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

Journal of Advanced Concrete Technology Vol. 4, No. 1, 159-177, February 2006 / Copyright © 2006 Japan Concrete Institute 159

Scientific paper

Direct Path-Integral Scheme for Fatigue Simulation of Reinforced Concrete in Shear Koichi Maekawa1, Kukrit Toongoenthong2, Esayas Gebreyouhannes3 and Toshiharu Kishi4

Received 10 September 2005, accepted 30 December 2005

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 ex-perimentally 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 assess-ment.

1. Introduction

The truss analogy of reinforced concrete (RC) with web reinforcement in shear was first proposed by Ritter and Morch in the early 20th century and has been worldwide used in practice. Afterwards, this load-carrying mecha-nistic analogy was upgraded to take into account the remaining force carried by cracked web concrete (de-noted 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 1982) and strut-tie models (Schlaich and Weischede 1982, Marti 1985, etc). This macro-behavioral modeling of RC members is the reflection of primary load-carrying mechanism.

This truss analogy was further extended to highly re-peated shear (e.g., Hawkins 1974, Okamura et al. 1981, Ueda and Okamura 1982), and at present, it serves some fatigue design codes for estimating stress amplitudes of web reinforcement. Figure 1 shows the typical relation of the repeated shear force and mean strain of web rein-forcement. Followings are experimentally obtained characteristics by Ueda and Okamura (1982).

(1) 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 de-creases according to the numbers of cycles but the re-duction rate is almost independent on the magnitude of shear forces.

(2) The shear capacity remains about 60% after 1 million cycles and 70-75% after 10 thousands cycles and

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. (1999) 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 fol-lowing hypothesis is assumed; (a) initial states of mate-rials 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 dete-rioration 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. 2. 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

1Professor, University of Tokyo, Japan. E-mail: [email protected] 2Engineer, Taisei Corporation, Japan. 3Research Fellow, University of Tokyo, Japan. 4Associate Professor, University of Tokyo, Japan.

160 K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No. 1, 159-177, 2006

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 dis-cussed but the authors accept the S-N diagram as a well-established knowledge for steel reinforcement.

2. Computational fatigue model of concrete in compression, tension and shear

Recent advance of computational mechanics enables engineers to conduct the direct integral of path-de-

pendent 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.

Vc

0.6Vc

-Vc

N=1N=106

time

shear force shear force

mean strain of web stirrup

focal point

Truss model

at N=1 (initial)

Truss model at N=1million

Vc 0.6 Vc

100cm

240cmfc’=35MPa, ft=20MPa

section A section B

double reinforced in web and axialdirections

D6@10cmfy= 345 MPa

D19@4fy=690 MPa

25cm

35cm D6@5cmfy= 345 MPa

75cm

25cm

Fig. 1 Extended truss model for RC members in highly repeated shear.

Fig. 2 Direct path-integral scheme for seismic and fatigue problems.

-800

-600

-400

-200

0

200

400

600

0 5 10 15 20

0

0 .2

0 .4

0 .6

0 .8

1

1 .2

1 .4

1 .6

1 .8

2

0 1 0 2 0 3 0 4 0 5 0

Fatigue-static problemshigh cycle and lower stresses

Seismic-dynamic problemslow cycles and higher stresses

Ground accelerationin time domain

Periodical external forcesin time domain

Tracing material hysteresis and damages by integrating rate-incremental constitutive models with regard to strain and time = Direct Integral Scheme

Tracing material hysteresis and damages by integrating rate-incremental constitutive models with regard to strain and time = Direct Integral Scheme

Path-dependent constitutive modelsLarge deformation, low cycle damagingreversed cyclic effect, short-term paths

Path-dependent constitutive modelssmaller deformation, high cycle damagingsingle sided, long-term paths with creep

dσ=F(low cycle plasto-damaging)dε+G(high rate) dt

dσ=F(high cycle plasto-damaging)dε+G(slow rate) dt

K. Maekawa, K. Toongoenthong, E. Gebreyouhannes and T. Kishi / Journal of Advanced Concrete Technology Vol. 4, No. 1, 159-177, 2006 161

2003), which has been used for seismic analysis in prac-tice as shown in Fig. 3.

On this scheme, 2D 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 1974, Raithby and Gal-loway 1974). Maekawa and El-Kashif (2004) 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.

2.1 High cycle damaging in tension As a phenomenological viewpoint, tensile fatigue of plain concrete is associated with crack propagation (Cornelissen and Reinhardt 1984, Reinhardt et al. 1986, Subramaniam et al. 2002) 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 me-chanics.

Tension nonlinearity (positive ε and σ) before and after cracking is assumed to be governed by the fractur-ing damage rooted in cracks, and no fatigue plasticity is

Cracked concrete

RC

Reinforcement

Local response

Mean stress of steel

Mean strain of steel

Averaged response ofsteel in concrete

Mean stressYield level

Crack locationMean strain

Local strain of steel

Local stress of steel

Average tensile strain

Crack widthCom

p. s

tren

gth

redu

ctio

n

1.0

Shear slip along a crack

Shea

r st

ress

tr

ansf

erre

d

Mean shear strain

Shea

r st

ress

tr

ansf

erre

d

Mean normal strain in x-dir.

Com

p.

stre

ngth

re

duct

ion 1.0

Comp. strain

Crack location

Local strain of concrete

Mean stress Damage zone

LOCAL RESPONSE MEAN RESPONSEMulti-system

Com

p.

stre

ngth

x1

y1

Crack 1 First crack

x1-y1:the firstcoordinate

θ

Crack 1

x1

y1

x1

y1

Crack 2

Adjustment of coordinate 1

θ

Crack 2

x1

y1

x2

y2 Crack 1

Introduction of coordinate 2

Crack 1 and 2 formquasi-orthogonal relation to

each other

Crack 1 and 2 form nonorthogonal relation to

each other8/2/ π−π>θ 8/2/ π−π<θ

X

C1

C2

α α

α

α

= 22.5α

Crack 2

α2

α2

Range of crack 2Range of crack 4Range of crack 1

Rang

e of c

rack

3Crack 3

Crack 4C3

C4

Based upon maximum four way independent direction

1st coordinateC1 C2 2st coordinate C3 C4

Crack 1

Fig. 3 Scheme of multi-directional fixed crack modeling.


simply assumed as,

εσ ⋅= To KE (1)

where, Eo is the initial stiffness of concrete solid. When we deal with compression-tension stress paths, the strain ε is equal to (ε−εp>0) where εp is defined as the plastic compressive strain along cracks. The tensile fracture parameter KT 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,

εε HdGdFdtdKT ++= (2)

The derivative denoted by H indicates the instanta-neous evolution of tension fracture, which is indicated by the tension stiffness formula for RC and tension soften-ing one for plain concrete (Maekawa et al. 2003) as,

max

max

)2(max

0,00

,)1(

εεεεεε

εεα αα

<≤=

=>

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛+−= +−

ordwhenHanddwhen

Ef

H cro

t

(3)

where, εcr is the crack strain defined equal to 2ft/Eo, and εmax as the maximum tensile strain in the past strain his-tory 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=0.4).

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 1983) in the scheme of multi-directional fixed crack formulation (Maekawa et al. 2003). The derivative of F indicates the time-dependent fracturing rate proposed by Hisasue (2005) as,

t

To

cr

crT

fKE

S

whenSF

whenKSF

εεε

εε

≡

≥⋅−=

<−⋅⋅−=−

−

max66

max235

,10

,)5.0(10

(4)

where, ft is the uniaxial tensile strength. The damaging rate is assumed dependent on the stress magnitude de-noted by S and the updated degraded stiffness as KT. This formulation derives from the uniaxial tension creep ex-periments of RC members after cracking, and corre-sponds to dry conditions exposed to normal climate (20 degree and 60% 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 vol-ume change by moisture ingredient.

Based on the experiment by Nakasu and Iwatate (1996), the cyclic fatigue damage in tension after crack-ing is idealized with regard to the derivative G as,

Fig. 4 Computed post-cracking tensile creep of RC.

0

500

1000

0 20 400

1

2

3

0 500 1000

15cm

297.8ｃｍ 15cm

３３４.２cement

９６１.４８１３.８sand gravelwater

１７５.７０．８３５

admixture

Single D19 bar used

Unit : kg/m3

Average strain (micro)

Ave

rage

stre

ss (M

Pa)

Elapsed time (day)

Ave

rage

stra

in (m

icro

)

experimentanalysis

experimentanalysis

Mix proportion of concrete usedwater cement sand gravel admixture 175.7 334.2 813.8 961.4 0.835


0

,,8,9~

0,0~

,

,~

max

20

<

−−≡=⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅⋅=

≥=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=

ε

εεε

γεεγε

εε

εε

σ

εσσ

ε

α

dwhen

mdd

dwhend

fwhere

dKGd

tp

o

m

tp

crtenv

env

tpT

(5)

This formula is applicable to the non-cracked state with m=20 and σtp/σenv=1.0. The evolution of fracture parameter results in the increasing strain when the am-plitude of stress is kept constant. The modeling formu-lated 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 60-70% of the static uniaxial strength. Roughly speaking, this is similar to the S-N curve for compression and ex-perimental reports also show this similarity of compres-sion and tension (Tepfers 1979, Kodama and Ishikawa 1982, ACI Committee-215 1982, RILEM Commit-tee-36-RDL 1984). 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 1.2-2.2 time of the one at the first cycle, and this matches the experimental facts by Saito and Imai (1983), Kaneko and Ohgishi (1991).

Figure 6 shows the comparison of analytical predic-tions derived from Eq. 5 and experiments after cracking. The uniaxial tension fatigue experiment of full amplitude was conducted by Nakasu et al. (1996) in use of RC slender column of 2% 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 50% after one-million

cycles for both experiment and analysis. This is also quantitatively close to the assumption by Ueda et al. (1999) for their fatigue tension stiffness.

2.2 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 2004a),

, e p o e CE Kε ε ε σ ε= + = (6)

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 (dam-age indicator) denoted by KC as above. It means that the concrete nonlinearity is represented by εp and KC. The basic equations can be totally differentiated with respect to time and the elasticity to form the path-integral evo-lutional laws as,

ee

ppp t

d εεεε

ε d dt ⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂= ,

ee

CCC

Kt

KdK ε

εd dt ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠

⎞⎜⎝

⎛∂

∂= (7)

The derivatives in terms of elastic strain (this is an indicator of the microscopic stress intensity) increment indicate the instantaneous nonlinearity and were pro-posed for plain concrete as,

0F when 0 p >=⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂

e

p

εε

,

0F when )/ /()/ ( p =∂∂∂∂−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂ppep

e

p FF εεεε

Fig. 5 Computed tensile fatigue life and strain responses.

0.5

0.6

0.7

0.8

0.9

1

10 100 1000 10000 100000 1000000

Cycles of fatigue

stre

ss a

mpl

itude

(∆σ/

ft’)

computed S-N line

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

1 10 100 1000 10000 100000 1000000

Cycles of fatigue

Com

pute

d st

rain

resp

onse

∆σ/ ft’=0.7

∆σ/ ft’=0.95 0.900.85

0.800.75


0Fwhen

)/ /()/ (

0,F when

k

k

=

+∂∂∂∂−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

<=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

λεε

λε

KFFK

K

keke

C

e

C

(8)

where, the instantaneous evolution functions of plasticity and damaging of unconfined plain concrete and the fa-tigue accumulated damage rate denoted by λ are given by,

1-55.0

exp038.0 p ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛−= epF

εε ,

( )( ){ }ββ 1.25-exp-10.73-exp −= KFk ,

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

207-1ln

0.351- eεβ

0g otherwise, 0, dwhen

101

6.0g

,9R

R,g)1(

)2230(

max,

,8

43

=<+

=

−−≡=

⋅⋅−⋅=

−

e

K

e

tpe

KK

ε

εεε

γγ

λ

(9)

where, R represents the effect of strain amplitude of high nonlinearity, γ indicates the normalized amplitude cor-responding to updated stress variation, and εe,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 2004).

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,

14

exp034.0, b

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠

⎞⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂=

∂

∂ e

b

ppp

tttεεε

φε

,

6-exp 2.1

6.0

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=

e

pFε

φ

1-exptK

tK

b⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎠⎞

⎜⎝⎛∂∂

=∂∂

kFKKξ ,

))5exp(1(5.045 ⎥⎦⎤

⎢⎣⎡ −−−= eεψξ

)( b

kn

FKtK

tK

−⎟⎠⎞

⎜⎝⎛∂∂

=⎟⎠⎞

⎜⎝⎛∂∂ ,

Fig. 6 Tension stiffness under uniaxial fatigue loading. -10

-5

0

5

10

15

20

25

0 500 1000 1500 2000

ft

Ave

rage

d te

nsile

stre

ss

Averaged tensile strain (µ)

11

106 106

103103

analysis

Cycle number (log)

Stre

ss re

duct

ion

rate

0

0.2

0.4

0.6

0.8

1

0 2 4 6

1000 2000Averaged tensile strain (µ)

1.5

1.0

0.5

Ave

rage

d st

ress

(MPa

)

105104

103

1

1

1

105104

103

Tension-stiffness curveσ=ft (εcr /ε)0.4

experiment by Nakasu et al.

experiment

analysis (high strain range)middle & low)


( )295.005.0 kn

FKtK

+−−=⎟⎠⎞

⎜⎝⎛

∂∂ ,

K-Fk <0.95, otherwise =0 (10)

These compressive constitutive equations were veri-fied in terms of material and structural member levels by El-Kashif and Maekawa (2004b). This model targets the space-averaged stress strain relation for the referential size of 20 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 1.0 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 de-sign specification model derived from plenty of fatigue

tests. For sensitivity check, slow rate of loading of 0.01 Hz is conducted as shown in Fig. 7. The fatigue strength is generally reduced by 20-30% of the slow loading when more time-dependent fracturing and plasticity may evolve (Award and Hilsdorf 1974, Raithby and Galloway 1974, Maekawa and El-Kashif 2004). Under small stresses, the loading rate effect becomes negligible, be-cause the time-dependency is much less in magnitude.

For verification of fatigue life with loading rates, the experiment by Award and Hilsdorf (1974) was used (see Fig. 8). The maximum stress is 90% 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 (=0.1fc’), 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 am-plitude (=0.5fc’), 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.

2.3 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. 2003). 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,

{ } 1.0,|)/(|1log1011

),(

10 ≥+−=

⋅=

∫ ωδ

ωδττ

dX

X or (11)

where, τor 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 accumu-lated intrinsic shear deformation. Provided that dispersed cracking be assumed in finite element, δ/ω can be re-placed with the average shear to normal strain ratio as γ/ε (Maekawa et al. 2003).

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

Fig. 7 Computed S-N diagram of fatigue compressive strength of concrete.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6

Max

. stre

ss/C

omp.

Stre

ngth

Log N

specified S-N diagram for design

0.01Hzanalysis rate ofloading = 1.0Hz0

50

100

150

200

250

300

0 0.0005 0.001 0.0015 0.002 0.0025 0.003

Compressive Strain

Compressive Stress (MPa)30

20

10

0.001 0.002 0.003

Fig. 8 Stress-rate effect of fatigue compressive strength of concrete.

0.001

0.01

0.1

1

1 10 100 1000 10000 100000

Cycles to Failure

Stre

ss R

ate

/ fc’

(/min

)

σmax/ fc’ = 0.9

∆σ/ fc’ (amplitude)= 0.9 ∆σ/ fc’ = 0.5

∆σ/ fc’ = 0.1

data byAward and Hilsdorf

analysis


and it does reduce the fatigue life of members dramati-cally. But, for RC beams in practice, single sided stress path is leading unlike columns. 2.4 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 constitu-tive 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 pro-vided. In order to accelerate the computation, the loga-rithmic 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 :

ek

e

k

ee

ppp

KFF

tdtdKK

dd

tdt

d

ελζε

ζ

εεεε

ζε

∆⎥⎥⎦

⎤

⎢⎢⎣

⎡⋅+⎟

⎠

⎞⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−+∆⎟⎠⎞

⎜⎝⎛=∆

∆⎟⎟⎠

⎞⎜⎜⎝

⎛+∆⎟⎟

⎠

⎞⎜⎜⎝

⎛=∆ ,

(12a)

Tension :

εεζζ ∆+∆⋅+∆⋅=∆ HGtFK (12b)

Crack shear transfer :

1.0,1log1011 10 ≥⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛∆⋅+−= ∑ ε

γζX (12c)

Fig. 9 Cyclic shear transfer stiffness and fatigue model.

0 1 2 3 4 50

1

2

3 τmax

=2.79 M Pa

δ , mm

10 40 100 1000

δ1000

X1000 cycle=δ1/δ1000

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

δ , m m

Nearly constant dialtion per cycling of load

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

δ , mm

ωo/σ

o/f

c = 0.53/0.037/32.6

ω , m mτ , MPa

τ , M Pa ω , m m

δ , mm

Continuous dialtion

0 1 2 3 4 50

1

2

Mod

erat

e Am

p...

9000098315τmax=2.26 MPa

ωo/σ

o/f

c = 0.70/0.06/33.1

Hig

h Am

plitu

de δ1

Continuous dilation

LoadLoadTwo directional

displacement transducers

Two directional displacement

transducersD22 steel barsD22 steel bars

Metal Platet = 25mmMetal Platet = 25mm 150150

280280

630630

Pin-Ø4mmPin-Ø4mm

Steel NutsSteel Nuts

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

( ){ }∫+ ωδ /1log10 dAccumulated normalized shear slip

Stif

fnes

s re

duct

ion

ratio

τmax=2.79 MPa1000 cycles

90000 cyclesτmax=2.26 MPa

model

983 cyclesτmax=2.26 MPa


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 corre-sponding 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. 12, 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 10 shows the computed strain and displace-ment histories of compression specimen and a RC beam. These were brought by simultaneously integrating Eqs. 6-10 with different time-integral magnification. For a cycle of loading, 20 discrete steps were assigned. Maximum stress is 80% of the static strength and the computed fatigue life of material is about 20,000 cycles. The maximum shear force is 160 kN for the beam fatigue simulation (section A in Fig. 1). Two different time in-tegrals are compared. One is the full-range (ζ=1.0) and the other is the binary sequence (ζ=1, 2, 5, 10, 20, 50, 100 …) as shown in Fig. 10. Both time splits give rise to almost the same results of strain and displacement (less than 5%). The computation time is approximately 50 times shortened by the acceleration scheme and the fea-sible computation becomes possible. It takes just 3-4 hours of PC-CPU time to complete RC beam fatigue simulation till one-million cycles.

Figure 11 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 90 kN and the web yielding of 150 kN.

The computed shear capacity after yielding is 190 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 (100 cycles). The mean tensile strain was recorded by using the displacement transducers of 300 mm gauge length crossing the main diagonal cracking. This verifi-cation 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. 1. 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 compu-tational fatigue analysis in line with the extended truss analogy, which has been already examined by systematic experiments (Ueda and Okamura 1982). 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. 3.1 Full amplitudes A simple supported RC beam was selected as the stan-dard specimen for discussion. The shear span to depth ratio is 3.2 with a T-section flange as shown in Fig. 1 in more detail. The nominal shear capacity of concrete is about 110 kN in analysis and almost the same as the calculated one by Okamura-Higai formulae (1980). The

-0.005

-0.004

-0.003

-0.002

-0.001

0

0 5000 10000 15000 20000

ζ=100

ζ=50

ζ=20ζ=10

ζ= 1,2,5

for whole time, ζ=1

Time (sec) =cycle number (1Hz)

Com

pres

sive

Stra

in

accelerated time step 0.00E+00

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

6.00E-04

7.00E-04

1 10 100 1000

Average strain of web steel

Number of cycles

for a whole, ζ=1

ζ=1005020105

ζ=12

(a) One-element material fatigue simulation (b) Fatigue simulation of reinforced concrete structures

Fig. 10 Magnified direct time and path-integral.


capacity by web reinforcement is 121 kN estimated by the practical truss model. The flexural yield may occur at 420 kN of the shear force.

Figure 12 and Fig. 13 show the analytical results on fully repeated shear versus average strain relation of stirrup up to 10,000 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, Fal-ghaly and Ueda (1981, 1982). It experimentally and analytically implies that the shear force carried by con-crete may decrease irrespective to the maximum force levels as shown in Fig. 13. This computed S-N relation of cracked web concrete shear in the RC beam is not so far from that proposed by Higai (1978), Okamura and Ueda (1982) except for the case of small amplitudes. The focal point of the unloading/reloading curves of the dia-gram is definitely identical in the analysis results as shown in Fig. 12. The unloading lines converge to the

unique point when reinforcement remains elastic. The dotted lines are prediction by the design formula of Okamura and Ueda (1982).

3.2 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. 1. The shear span to depth ratio is 3.2, which is the same as that discussed in section 3.1. The nominal shear capacity of concrete is about 90 kN and that of web reinforcement is expected to be 60 kN. The flexural yield may occur at 210 kN of the shear force. This case arranges half reinforcement of that of the pre-vious section. Figure 14 shows the shear force and mean stirrup strain relation. The amplitude is unchanged as 50 kN but the minimum shear force is variable. The ratio of the minimum shear to the maximum one, denoted by r, ranges from 0.44 to 0.6. Based upon systematically ar-

Fig. 11 Comparison of analytical and experimental results.

0

50

100

150

200

250

300

350

0 0.001 0.002 0.003 0.004 0.005

Average strain of RC web with diagonal cracks

100

50

150

Shea

r for

ce (k

N)

experiment

analysis

0

0.001

0.002

0.003

0.004

0.005

0.006

0 2000 4000 6000 8000 10000

Aver

age

stra

in o

f RC

web

w

ith d

iago

nal c

rack

s

20 40 60 80 100Number of cycle

experiment

analysis

0

1

2

3

4

20001000 1500

Log N

average strain of web (µ)

20

0

Vc0.75Vc

-Vc

yielding

shear focal point

modified truss model

extended truss model(0.75Vc 10,000 cycles

(large amplitude)(Middle)(Small)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4

Vc/V

c,m

ax(N

=1)

computed results

Design S-N curve for(Vmin/Vmax)=0.0

Small amplitudeMiddlelarge

Log N

Fig. 12 Shear force versus mean strain of RC web zone with distributed stirrup.

Fig. 13 Extracted S-N diagram of concrete in shear.


ranged experiments, Higai (1978) proposed the empirical fatigue shear capacity formula, which was also incorpo-rated in the extended truss model by Ueda and Okamura (1982) as,

NrrVV

co

c1010 log|)|1(*036.0log ⋅−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ (13)

where, r=Vc,min/Vc,max, N is the number of fatigue cycles and Vco is defined as static shear capacity of concrete. This formula and the FE computed results are compared as shown in Fig. 14. The computed fatigue life and the focal point are roughly equivalent to the empirical for-mula by Eq. 13, 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 1.0 Hz similar to the experiments. Figure 15 shows the computed S-N curve normalized by the computed static shear capacity. Gen-erally, the analysis coincides with the empirical S-N formula with reasonable accuracy. The crack pattern after many cycles is shown in Fig. 15a. 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. 15b, c). The computed fatigue strength of the slender beam is close to

0000

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

0000

2000

4000

6000

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

Middle amplitude

0

1

2

3

4

0 0.0004 0.0008 0.0012 0.0016 0.002

Log

N

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

Design S-N curve for(Vmin/Vmax) = 0.5

Computed values failure

Vc/

Vc,

max

logN

Average strain of web

120

80

40

-40

-80

Shea

r for

ce (k

N)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

0

1

2

3

4

5

6

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018

Log

N

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018

Low amplitude

Design S-N curve for(Vmin/Vmax)=0.65

Vc/V

c,m

ax

logNAverage strain of web

120

80

40

-40

-80

Shea

r for

ce (k

N)

Fig. 14 Repeated shear force and stirrup strain relation under smaller amplitudes.


RC beam without web reinforcement

0

1000

2000

3000

4000

5000

6000

7000

8000

0 2 4 6 8 10 12Midspan deflection (mm)

reference

171 cycles 581 cycles

y

80

60

40

20

She

ar fo

rce

(kN

)

(a) Crack pattern under fatigue loading.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7

V c/V

co

log N

FE analysis

maxmin /

log|)|1(*036.0log

VVr

NrrVV

co

c

=

⋅−−=⎟⎟⎠

⎞⎜⎜⎝

⎛

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7

FE analysis

V c/V

co

log N

maxmin /

log|)|1(*036.0log

VVr

NrrVV

co

c

=

⋅−−=⎟⎟⎠

⎞⎜⎜⎝

⎛

(b) In case of the slender RC beam with a/d=3.2: full amplitude r=0

(c) In case of the short RC beam with a/d=1.67: full am-plitude r=0

0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8

maxmin /

log|)|1(*036.0log

VVr

NrrVV

co

c

=

⋅−−=⎟⎟⎠

⎞⎜⎜⎝

⎛

in case where Vc/Vco=0.7

analysisdesign formula

Index of amplitude: r= Vmin / Vmax

log

N

(d) Effect of amplitude for the slender RC beam (a/d=3.2)

Fig. 15 Finite element analysis of RC without web reinforcement under fatigue loading.


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 1978). Figure 15d 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 Ta-ble 1 and the sensitivity analysis results in Fig. 16.

The web strain becomes larger due to the rapid dete-rioration of shear transfer and the pattern of unload-ing-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 degenera-tion 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 2. Figure 17 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 ten-sion-stiffness has never been discussed although static mechanistic investigation was reported (Ueda et al. 1995). 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 inte-grate 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. 18. The influence of shear transfer on the fatigue evolution is compara-

2000

0000

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

0000

2000

0 0.0002 0.0004 0.0006 0.0008

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.1 1 10 100 1000 10000

Number of cycle

Average strain of web steelShear force (kN)

standard case

accelerated shear transfer degradation

accelerated shear transfer degradation


standard case

120

80

40

-40

-80

-120

Fig. 16 Sensitivity fatigue analysis of RC beams with regard to shear transfer.

Table 1 Sensitivity analysis with regard to fatigue evolution of shear transfer.

Original modeling Sensitivity analysis for trial

{ } 1.0,|)/(|1log1011

),(

10 ≥+−=

⋅=

∫ εγ

γεττ

dX

X or { } 01.0,|)/(|101log

1011

),(

610 ≥+−=

⋅=

∫ εγ

γεττ

dX

X or


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 rein-forced RC beam Another series of sensitivity analyses were conducted with the non-web reinforced beam (see Fig. 15a). The standard material models lead to the fatigue capacity, which is almost consistent with the empirical formula as shown in Fig. 15b. 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. 19.

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 ex-amined by assuming that the fatigue factor denoted by X is unity all the time in Eq. 11. The shear transfer fatigue modeling is also significant as shown in Fig. 19, but its

Fig. 17 Sensitivity fatigue analysis of RC beams with regard to tension stiffness.

2000

8000

4000

0

4000

8000

2000

0 0.0005 0.001 0.0015 0.002

0

0.0005

0.001

0.0015

0.002

0.0025

0.1 1 10 100 1000 10000

Shear force (kN) Average strain of web steel

Number of cycle

accelerated tensionstiffness degradation

standard case

accelerated tension stiffness degradation

deviating from focal pointafter yield of web reinforcement

120

80

40

-40

-80

-120

Average strain of web steel before yield

Fig. 18 Sensitivity fatigue analysis of RC beams with regard to both tension and shear.

2000

8000

4000

0

4000

8000

2000

0 0.001 0.002 0.003 0.004 0.005

0

0.001

0.002

0.003

0.004

0.005

0.1 1 10 100 1000 10000


Number of cycle

Shear force (kN)


accelerated tension stiffness and shear

degradation

accelerated tension stiffness and shear

degradation

accelerated tensionstiffness degradation

120

80

40

-40

-80

-120

deviating from focal pointafter yield of web reinforcement

Table 2 Sensitivity analysis with regard to fatigue evolution of tension stiffness.

Original modeling Sensitivity analysis for trial

0,,9~

0,0~~)1(10

max

8

496

<−

−≡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅=

≥=⋅−⋅=

εε

εεγ

εεγε

εεεε

dwhendd

dwhenddKKGd

tp

o0,,9~

0,0~~)1(10

max

8

4915

<−

−≡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅=

≥=⋅−⋅=

εε

εεγ

εεγε

εεεε

dwhendd

dwhenddKKGd

tp

o


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 de-pendent 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 2004, Toongoen-thong and Maekawa 2005a, 2005b). The authors com-bined this procedure with the fatigue modeling of mate-rials in order to examine the function of direct path in-tegral scheme.

4.1 Corrosive cracking along main reinforce-ment - 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 rein-forcement (Matsuo et al. 2004), and some predictive methods were proposed (Toongoenthong and Maekawa 2005a). The increased shear capacity due to the corrosive cracking is also shown in Fig. 20 together with the nu-merical 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 70% of the static strength of the non-damaged beam (no flange in Fig. 1). The growing mid-span deflection and the induced shear cracking at each cycle are shown in Fig. 21. 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. 581 cycles for the sound beam, and 25,000 cycles for the pre-damaged beam. The extended fatigue life of cor-roded 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 propaga-tion and prolong the fatigue life.

4.2 Corrosive cracking along tensile rein-forcement at anchorage area - with shear reinforcement - As shown in Fig. 22, 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

Fig. 19 Fatigue life sensitivity analysis of RC beams without web reinforcement.

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 1 2 3 4 5 6 7

maxmin /

log|)|1(*036.0log

VVr

NrrVV

co

c

=

⋅−−=⎟⎟⎠

⎞⎜⎜⎝

⎛

V c/V

c o

log N

tension fatigue ignored

shear transfer fatigueignored

compression fatigue ignored

Fig. 20 Shear force versus displacement relation of corroded RC beam without stirrups.

0

000

000

000

000

000

000

000

0 2 4 6 8 10

Midspan deflection (mm)

No-crk

pre-crk 0.5 mm

120

80

40Shea

r for

ce (k

N)

No crack without stirrups

0.5 mm crack without stirrups


corrosive cracking is controlled to 0.5 mm by introduc-ing the forced corrosion strain. Here, the strain localiza-tion of plain concrete is assumed normal to the main reinforcement in line with the multi-directional fixed crack approach (Toongoenthong and Maekawa 2005a). 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 numeri-cally investigated under the amplitude of 10.0 tonf with

the loading range of 15-65% of the maximum capacity. Figure 23 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 sec-tion-2 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-1 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 24 shows the response of cycle number and the stress level normalized by the yield strength inside the web reinforcement.

Fig. 21 Fatigue life of corroded RC beam without web reinforcement.

0

000

2000

3000

4000

5000

6000

7000

8000

0 2 4 6 8 10 12Midspan deflection (mm)

reference

crack 0.5mm inside

80

60

40

20

She

ar fo

rce

(kN

)

Midspan deflection (mm)171 cycles 581 cycles

a) Sound beam without corrosive cracking

b) Pre-damaged beam with corrosive cracking

581 cycles 25,000 cycles

Fig. 22 Monotonic loading response of anchorage damaged RC beam.

0

5000

0000

5000

0000

5000

0 5 10 15 20

reference

crk at anchorage

200

100Load

(kN

)

Displacement (mm)

No crack with stirrups

0.5 mm crack at anchorage


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 con-firmed. 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

(a) Response of section-1

(b) Response of section-2

Fig. 23 Fatigue response of stirrups around the damaged anchorage.

Average1

0

000

000

000

000

000

000

000

0 0.001 0.002 0.003 0.004 0.005

crack around anchorageno crk

120

80

40Shea

r for

ce (k

N)

Average web strain

Crack width 0.5 mm anchorageAvg.1 Avg.2

at 100th cycle

Average2

0

00

00

00

00

00

00

00

0 0.001 0.002 0.003 0.004 0.005

crack around anchorage

no crk

120

80

40Shea

r for

ce (k

N)

Average web strain

at 1,000th cycle

at 20,000th cycle

Average1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 10 100 1000 10000 100000

Number of Cycles

Norm

aliz

ed

steel

str

ess

crk at anchorage

no crk

Average 2

0

0.2

0.4

0.6

0.8

1

1.2

1 10 100 1000 10000 100000

Normalized avgerage steel stress

Num

ber

of C

ycle

s

crk at anchorage

no crk

Number of Cycles

Nor

mal

ized

ste

el s

tress

Fig. 24 Stress history for assessment of fatigue rupture of web reinforcement.


future investigation and development. (1) 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.

(2) The modeling used in this study may cover drying conditions of about 60% 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 experi-ments. This study was financially supported by Grant-in-Aid for Scientific Research (S) No.15106008. References ACI Committee 215 (1982). “Fatigue of Concrete

Structures.” Shah, S. P. ed., American Concrete Institute.

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