ᇹ

‫ׅ‬ᢊែ೛࠿༿⇝∙∃⇞⇌∆ᛯ૨‫إ‬ԓᨼ‒ ‫ם‬ங‫˟ܖ‬ㄽᩥDevelopment of an Effective Numerical Model for Fatigue Analysis of RC
Bridge Slabs
Arslan Qayyum KHAN*, Pengru DENG**, Takashi MATSUMOTO****Doctoral student, Graduate School of Engineering, Hokkaido University(Kita 13, Nishi 8, Kita-ku, Sapporo 060-8628, Japan)**Assistant Professor, Faculty of Engineering, Hokkaido University(Kita 13, Nishi 8, Kita-ku, Sapporo 060-8628, Japan)***Professor, Faculty of Engineering, Hokkaido University(Kita 13, Nishi 8, Kita-ku, Sapporo 060-8628, Japan)
This study presents an effective numerical method to predict the fatigue
behaviors of RC bridge slab due to moving wheel load. The bridging stressdegradation concept is employed as a fundamental root of the propagation of
cracks that prompts the fatigue failure of RC bridge slab. The smeared model
approach is adopted in the numerical model established and finite element method(FEM) is used in order to solve the model. The fatigue behaviors of RC slab,
centre displacement evolution and propagation of cracks, predicted by thenumerical method are compared with those of experimental results. The numericalresults show a good agreement with the experimental results, and this numericalmethod can be used for the fatigue analysis of RC slabs under moving wheel load.
      Keywords㸸RC slab, fatigue, bridging stress degradation, moving wheel load

was used [3].

1. Introduction

For the first time, the bridging stress degradation concept
Reinforced concrete (RC) bridge slab is amongst the mostwas introduced as a fundamental root of fatigue crack

critical part in bridge susceptible to fatigue failure since itpropogation in RC and fiber reinforced concrete (FRC) beams

directly sustains repetitive moving wheel loads. Thus, it is very

[4].
The bridging stress degradation was employed in fatigue

important to be able to accurately evaluate the load-carrying

analysis of RC slabs reinforced with plain bars. The model

capacity and fatigue behaviors of RC bridge slab.
In the past, many researchers have performed experiments to
examine the fatigue behaviors of RC slabs. Recently, RC

captured the cracking pattern of RC slab and strain in plain
reinforcement bars [5].

structural members subjected to repetitive loading have alsoThis study presents a numerical model to predict the

been numerically analysed to predict the fatigue life. Ueda et al.

fatigue characteristics of RC bridge slab subjected to repetitive

used finite element method (FEM) to analyze the fatigue

loading. The experimental study by Civil Engineering[1]characteristics of steel-concrete sandwich beams .

Research Institute (CERI) for Cold Region is used to verify the

RC beams without shear reinforcement were analysed andanalytical results. The numerical model of RC slab is

the fatigue crack propagation concept was proposed to predict

established using MSC/MARC software by smeared model[2]

the fatigue flexural performances of RC beams . A study on amethodology and FEM is used to get the solution of the model.

numerical fatigue simlution of RC slabs under moving load

The bridging stress degradation concept is considered as awas carried out and a direct path-integral method with fatigue

major mechanism for the propagations of cracks in RC slab.

constitutive models for concrete in tension and compression

Fatigue life, centre displacement evolution and propogation of

1 - 319

cracks are presented and compared with experimental ones inthis study. The numerical results show satisfactory agreementwith the experimental results.

σft
N increase

2. Methodology
2.1 Concrete model
Nonlinear behavior of concrete both in tension andcompression is used as shown in Table 1. The bridging stressεtdegradation concept, the most important part in this current
fatigue analysis, is introduced to represent the cracking

εFigure 2 Hysteretic behavior of concrete under cyclic

behavior of concrete.

loading

Table 1 Nonlinear constitutive laws of concrete [6]Compression

Tension*propogation and reduction in bridging stress is defined as
bridging stress degradation which depends on maximum

0 t H t Hm

Vfc

H § H ·
¨2  ¸H m ¨© H m ¸¹

Ht t H t 0Vtensile strain (εt) and number of cycles (N) [4].

Ec H

The equation of bridging stress degradation proposed byZhang et al. [7]:Hm t H t Hu

Vfc

Hu  H
Hu  Hm

H t HtV§H ·ft ¨ t ¸
©H ¹

VN
V1

0.4

1  0.08  4 u G max log N

Where δmax, is rewritten in the form of tensile strain (εt) by

Ec = modulus of elasticity of concrete,considering unit element length (δmax= εt × l).

fc’ = concrete compressive strength,

The stress strain behavior under cyclic loading is displayed

εm = fc/2Ec =concrete strain corresponding of fc’
ft = tensile strength and εt = ft/Ec = strain at tensile strengthin Figure 2. Since loading and reloading paths are same, no*The multiplier 0.4 can be used for the concrete with deformed bar

plastic strain is occurred.

In bridging stress degradation concept, a crack begins with

2.2 Reinforcement bar model

length (a) and width (w) as displayed in Figure 1(a). The crack

The smeared model is adopted for reinforcement bar in thegoes through a process of opening and closing which causes in

RC bridge slab. A bilinear curve with yield stress (fy) is used forthe reduction of bridging stress in repetitive moving loads.

representation of stress strain behavior of the reinforcement bar.

Thus, the existing crack progresses with increased length (da)The Giuffré-Menegotto-Pinto model [8] is utilised to represent

and increased width (dw) as shown in Figure 1(b). This crack

the hysteretic behavior of reinforcement bar under repetitive
loading as following:

σ1

σN

V

a+da

f y

H

a

w
1st cycle

w+dw

R

Nth cycle

Figure 1 Crack propagation due to bridging stress:

Ro 

H

H y ª

1 H

H
H y

§ H · º¸ »«1  ¨«¬ © H y ¹ »¼
R

1R

a1[ max
a 2  [ maxWhere H is hardening parameter, Ro, and R are transition(a) first cycle (b) after N cycles

2- 320 -

D16@ 250mm

Transverse
Direction

1325mmfy

Moving load zone 190 x 2000 mm

D16@ 125mm

3300mmσEsεILongitudinal Direction

II

230mm

II

D19@125mm

D19@ 250mm

Figure 4 Slab geometry

Figure 3 Hysteretic behavior of reinforcementbar under cyclic loadingparameters between elastic hardening for the first and Nth
cycle (Ro = 20), ζmax is the maximum excursion in the plastic
range, a1 and a2 are the parameters for change of R withrepetitive load history, equal to 18.5 and 0.00015, respectively.
The yield strength reduces as number of cycles increases
shown in Figure 3.
2.3 Slab details

Figure 5 Loading pattern

The slab used in this study is haunched RC slab withdimensions 3300 × 2650 × 230 mm. In tension zone, the slab is

the start, 150 kN load was applied and then the load was

reinforced with D19@125 mm along transverse direction andincreased up to 240 kN with the interval of 30 kN.

D16@125 mm along longitudinal direction. Similarly,
D19@250 mm along transverse direction and D16@250 mm

2.4 Analytical procedure
The finite element software MSC/MARC was used toalong longitudinal direction are provided in compression

model the slab, a 3D model using solid elements, as displayedzone.The slab details are shown in Table 2.

in Figure 6 (a). Due to symmetry in loading and boundary
Table 2 Slab Details
Slab Dimension (L × W)
3300 × 2650 mm
Slab thickness
230 mm
D19@125mm (Bottom)
Transverse reinforcement
D19@250mm (Top)D16@125mm (Bottom)
Longitudinal reinforcement
D16@250mm (Top)Concrete
Ec = 21727 MPa fc' = 32.4 MPA
Steel
Es= 200000 MPa fy = 345 MPA
The moving load zone was of 2000 × 190 mm and the load
was applied along the longitudinal direction of the slab. The
geometry of slab, reinforcement details and moving load zone
is displayed in Figure 4.
The step loading type was adopted as shown in Figure 5. At

conditions, the half slab was analysed. Firstly, the moving load
was applied at the central elements of slab. Then thoseelements were unloaded and the adjacent elements on the rightside were loaded. The same procedure was continued until thecompletion of one cycle of the moving load and one cycle ofmoving load was considered when the load is again applied on
the central elements of slab.
After completion of one cycle of moving load, a number ofcracked elements were appeared as shown in Figure 6 (b).
According to bridging stress degradation concept, theconstitutive law for the cracked elements was modified. It led
to the loss in tensile strength in those cracked elements causing
a stress concentration at crack tip and new cracked elements
were apperead around the stress concentration as displayed in3- 321 -

4.0

EXP
ANA

3.5

Displacement (mm)Moving load

3.0
2.5
2.0
1.5
1.0
0.5
0.0

0

200000 400000 600000 800000 1000000

No. of Cycles(a) Analytical model

Figure 7 Centre displacement evolution

Displacement (mm)4.0

Cracked elements(N = 1)

3.5

EXP
ANA

3.0

ANA

[9]

2.5
2.0
1.5
1.0
0.5
0.0

(b) 1st cycle (N0200000 400000 600000 800000 1000000

No. of Cycles

Figure 8 Centre displacement evolution
fatigue analysis at different loading cycles are compared withthe displacements observed in the experiment. As number of
Crackedelements (N=2)

cycles is increased, the crack elements are increased which
result into reduction of RC slab stiffness. The reduction in RC

Cracked elements(N=1)

slab stiffness leads to increase in centre displacement of RC
slab. The RC slab center displacement evolution shows
acceptable agreement with the experimental result as shown in(c) 2nd cycle (N=2)

Figure 7.

Figure 6 Analytical procedure
3.2 Effect of calculation step
Figure 6 (c).

Calculation step plays a significant role in numerical model

The overall RC slab stiffness was decreased with theand FEM analysis. In previous study, a comparison is madeincrease of cycles of moving load due to bridging stressbetween experimental and numerical results for centre

degradation and crack propogation. The procedure wasdisplacement evolution [9].

continued until fatigue failure occurred and the numericalresults were recorded in each cycle of moving load.

In this study, smaller calculation step is used, and centre
evolution displacement is compared with experimental results
as well as the numerical model results having larger calculation

3. Results and discussions

step. In the Figure 8, it can be seen that the numerical modelusing smaller calculation step is showing better agreement with

3.1 Center displacement evolution
The displacements at centre of the RC slabs obtained during

the experiment work as compared with the model having larger
calculation step.

4- 322 -

Cycle 1


M.D

T.D

L.D

Cycle 1005000


1

2000

5000

55000

1050002050001005000

M.D: Moving Direction L.D: Longitudinal Direction T.D: Transverse Direction

Figure 10 Maximum principal strain distribution

Figure 9 Propogation of cracked elements

3.4 Maximum principal strain distribution

3.3 Propagation of crack elements
This numerical model considers primary aspect that the
propagation of cracked elements and degradation of bridging
stress are the core reasons of fatigue failure. Thus, it issignificant to show the propagation of cracked elements.
Uncracked elements are shown by white color. The cracked
elements after first cycle of moving load are displayed by red
color. Similarly, the cracked elements caused by different
number of cycles are indicated with different colors as shownin Figure 9.

The comparison of maximum principal strain distribution on
bottom surface of RC slab after 1st cycle and last cycle isdisplayed in Figure 10. The strain is increased in diagonal
direction from loading point towards the supporting corner.
The increase in tensile strain in diagonal direction provides
justification of propagation of cracked elements in diagonal
direction as shown in Figure 9.
4. Conclusions

From the figure, it can be observed that the cracked elements
are spread all over the longitudinal direction of RC slab,
wheras the cracked elements are distributed from mid span of
RC slab towards transverse direction with less degradation.
Since the moving load effect was transferred along thelongitudinal direction, the cracked zone is larger in thelongitudinal direction as compared to the transverse direction.
The cracked volume of RC slab after first cycle of moving load,
shown by red color, is 5.75% and the total cracked volume
after final loading cycle, 1005000, is 44.2% as presented inTable 3. In the analysis, the average degradation ratio of RC
slab is 442.3 mm3/cycles. Furthermore, the cracked elements
are propagated in vertical direction up to 0.63 times the
thickness of the RC slab.

RC Slabpredict the fatigue behavior of RC slab under a movingload.The propagation of cracked elements due to bridging
stress degradation is considered as the main cause of fatigue
failure.
The cracked zone is larger in the longitudinal direction ascompared to the transverse direction of the RC slab.
This method was verified using the experimental results, andit showed a good agreement. This numerical model using
smaller calculation step is showing closer results with theexperiment work as compared with the model having larger
calculation step. Thus, the fatigue behaviors of RC slabs undermoving load can be predicted by this numerical method.
Acknowledgement

Table 3 Cracked elements volume
SlabIn this paper, bridging stress degradation concept is used to

Cracked Volume (%)

Average degradationst3

1 cycleLast cycle

ratio (mm /cycles)

5.75

44.2

442.3

We are grateful to Dr. K. Kakuma from Civil Engineering
Research Institute (CERI) for Cold Region for sharing theexperimental data.

5- 323 -

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