cycles and higher stresses Fatigue-static problems high cycle and lower stresses Tracing Tracing material material hysteresis and and damages by by integrating rate-incremental constitutive models models with with regard regard to to strain strain and and time time = Direct Direct Integral Integral Scheme dσ=f(low cycle plasto-damaging)d +G(high rate) dt Path-dependent constitutive models Large deformation, low cycle damaging reversed cyclic effect, short-term paths dσ=f(high cycle plasto-damaging)d +G(slow rate) dt Path-dependent constitutive models smaller deformation, high cycle damaging single sided, long-term paths with creep Ground acceleration in time domain Periodical external forces in time domain Fig. Direct path-integral scheme for seismic and fatigue problems. assessment of initially damaged RC members as the coupled problem of fatigue and ambient actions. In this paper, fatigue life model of reinforcement is not discussed but the authors accept the S-N diagram as a well-established knowledge for steel reinforcement.. Computational fatigue model of concrete in compression, tension and shear Recent advance of computational mechanics enables engineers to conduct the direct integral of path-dependent constitutive models by tracing microscopic material states under all cyclic paths of stresses and strains. In the past decade, this scheme has become successful for simulating dynamic response of structures under seismic actions, and partly specified as a tool to assess structural performances in design. Here, the cyclic degradation is expressed by incremental plasticity and damaging. In this study, the direct path-integral scheme is aimed as a more versatile means of fatigue analysis. Then, the authors started the research from the multi-directional fixed crack model (Maekawa et al.

3 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, 6 6 Reinforcement Local strain of steel Crack location Mean strain Yield level Mean stress Local stress of steel Mean stress of steel Local response Averaged response of steel in concrete Mean strain of steel y x Crack First crack x -y :the first coordinate Crack and form Crack and form non quasi-orthogonal relation to orthogonal relation to each other each other θ > π/ π/8 θ < π/ π / 8 Adjustment of coordinate Introduction of coordinate RC x y x θ Crack x x y θ Crack y Local strain of concrete y Crack Crack Cracked concrete Multi-system Shear stress transferred Crack location Mean stress Damage zone Comp. strength reduction Shear slip along a crack. Crack width LOCAL RESPONSE Shear stress transferred Comp. strength reduction Average tensile strain Mean shear strain. Mean normal strain in x-dir. Comp. strength Comp. strain MEAN RESPONSE Range of crack 3 C Range of crack Range α α of α crack α α 4Range of α X Crack 4 crack Crack 3 Crack C3 Crack C C st coordinate C3 C4 st coordinate Based upon maximum four way independent direction α =.5 C4 C Fig. 3 Scheme of multi-directional fixed crack modeling. 3), which has been used for seismic analysis in practice as shown in Fig. 3. On this scheme, D space averaged constitutive model is formed by combining one-dimensional stress-strain relations of cracked concrete in tension, compression and shear. Each constituent modeling is strain path dependent. Here is an important aspect that all component behaviors are time dependent as well. Then, the apparent effect of cyclic actions under high stress magnitude is chiefly attributed to the time-dependent plasticity and fracturing. All fatigue experiments more or less unavoidably include the time-dependent effect. For example, the apparent fatigue life of materials may vary according to the stress frequency (Award and Hilsdorf 974, Raithby and Galloway 974). Maekawa and El-Kashif (4) extracted the accumulated fatigue damage of concrete solid by experimentally subtracting the component of time-dependent deformation and re-formulated more generic constitutive model for compression. This fatigue compression model is hereafter used inside the frame of multi-directional crack modeling (Fig. 3). Similar to the compression model, the tension stiffening/softening and shear transfer models have to be enhanced to consider the cumulative fatigue-time damaging for the direct path-integral scheme. In this study, simple formulae for tension and crack shear are also performed with respect to incremental time and strains, and installed together with the compressive fatigue model.. High cycle damaging in tension As a phenomenological viewpoint, tensile fatigue of plain concrete is associated with crack propagation (Cornelissen and Reinhardt 984, Reinhardt et al. 986, Subramaniam et al. ) and the apparent reduction of tension stiffness of RC element after the first cracking is attributed to the fatigue of bond and subsequent new sectional cracking into tension members. In order to represent these microscopic phenomena in structural analyses, the space-averaging scheme is applied for formulation. Then, the overall stiffness and average strain are adopted as the main state variables rather than the running ligament of a single crack in fracture mechanics. Tension nonlinearity (positive and σ) before and after cracking is assumed to be governed by the fracturing damage rooted in cracks, and no fatigue plasticity is

4 6 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, 6 simply assumed as, σ E () = o K T where, Eo is the initial stiffness of concrete solid. When we deal with compression-tension stress paths, the strain is equal to ( p >) where p is defined as the plastic compressive strain along cracks. The tensile fracture parameter K T is a scalar to stand for path-dependent instantaneous fracturing, time-dependent tension creep and fatigue accumulated damage. For application to the integral scheme with regard to time, the total incremental form yields, dk T = Fdt + Gd + Hd () The derivative denoted by H indicates the instantaneous evolution of tension fracture, which is indicated by the tension stiffness formula for RC and tension softening one for plain concrete (Maekawa et al. 3) as, f α ( α ) ( α) t + H = + cr max, Eo when d > and = max H =, when d or < max (3) where, cr is the crack strain defined equal to f t /E o, and max as the maximum tensile strain in the past strain history measured from the compressive plastic strain. The derivative H characterizes the envelope of tension stress-strain curve after cracking. This formula covers the tension stiffness of RC associated with bond (α=.4). If the element covers plain concrete, this formulation also describes the tension softening or bridging stress across the fracture process zone of concrete. In this case, the value of α depends on the fracture energy of concrete in tension and the sizes of finite elements (Bazant and Oh 983) in the scheme of multi-directional fixed crack formulation (Maekawa et al. 3). The derivative of F indicates the time-dependent fracturing rate proposed by Hisasue (5) as, F = F = 5 6 EoKT S f t S 3 6 S, ( K T when.5), max when cr max < cr (4) where, f t is the uniaxial tensile strength. The damaging rate is assumed dependent on the stress magnitude denoted by S and the updated degraded stiffness as K T. This formulation derives from the uniaxial tension creep experiments of RC members after cracking, and corresponds to dry conditions exposed to normal climate ( degree and 6% RH) as shown in Fig. 4. Here, the self-equilibrated stress is induced by drying and the cracking stress is apparently degraded if the initial stress at the analysis is assumed null. Then, the equivalently reduced tensile strength is used for structural concrete. This modeling macroscopically represents the creep and delayed cracking of concrete, local bond creep and volume change by moisture ingredient. Based on the experiment by Nakasu and Iwatate (996), the cyclic fatigue damage in tension after cracking is idealized with regard to the derivative G as, 5cm Average strain (micro) 5 experiment analysis 4 Elapsed time (day) 97.8cm Single D9 bar used Mix proportion of concrete used Average stress (MPa) 3 5 Average strain (micro) 5cm water cement sand gravel admixture Unit : kg/m 3 experiment analysis Fig. 4 Computed post-cracking tensile creep of RC.

5 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, 6 63 Gd = K T σ σ tp env ~ m d d = 9 γ, o when d < d ~,, cr where σ env = ft tp d ~ =, when d α m = 8, γ max tp, (5) This formula is applicable to the non-cracked state with m= and σ tp /σ env =.. The evolution of fracture parameter results in the increasing strain when the amplitude of stress is kept constant. The modeling formulated is applied to both smeared crack tension stiffness and softening modeling as discussed in the following sections. Figure 5 shows the computed S-N diagram for tensile fatigue strength derived from Eq. 5 before cracking. The fatigue strength of one-million cycles is approximately 6-7% of the static uniaxial strength. Roughly speaking, this is similar to the S-N curve for compression and experimental reports also show this similarity of compression and tension (Tepfers 979, Kodama and Ishikawa 98, ACI Committee-5 98, RILEM Committee-36-RDL 984). The computed strain response under the high cyclic tension is also shown in Fig. 5. Just before the fatigue rupture, the tensile maximum strain reaches approximately.-. time of the one at the first cycle, and this matches the experimental facts by Saito and Imai (983), Kaneko and Ohgishi (99). Figure 6 shows the comparison of analytical predictions derived from Eq. 5 and experiments after cracking. The uniaxial tension fatigue experiment of full amplitude was conducted by Nakasu et al. (996) in use of RC slender column of % reinforcement ratio. The averaged concrete stress transferred by bond was extracted as below. Roughly speaking, the average transferred tension stress through bond is reduced to 5% after one-million cycles for both experiment and analysis. This is also quantitatively close to the assumption by Ueda et al. (999) for their fatigue tension stiffness.. Compression fatigue model The compression model of concrete is formulated in the scheme of elasto-plastic and fracturing concept and we have (El-Kashif and Maekawa 4a), = +, σ = E K (6) e p o e C where, total strain is the sum of elastic and plastic strain components and the compressive stress is related to the elastic strain and compression fracture parameter (damage indicator) denoted by K C as above. It means that the concrete nonlinearity is represented by p and K C. The basic equations can be totally differentiated with respect to time and the elasticity to form the path-integral evolutional laws as, d p dk p p = dt + d e t, e C K C K C = dt + d e t (7) e The derivatives in terms of elastic strain (this is an indicator of the microscopic stress intensity) increment indicate the instantaneous nonlinearity and were proposed for plain concrete as, p = when Fp >, e p = ( Fp / e ) /( Fp / p ) when Fp = e stress amplitude ( σ/ ft ) computed S-N line Cycles of fatigue Computed strain response.e-4.5e-4.e-4 5.E-5.E+ σ/ ft = σ/ ft =.7 Cycles of fatigue Fig. 5 Computed tensile fatigue life and strain responses.

6 64 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, 6 Averaged stress (MPa) Averaged tensile stress f t experiment by Nakasu et al. Tension-stiffness curve σ=f t ( cr /).4 Averaged tensile strain (µ) 5 5 Averaged tensile strain (µ) analysis Stress reduction rate Fig. 6 Tension stiffness under uniaxial fatigue loading analysis (high strain range) middle & low) experiment 4 6 Cycle number (log) K C e = λ when F <, K C = ( Fk / e ) /( Fk / K) + λ e when F = k k (8) where, the instantaneous evolution functions of plasticity and damaging of unconfined plain concrete and the fatigue accumulated damage rate denoted by λ are given by, p.38 exp -.55 e F = p, {-.73β ( - exp( -.5β ))} F k = K exp, β = λ = K R = 9γ, 7 ln e 3 ( K) 8.6 g = (3K + when d <, e 4 ) g R, γ e, tp e,max otherwise, g = (9) where, R represents the effect of strain amplitude of high nonlinearity, γ indicates the normalized amplitude corresponding to updated stress variation, and e,tp denotes the turning point of compressive elastic strain similar to the formulation for tension in Eq. 5. Originally, this formulation was applied to large magnitude of stresses for low cycle fatigue and its applicability was extended to high-cycle by introducing the parameter g in Eq. 9 (Maekawa and El-Kashif 4). Time-dependent plastic and fracturing are significant under higher stresses. Creep rupture in compression can be idealized as the combination of fracturing and plastic evolution accompanying the increasing elasticity. Here, we have the rate functions of plastic and damaging as, p t p = φ t p, t.6 exp F - 6 p φ =. e K K = t t ξ b b b e =.34 exp 4 exp K ξ -, K F k.5( exp( 5 )) 45 ψ = e K t b K = t n ( K F k ),,

7 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, 6 65 M ax. stress/comp.strength K t n (.95 K + F ) =.5 k K-F k <.95, otherwise = () These compressive constitutive equations were verified in terms of material and structural member levels by El-Kashif and Maekawa (4b). This model targets the space-averaged stress strain relation for the referential size of cm. As the compressive softening accompanies the localization, the computational model is adjusted with consistency of fracturing energy and the element size. The computed S-N diagram is shown in Fig. 7. The full cyclic stress path was outlined by large numbers of discrete time steps. Failure was identified by the drastic increase of strain rate and following collapse of static equilibrium. The minimum stress is programmed as zero (single side amplitude) and the computational rate of fatigue stress was. Hz. The constant amplitude was applied to the concrete finite element till the failure. The popular design values of S-N diagram are also shown together. The computed S-N diagram is close to the design specification model derived from plenty of fatigue Log N Fig. 7 Computed S-N diagram of fatigue compressive strength of concrete. Stress Rate / fc (/min) Compressive Stress (MPa) Compressive Strain σ/ fc (amplitude)=.9, specified S-N diagram for design.hz σ/ fc =.5 analysis rate of loading =.Hz data by Award and Hilsdorf. σ/ fc =. analysis σ max / fc =.9. Cycles to Failure Fig. 8 Stress-rate effect of fatigue compressive strength of concrete. tests. For sensitivity check, slow rate of loading of. Hz is conducted as shown in Fig. 7. The fatigue strength is generally reduced by -3% of the slow loading when more time-dependent fracturing and plasticity may evolve (Award and Hilsdorf 974, Raithby and Galloway 974, Maekawa and El-Kashif 4). Under small stresses, the loading rate effect becomes negligible, because the time-dependency is much less in magnitude. For verification of fatigue life with loading rates, the experiment by Award and Hilsdorf (974) was used (see Fig. 8). The maximum stress is 9% of the uniaxial compressive strength and three levels of amplitudes are applied under different stress rates. The degraded fatigue cycles can be seen under the slower rate of stresses and the proposed constitutive model basically matches the facts. In the case of small amplitude (=.f c ), it should be noted that time-dependent plasticity and damaging grow to be large and a small stress deviation may result in large difference of apparent life because the time-averaged stress is high. In the case of middle amplitude (=.5f c ), the nonlinear creep generally gets less during the stress repetition. With consideration of these behavioral characters and reproducibility of experiments, this nonlinearity is thought to be fairly simulated..3 Shear transfer fatigue It is known that the shear transfer across crack planes is degenerated under repeated slip. The crack roughness and contact friction are thought to be degraded after the high cycles. According to the contact density modeling of rough cracks, the transferred shear can be formulated in terms of intrinsic shear slip or shear strain normalized by the crack opening or averaged normal strain (Maekawa et al. 3). The authors conducted fatigue shear transfer experiments to quantify the degenerated shear stiffness under high cycles on a single crack plane (See Fig. 9), and here propose the stiffness reduction rate for extending the applicability of the original contact density modeling as, τ = X τ or ( δ, ω) X = log { + d( δ / ω) },. () where, τ or is the transferred shear stress calculated by the original contact density model, (δ,ω) are the crack shear slip and width, and X is the fatigue modification factor to express the stiffness reduction with regard to accumulated intrinsic shear deformation. Provided that dispersed cracking be assumed in finite element, δ/ω can be replaced with the average shear to normal strain ratio as γ/ (Maekawa et al. 3). The shear transfer fatigue is experimentally known to be predominant when the stress path is reversed cyclic. The above equation is thought to cover the single sided stress amplitude under a dry condition. The reversed cyclic paths may arise under moving loads such as slabs

8 66 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, 6 High Amplitude τ, MPa ω o /σ o /f c =.53/.37/3.6 3 X cycle =δ /δ 4 δ δ ω, mm. τ max =.79 MPa Nearly constant dialtion per cycling of load Moderate Amp δ, mm τ, MPa ω o /σ o /f c =.7/.6/ δ, mm ω, mm τ max =.6 MPa Continuous dialtion dilation Two directional displacement transducers Pin-Ø4mm δ, mm Load 63 Steel Nuts D steel bars Metal Plate t = 5mm δ, mm τ max =.79 MPa.8 cycles Stiffness reduction ratio. 9 cycles τ max =.6 MPa model Accumulated normalized shear slip Fig. 9 Cyclic shear transfer stiffness and fatigue model cycles τ max =.6 MPa log { + d ( δ / ω )} and it does reduce the fatigue life of members dramatically. But, for RC beams in practice, single sided stress path is leading unlike columns..4 Structural analysis Well established nonlinear dynamic analyses for RC structures can be extended to fatigue analyses in time-space domains by direct path-integral of constitutive modeling. The essence is the material modeling as explained in the previous sections. As the fatigue options of concrete compression, tension and shear transfer are just added to the original modeling in terms of evolution derivatives, these extended material models also cover the high nonlinear behaviors. Then, the fatigue analysis system consequently shares the same scheme as that of nonlinear failure analysis under seismic loads. Repeated loads of high cycles can be directly input as the nodal forces in full time steps. However, it takes much time of computation when tons of cycles are provided. In order to accelerate the computation, the logarithmic time derivative can be applied for path-integrating the constitutive models. The time-dependent and damaging rates with respect to strain paths are simply integrated in the numerical scheme as, Compression : d p d p p = ζ t e dt + d, e (a) dk Fk F k K = ζ t + + ζ λ e dt e K Tension : K = ζ F t + ζ G + H (b) Crack shear transfer : log γ X =,. + ζ (c)

9 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, 6 67 where, ζ is the integral acceleration factor. As material nonlinearity in structures greatly evolves at the beginning of fatigue loadings, the value of ζ should be unity for accuracy, i.e., the direct exact paths of forces and corresponding internal stresses shall be tracked in the real time scale. After many cycles, plastic and fracturing rates and the fatigue damaging evolution tend to be much reduced and the increment of nonlinearity via cycle becomes exponentially small. At this stage, the value of ζ can be larger for computational efficiency for each load step. The single cycle of computational load is equivalent to ζ cycles. In Eq., just time and strain path integral are magnified by ζ but it should be noted that the terms of instantaneous nonlinearity associated with evolution boundaries (plasticity and damaging) are not magnified by ζ, but the path is strictly mapped out. Figure shows the computed strain and displacement histories of compression specimen and a RC beam. These were brought by simultaneously integrating Eqs. 6- with different time-integral magnification. For a cycle of loading, discrete steps were assigned. Maximum stress is 8% of the static strength and the computed fatigue life of material is about, cycles. The maximum shear force is 6 kn for the beam fatigue simulation (section A in Fig. ). Two different time integrals are compared. One is the full-range (ζ=.) and the other is the binary sequence (ζ=,, 5,,, 5, ) as shown in Fig.. Both time splits give rise to almost the same results of strain and displacement (less than 5%). The computation time is approximately 5 times shortened by the acceleration scheme and the feasible computation becomes possible. It takes just 3-4 hours of PC-CPU time to complete RC beam fatigue simulation till one-million cycles. Figure shows the computed and experimental mean strains of RC web with stirrups and the applied shear. Design formula for shear capacity predicts the diagonal cracking shear of 9 kn and the web yielding of 5 kn. The computed shear capacity after yielding is 9 kn by the push-over analysis. First, the authors checked the cyclic responses under rather high stresses in experiment. Yielding of web reinforcement partly occurred at the first cycle of loading and the number of repetition was small ( cycles). The mean tensile strain was recorded by using the displacement transducers of 3 mm gauge length crossing the main diagonal cracking. This verification is effective for comparatively higher stress states under plastic conditions. 3. Verification: Extended Truss Model In the frame of extended truss model, the concept of focal point on the unloading-reloading paths is much useful for limit state design as shown in Fig.. The presence of the focal point on the diagram of shear force versus web stress (strain) implies that the residual strain proceeds according to the cycles of loads, and the stiffness with regard to shear force and web steel strain declines. It is clear that the path-dependency of tension stiffness and the shear transfer will cause the residual deformation of web reinforcement when applied shear be removed. Thus, this chapter will discuss the applicability of the computational fatigue analysis in line with the extended truss analogy, which has been already examined by systematic experiments (Ueda and Okamura 98). It can be hardly discussed solely by the experimental approach, but the numerical sensitivity simulation and its comparison with the reality may lead to the internal mechanism. 3. Full amplitudes A simple supported RC beam was selected as the standard specimen for discussion. The shear span to depth ratio is 3. with a T-section flange as shown in Fig. in more detail. The nominal shear capacity of concrete is about kn in analysis and almost the same as the calculated one by Okamura-Higai formulae (98). The Compressive Strain Time (sec) =cycle number (Hz) for whole time, ζ=,,5 -. ζ= ζ= ζ= ζ=5 accelerated time step ζ= (a) One-element material fatigue simulation Average strain of web steel for a whole, ζ= 7.E-4 6.E-4 5.E-4 4.E-4 3.E-4.E-4.E ζ=.e+ ζ= Number of cycles (b) Fatigue simulation of reinforced concrete structures Fig. Magnified direct time and path-integral.

10 68 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, analysis experiment Average strain of RC web with diagonal cracks experiment analysis Average strain of RC web with diagonal cracks Number of cycle Fig. Comparison of analytical and experimental results. Log N 4 3 Vc.75Vc -Vc (Small) shear focal point (Middle) 5 (large amplitude) yielding modified truss model extended truss model (.75Vc, cycles average strain of web (µ) Vc/Vc,max(N=) Design S-N curve for (Vmin/Vmax)=. computed results Small amplitude Middle large 3 4 Log N Fig. Shear force versus mean strain of RC web zone with distributed stirrup. capacity by web reinforcement is kn estimated by the practical truss model. The flexural yield may occur at 4 kn of the shear force. Figure and Fig. 3 show the analytical results on fully repeated shear versus average strain relation of stirrup up to, cycles. Approximately, the growth of stirrup strains is not highly dependent on the shear force amplitude when the maximum shear is not so close to the diagonal cracking capacity denoted by Vc. This result is similar to the reality as pointed out by Okamura, Falghaly and Ueda (98, 98). It experimentally and analytically implies that the shear force carried by concrete may decrease irrespective to the maximum force levels as shown in Fig. 3. This computed S-N relation of cracked web concrete shear in the RC beam is not so far from that proposed by Higai (978), Okamura and Ueda (98) except for the case of small amplitudes. The focal point of the unloading/reloading curves of the diagram is definitely identical in the analysis results as shown in Fig.. The unloading lines converge to the Fig. 3 Extracted S-N diagram of concrete in shear. unique point when reinforcement remains elastic. The dotted lines are prediction by the design formula of Okamura and Ueda (98). 3. Middle and small amplitudes The verification is performed with different minimum and maximum shear forces. For dealing with several dimensions and detailing, the authors used the simple supported RC beam with the rectangular cross-section hereafter as shown in Fig.. The shear span to depth ratio is 3., which is the same as that discussed in section 3.. The nominal shear capacity of concrete is about 9 kn and that of web reinforcement is expected to be 6 kn. The flexural yield may occur at kn of the shear force. This case arranges half reinforcement of that of the previous section. Figure 4 shows the shear force and mean stirrup strain relation. The amplitude is unchanged as 5 kn but the minimum shear force is variable. The ratio of the minimum shear to the maximum one, denoted by r, ranges from.44 to.6. Based upon systematically ar-

11 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, Log N 4 3 Vc/Vc,max.6.4. Computed values Design S-N curve for (Vmin/Vmax) = logn failure Middle amplitude Average strain of web Vc/Vc,max Design S-N curve for (Vmin/Vmax)= logn Log N Low amplitude Average strain of web -8 Fig. 4 Repeated shear force and stirrup strain relation under smaller amplitudes. ranged experiments, Higai (978) proposed the empirical fatigue shear capacity formula, which was also incorporated in the extended truss model by Ueda and Okamura (98) as, log V V =.36*( r r ) c log co N (3) where, r=v c,min /V c,max, N is the number of fatigue cycles and V co is defined as static shear capacity of concrete. This formula and the FE computed results are compared as shown in Fig. 4. The computed fatigue life and the focal point are roughly equivalent to the empirical formula by Eq. 3, but in the case of small amplitudes just after diagonal cracking, computed evolution of stirrup mean strain tends to be slow. In fact, this tendency is also seen in the experiment but in the practical design, this drift is simply ignored for safer side. For further verification, non-web reinforced concrete beams were selected for computing the S-N curve of fatigue life under full and partial amplitudes. The target is the beam having the same rectangular section and detailing as those of the RC beam, but the web steel is removed. The rate of loading is. Hz similar to the experiments. Figure 5 shows the computed S-N curve normalized by the computed static shear capacity. Generally, the analysis coincides with the empirical S-N formula with reasonable accuracy. The crack pattern after many cycles is shown in Fig. 5a. It is clear that the failure mode is the typical shear accompanying diagonal cracking. The S-N diagram of full amplitude is computed for both slender and short beams (Fig. 5b, c). The computed fatigue strength of the slender beam is close to

12 7 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, 6 y RC beam without web reinforcement reference Midspan deflection (mm) 7 cycles 58 cycles V c /V co (a) Crack pattern under fatigue loading..9 FE analysis.9 FE analysis Vc V.4 log.36*( r r ) logn.3 V =.4 c log.36*( r r ) logn co.3 V = co. r = Vmin / V. max.. r = Vmin / Vmax log N log N (b) In case of the slender RC beam with a/d=3.: full amplitude r= log N V log V r = V c co min / max V c /V co =.36 * ( r r ) log N V design formula (c) In case of the short RC beam with a/d=.67: full amplitude r= in case where V c /V co =.7 analysis Index of amplitude: r= V min / V max (d) Effect of amplitude for the slender RC beam (a/d=3.) Fig. 5 Finite element analysis of RC without web reinforcement under fatigue loading.

13 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, 6 7 the empirically derived S-N formula. The short beam exhibits a little higher fatigue capacity as previously reported, and the analysis matches the tendency of the fact (Higai 978). Figure 5d shows the fatigue life related to amplitude. According to the reduced amplitude, the logarithmic fatigue life is increased linearly in computation. Design formula and the computed one are close to each other. 3.3 Effect of shear transfer There is a fact that the diagonal crack plane becomes smooth after many cycles. Repeated shear slip may cause the local frictional failure at contact point and the crack roughness is erased. Thus, the loss of shear transfer mechanism definitely amplifies the web deformation and it must be a part of fatigue deterioration of RC beams. But, it is hard to quantitatively verify the effect of shear transfer deterioration solely by experimental approach. Then, the numerical sensitivity analysis was conducted with provisionally intensified factor for the shear slip accumulation by one-million times as underlined in Table and the sensitivity analysis results in Fig. 6. The web strain becomes larger due to the rapid deterioration of shear transfer and the pattern of unloading-reloading paths is much changed. The focal point of unloading-reloading paths exists at the beginning of cycles. But, comparatively large plastic deformation of RC web reinforcement is generated under higher cycles and the fixed focal point no longer exists. This unrealistic response caused by accelerated shear transfer fatigue means that the fatigue nonlinearity of RC in shear cannot be explained solely by the deterioration of rough crack shear transfer. 3.4 Effect of tension stiffness Another source of fatigue deterioration is the degeneration of local bond or tension stiffness in a macroscopic point of view. Deterioration of bond around the web reinforcement may allow further propagation of diagonal shear cracking and increase in crack width, which may lead to the reduced shear transfer along cracks. Then, we have the sensitivity analysis by assuming fictitiously great evolution of tension-stiffness damage as underlined in Table. Figure 7 shows the shear force versus strain of web reinforcement relation. It is understandable that the high evolution of tension stiffening damage causes the larger strain of web steel. Here, the point of interest is the firm focal point like the fact before yielding. This mechanism of structural fatigue rooted in tension-stiffness has never been discussed although static mechanistic investigation was reported (Ueda et al. 995). The bond of web reinforcement is thought to have less direct effect on capacity. Contrary to the tension stiffness, the shear transfer has been regarded as the shear carrying ability in a direct manner. But, when we integrate these two sorts of sensitivity analyses, tension stiffness fatigue cannot be ignored in discussion as well. Additional shear transfer decay is thought to accelerate the damage evolution as shown in Fig. 8. The influence of shear transfer on the fatigue evolution is compara- τ = X = Table Sensitivity analysis with regard to fatigue evolution of shear transfer. Original modeling τ (, γ ) Sensitivity analysis for trial τ = X τ (, γ ) X or or 6 log { + d( γ / ) },. X = log { + d( γ / ) }, standard case accelerated shear transfer degradation Average strain of web steel Average strain of web steel accelerated shear transfer degradation standard case. Fig. 6 Sensitivity fatigue analysis of RC beams with regard to shear transfer. Number of cycle

14 7 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, 6 ~ 8 d = 9 γ Table Sensitivity analysis with regard to fatigue evolution of tension stiffness. Original modeling Gd = K ( K) d ~ d ~ =, when d d, o γ max tp, when d < Gd = d ~ =, 5 ~ 8 d d = 9 γ, o Sensitivity analysis for trial 9 4 K ( K) d ~ when d tp γ max, when d < Average strain of web steel accelerated tension stiffness degradation Average strain of web steel before yield 8-8 deviating from focal point after yield of web reinforcement accelerated tension stiffness degradation standard case. Number of cycle Fig. 7 Sensitivity fatigue analysis of RC beams with regard to tension stiffness. 8 4 accelerated tension stiffness and shear degradation Average strain of web steel.5 accelerated tension stiffness and shear.4 degradation Average strain of web steel accelerated tension stiffness degradation deviating from focal point after yield of web reinforcement. Number of cycle Fig. 8 Sensitivity fatigue analysis of RC beams with regard to both tension and shear. tively large in the case where the tension stress transfer across cracking is much deteriorated. Thus, the combined effect should be focused. 3.5 Sensitivity to fatigue life of non-web reinforced RC beam Another series of sensitivity analyses were conducted with the non-web reinforced beam (see Fig. 5a). The standard material models lead to the fatigue capacity, which is almost consistent with the empirical formula as shown in Fig. 5b. Even if the fatigue evolution of compression is forced to zero and other models remain unchanged, the S-N curve of RC is not much influenced as shown in Fig. 9. But, if the tension fatigue model before and after cracking is solely avoided, the computed fatigue life is substantially prolonged. It means that the tension fatigue modeling plays a substantial role. The influence of shear transfer fatigue on the structural fatigue life is also examined by assuming that the fatigue factor denoted by X is unity all the time in Eq.. The shear transfer fatigue modeling is also significant as shown in Fig. 9, but its

15 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, 6 73 V c /V co Vc log =.36*( r r ) logn V r = V co min / Vmax compression fatigue ignored shear transfer fatigue ignored tension fatigue ignored log N Fig. 9 Fatigue life sensitivity analysis of RC beams without web reinforcement. influence on the structural capacity seems comparatively smaller than the tension fatigue. The large amplitude or the maximum shear produces low cycle fatigue. In this case, the fatigue models of tension, compression and shear seem to play a minor role. In fact, the apparent cumulative damaging is chiefly caused by the time dependent nonlinear evolution already formulated in the original modeling. Analytically, the creep rupture may occur even when the shear force is kept constant as zero-amplitude according to time. 4. Corrosive cracking damage on fatigue life Fatigue behavior of RC structures may be influenced by corrosive cracking formed along main reinforcement. In this chapter, an effect of longitudinal cracking on fatigue strength is discussed. The corrosion gel formation and its mechanical effect are incorporated into the nonlinear analysis (Coronelli and Gambarova 4, Toongoenthong and Maekawa 5a, 5b). The authors combined this procedure with the fatigue modeling of materials in order to examine the function of direct path integral scheme. 4. Corrosive cracking along main reinforcement - without shear reinforcement - Elevated shear capacity is experimentally known when corrosive cracking is induced along main reinforcement within the shear span of a RC beam with no web reinforcement (Matsuo et al. 4), and some predictive methods were proposed (Toongoenthong and Maekawa 5a). The increased shear capacity due to the corrosive cracking is also shown in Fig. together with the numerical crack pattern at the failure. It is clear that the shear load carrying mechanism is definitely changed. The corrosion cracking may reduce the bond of main reinforcement and form the tied-arch action which results in the increased static shear capacity. This phenomenon was also experienced and the analytical result follows this fact. The behavior under fatigue loading is numerically investigated under the amplitude of 7% of the static strength of the non-damaged beam (no flange in Fig. ). The growing mid-span deflection and the induced shear cracking at each cycle are shown in Fig.. The large initial deflection of the first cycle for the damaged beam is caused by the pre-corrosive cracking. The numerical crack pattern is also shown for both sound and pre-damaged beams. The diagonal shear crack takes place subsequently for the pre-damaged beam, i.e. 58 cycles for the sound beam, and 5, cycles for the pre-damaged beam. The extended fatigue life of corroded RC beams without shear reinforcement is not quantitatively formulated in maintenance procedure but similar experience has been earned with ASR damaged slabs. ASR expansion may induce laminated cracking along member axes and they arrest shear crack propagation and prolong the fatigue life. 4. Corrosive cracking along tensile reinforcement at anchorage area - with shear reinforcement - As shown in Fig., the monotonic loading analysis is compared between the sound beam and the one with corrosive pre-cracking in the anchorage zone, where the web reinforcement is placed analytically. The width of 8 4 No-crk pre-crk.5 mm Midspan deflection (mm) No crack without stirrups.5 mm crack without stirrups Fig. Shear force versus displacement relation of corroded RC beam without stirrups.

16 74 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, reference crack.5mm inside Midspan deflection (mm) 7 cycles 58 cycles a) Sound beam without corrosive cracking 58 cycles 5, cycles b) Pre-damaged beam with corrosive cracking Fig. Fatigue life of corroded RC beam without web reinforcement. 5 Load (kn) 5 5 reference crk at anchorage 5 5 Displacement (mm) No crack with stirrups.5 mm crack at anchorage Fig. Monotonic loading response of anchorage damaged RC beam. corrosive cracking is controlled to.5 mm by introducing the forced corrosion strain. Here, the strain localization of plain concrete is assumed normal to the main reinforcement in line with the multi-directional fixed crack approach (Toongoenthong and Maekawa 5a). The monotonic loading also shows less influence on the load carrying capacity and stiffness of the damaged beam compared to the referential one. The web reinforcement may successfully restrict the bond crack propagation at the anchorage zones. Although the initial longitudinal damage exists around anchorage, the anchorage failure along the corrosive crack plane is prevented by enough shear reinforcement. Thus, the truss shear resisting mechanism can effectively function. The behavior under high repetition of load is numerically investigated under the amplitude of. tonf with the loading range of 5-65% of the maximum capacity. Figure 3 shows the accelerated damage initiated from the corrosive crack tip and the rapid evolution of average web strain can be seen at the member sections. The section- does not have crack intersection at the beginning of loading and gives rise to small deformation. In this case, the stable focal point is not identical in analysis, too. The apparent reduction in concrete shear capacity was obtained with the increase in number of loading cycles at section- near the corrosive cracking tip. To investigate the risk of stirrup rupture, the average web strain is converted to the induced stress in the web reinforcement. Figure 4 shows the response of cycle number and the stress level normalized by the yield strength inside the web reinforcement.

17 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, Average Crack width.5 mm anchorage Avg. Avg. at th cycle crack around anchorage no crk Average web strain (a) Response of section- Average at,th cycle 8 4 crack around anchorage no crk at,th cycle Average web strain (b) Response of section- Fig. 3 Fatigue response of stirrups around the damaged anchorage. Normalized steel stress Average crk at anchorage no crk Number of Cycles Average Normalized Number avgerage of Cycles steel stress Fig. 4 Stress history for assessment of fatigue rupture of web reinforcement. Normalized Number of steel Cycles stress crk at anchorage no crk 5. Conclusions Similar to the nonlinear dynamic analysis method for seismic resistance, the simplified fatigue constitutive modeling was incorporated into the direct integral scheme of loading paths, and this approach was verified to be consistent with the fatigue life of material and structural members in shear. Experimental verification was also conducted in comparison with the experimental facts and well-known design formulae, and the extended truss model for fatigue design was mechanistically confirmed. The coupling of the fatigue analysis with the corrosion of reinforcement or initial defects was tried and qualitatively examined. The followings are the issues of

18 76 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No., 59-77, 6 future investigation and development. () The applicability of each fatigue material modeling shall be extended for more generalization. The shear transfer fatigue modeling, which has much to do with the presence of the focal point, can be further promoted for flatter crack planes of high strength concrete. High cyclic tension stiffness under large mean strain close to the yield one of reinforcement should be investigated. () The modeling used in this study may cover drying conditions of about 6% RH under room temperature. For dealing with natural environmental conditions, dry-wet cycles should be encompassed especially for fatigue analysis of RC slabs. The coupling of moisture, corrosion and fatigue will be the target of investigation. Acknowledgments The authors express their gratitude to Ms. T. T. Myo and Mr. K. Hisasue for their assistance to conduct experiments. This study was financially supported by Grant-in-Aid for Scientific Research (S) No.568. References ACI Committee 5 (98). Fatigue of Concrete Structures. Shah, S. P. ed., American Concrete Institute. Award, M. E. and Hilsdorf, H. K. (974). Strength and deformation characteristics of plain concrete subjected to high repeated and sustained loads. ACI publication SP 4-, -3. Bazant, Z. P. and Oh, B. H. (983). Crack band theory for fracture of concrete, Materials and Structures, RILEM, 6 (93), Collins, M. P. and Vecchio, F. (98). The response of reinforced concrete to in-plane shear and normal stresses. University of Toronto. Cornelissen, H. A. W. and Reinhardt, H. W. (984). Uniaxial tensile fatigue failure of concrete under constant-amplitude and program loading. Magazine of Concrete Research, 36 (9), 6-6. Coronelli, D. and Gambarova, P. (4). Structural assessment of corroded reinforced concrete beams: modeling guidelines. Journal of Structural Engineering, ASCE, 3 (8), 4-4. El-Kashif, K. F. and Maekawa, K. (4a). Time-dependent nonlinearity of compression softening in concrete. Journal of Advanced Concrete Technology, (), El-Kashif, K. F. and Maekawa, K. (4b). Time-dependent post-peak softening of RC members in flexure. Journal of Advanced Concrete Technology, (3), Hawkins, N. M. (974). Fatigue characteristics in bond and shear of reinforced concrete beams. ACI publication SP-4, Higai, T. (978). Fundamental study on shear failure of reinforced concrete beams. Proc. of JSCE, No.79, 3-6. Hisasue, K. and Maekawa, K. (5). Time-dependent tension stiffness of reinforced concrete and effect of drying shrinkage. Transaction of JSCE, V, Kaneko, R. and Ohgishi, S. (99). Direct-tension and splitting tensile fatigue characteristics on plain concrete. Journal of Structural and Construction Engineering, Architectural Institute of Japan, 49 (), Kodama, K. and Ishikawa, Y. (98). Study on fatigue characteristics of concrete. Proc. of JCI, 4, Marti, P. (985). Basic tools of reinforced concrete beam design. ACI Journal, Proceedings, 8 (), Maekawa, K. and El-Kashif, K. F. (4). Cyclic cumulative damaging of reinforced concrete in post-peak regions. Journal of Advanced Concrete Technology, (), Maekawa, K, Pimanmas, A. and Okamura, H. (3). Nonlinear Mechanics of Reinforced Concrete. Spon Press, London. Nakasu, M. and Iwatate, J. (996). Fatigue experiment on bond between concrete and reinforcement. Transaction of JSCE, V-46, Okamura, H., Farghaly, S. A. and Ueda, T. (98). Behaviors of reinforced concrete beams with stirrups failing in shear under fatigue loading. Proc. of JSCE, No.38, 9-. Okamura, H. and Higai, T. (98). Proposed design equation for shear strength of reinforced concrete beams without web reinforcement. Proc. of JSCE, No.3, 3-4. Raithby, K. D. and Galloway, J. W. (974). Effect of moisture condition, age and rate of loading on fatigue of plain concrete. ACI publication SP 4-, Schlaich, J. and Weischede, D. (98). Detailing of Concrete Structures. Bulletin d Information 5, Comité Euro-International du Béton, Paris, 63. Reinhardt, H. W., Cornelissen, H. A. W. and Hordijk, D. A. (986). Tensile tests and failure analysis of concrete. Journal of Structural Engineering, ASCE, RILEM Committee 36-RDL (984). Long term random dynamic loading of concrete structures. Materials and Structures, 7 (9), RILEM, -8. Saito, M. and Imai, S. (983). Direct tensile fatigue of concrete by the use of friction grips. ACI Journal, Sept.-Oct., Subramaniam K. V., Popovics, J. S. and Shah, P. (). Fatigue fracture of concrete subjected to biaxial stresses in the tensile C-T region. Journal of Engineering Mechanics, ASCE, 8 (6), Tepfers, R. (979). Tensile fatigue strength of plain concrete. ACI Journal, August, Toongoenthong, K. and Maekawa, K. (5a). Multi-mechanical approach to structural performance assessment of corroded RC members in shear. Journal of Advanced Concrete Technology, 3 (), 7-. Toongoenthong, K. and Maekawa, K. (5b).

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