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‫ׅ‬ᢊែ೛࠿༿⇝∙∃⇞⇌∆ᛯ૨‫إ‬ԓᨼ‒ ‫ם‬ங‫˟ܖ‬ㄽᩥLife Prediction of RC Bridge Deck Slabs in a Punching Shear Failure Mode
Pengru DENG*㸪Takashi MATSUMOTO***༤㸦ᕤ㸧, ໭ᾏ㐨኱Ꮫࣙᩍ, ኱Ꮫ㝔ᕤᏛ◊✲㝔㸦ࠛ060-8628 ᮐᖠᕷ໭༊໭ 13 す 8㸧**Ph.D., ໭ᾏ㐨኱Ꮫᩍᤵ, ኱Ꮫ㝔ᕤᏛ◊✲㝔㸦ࠛ060-8628 ᮐᖠᕷ໭༊໭ 13 す 8㸧Focusing on the propagation of critical cracks, a theoretical life prediction
method for RC bridge deck slabs in a punching shear failure mode is developed
based on fracture mechanics. In this method, the crack propagation is considered
as a result of a concrete bridging degradation and a rebar/concrete interface bondslip degradation. From the analysis, the growth of the critical punching shear
cracks can be obtained, which is then employed in determining the fatigue life
following some shear failure criteria. Applicability and reliability are verified witha good accuracy achieved in the fatigue analysis of RC slabs tested in CERI.

      Keywords㸸 RC slabs, wheel running load, punching shear, fatigue

mechanisms cannot be reflected. As for the numerical

1. Introduction

approaches, they are very time-consuming due to the widely
In last few decades, a brittle and catastrophic punching shear
failure mode has been usually reported around the world on

distribution cracked elements. The computing time can evenreach 1~2 days/cycle before failure.

RC bridge deck slabs. From existing researches, it has been

Therefore, in this study, an analytical fatigue life prediction

confirmed that this detrimental phenomenon stems from amethod is proposed focusing on the propagation of punchingneglection of a shear capacity, especially a fatigue shear

shear cracks of a critical RC beam simplified from an RC slab

capacity, of RC slabs in previous design codes. Aoordingly, RC

under cyclic moving loads according to the degradation

bridge slabs have been designed lacking rebars and thickness.

process and failure crack pattern. From this method, fatigue

So far, extensive experimental works have been conductedcrack growth which is assumed as a result of concrete bridging

to uncover this punching shear failure mode with a relatively

degradation and rebar/concrete interface bond-slip degradation

simple experimental moving load set up as shown in Figure 1.

can be obtained as well. Compared with existing empirical and

The punching shear failure mode and shortened real fatigue life

numerical approaches, this method can not only account for thehave been reproduced under this loading. According to thedegradation mechanisms but also save a lot of computing time

experimental observations like the failure crack pattern, an(about 10 minutes/cycle). This time-saving characteristic

empirical life prediction equation was formulated throughmakes it extremely suitable for parameteric study and futher[1]

fitting a huge set of experiment data statistically . In addition,

design code improvement.

taking advantage of FEM, the fatigue behaviors of RC slabs
have been investigated through numerical analysis for the

2. Problem simplification[2]

cyclic moving load . From these numerical analyses, it was
concluded that the modeling of tension fatigue of concrete andReferencing to a typical failure crack pattern shown inrebar/concrete interface bond slip characteristic should be

Figure 1, the cracking process can be explained as follows: (1)considered carefully.

The moving load leads to an initiation of cracks along the

However, the existing researches possess their own

direction vertical to the bridge axis because RC bridge slabs are

disadvantages. As the empirical life prediction equations werenormally with a smaller reinforcement ratio along bridge axis

from statistically fitting experiment data, the inner degradation

direction; (2) With continuing cycles of moving load, the1 - 65

cracks vertical to the bridge axis become more dense and fullyzone (εrb) can be related to α and β asdeveloped as observed from a b-b cross-section in Figure 1§

H II x

These cracks propagate almost parallelly to each other; (3) The

H t ¨1 ©

parallel cracks cut down the shear transferring capability
dramatically. Each region isolated by a couple of parallel

H ru

cracks resembles an RC beam. Obviously, the beam located in

H rb

(1)

h  c  db 2  D h

H t

(2)

E D hthe midspan is the most critical one; (4) In the critical beam, alarge shearing stress concentrates around the loading area. Thisx Dh ·¸
E h Dh ¹Hth  c  db 2

shear stress leads to propagation of a couple of punching shear(3)

E D hcracks and the final brittle punching shear failure. Hence, thewhere c and db are the cover depth and rebar diameter,

failure of an RC slab under cyclic moving loads is determined

respectively. εt is cracking strain of concrete.

by the failure of a critical RC beam.

b
Moving load lane

In addition, according to the cracking process and failure

b-b cross sectionmode, Matsui proposed a method of determing the geometriesof the critical RC beams as shown in Figure 2, where h and l
are the depth and span of the RC beam, respectively; lw and b
are the length and width of the wheel/beam contact area; de isaathe effective depth for tensile rebar [1]. And then some empiricallife prediction equations have been formulated using the

Critical RC
beampunching shear capacity of this critical beam as a normalized
parameter. In these empirical equations, the only parameterbused is the punching shear capacity of the critical beam, whichindicates that the fatigue life of the RC slab depends on thea-a cross section

fatigue life of the critical RC beam.

Punching shear cracks

Therefore, according to the theoretical bases of existing

Figure 1 A typical failure crack pattern of an RC slab

researches and the fatigue behavior of RC slabs under cyclic

under cyclic moving load

moving loads, the life prediction of an RC bridge slab issimplified into the life prediction of a critical RC beam

Punching shear
crack line

lw

focusing on the propagation of the punching shear cracks as
45°shown in Figure 2.

de b deh45°l

3. Methods

Figure 2 Geometry of the simplified RC beam and thepunching shear cracks

For the critical RC beam subjected to loads, Figure 3 shows
sectional forces and stresses acting on the punching shear crackσIIMAcross-section. For every loading cycle, the cracking states can

VA

Tru

τII

be calculated, and the failure moment can be determined. One
can obtain the forces and stresses from actions (applied loads)hand reactions (concrete and rebar stresses).

σt

σI

βh
αh

3.1 Basic assumptions

Trb

x
Vrb

(1) Plane cross-section assumption: From this assumption,
the strain distribution (εII) at the uncracked concrete and thenormal strain of rebar on both compression (εru) and tension

2- 66 -yFigure 3 Sectional stresses and forces

(2) The cracked cross-section is assumed to rotate around thesolving Eq. (6) to Eq. (8).

[3]neutral axis . From the tensorial consideration of strains at thecrack location, the normal force (Vrb) of longitudinal rebars in

3.3 Formulate sectional stresses and forces with cracking state

the beam can be related to their shear force (Trb) as:

parameters

0.4Trb tan I

Vrb(1) Sectional forces and stresses due to applied loads:(4)

According to the transferring characteristic of moment and(3) The crack has a linear crack opening profile [4]. From thisshearing force, sectional moment (MA) and shearing force inassumption, the crack width (w(x)) at any location (x) along thethe critical RC beam can be obtained from FEM of the RC slab

crack can be expressed by the crack mouth opening

and analysis of the RC beam, respectively. And then, stresses

displacement (CMOD, δ) and crack depth ratio (α) as

can be easily expressed with α, β and δ from sectional analysis.

§©

G ¨1 

w x

x ·D h ¸¹(5)

With these material strains and crack opening displacments,
materials stresses and concrete bridging stress can be expressedwith cracking state parameters, i.e. α, β and δ following
appropriate material stress-strain and concrete bridging models,
respectively. Mathematically, to obtain α, β and δ after every
loading cycle, at least three independent equations withunknown of α, β and δ should be formulated.

equilibrium equation along x axis and one moment equilibrium
equation. In addition, a CMOD decomposition equation can beestablished following [5]. The three equations are given as:

³

0

h

Dh

³0 V I x ª¬ h  x  D h  x º¼ dx  Trb  h  as 
³ V II x h  x dx  Vrb  D h  as  Tru  asDhG

uncracked concrete.
(3) Stresses from cracked concrete: In the first cycle, stresses
from cracked concrete is expressed with α, β and δ following
an empirical concrete bridging model [4]. From 2nd loading
cycle to the final failure, the degradation due to a crack opening
and closing process is accounted for based on a concrete

incorporating multi-crack effect and bond-slip effect. And then,the rebar stresses are obtained through substituting rebar strain
in Eq. (3) into this modified rebar stress-strain relation. Fromvariation due to bond slip degradation is considered throughintroducing a bond slip degradation model.

(7)

MA

G A D  G I V I x ,D  G rb Trb ,Vrb ,D  G s Trb

in this study, firstly the rebar stress-strain relation is modified

2nd loading cycle to the final failure, the rebar stresses
(6)

0

Dh

h

and elastic modulus of concrete. No degradation is assumed for(4) Rebar stresses: To account for a tension stiffening effect,

Referencing to the Figure 3, one can establish one force

V I x dx  ³ V II x dx  Trb  Tru

α, β and δ through multiplying the concrete strain from Eq. (1)bridging degradation model.

3.2 Governing equations with cracking state parameters

Dh(2) Stresses from uncracked concrete can be expressed with

3.4 Failure moment prediction with cracking state parameters
As concrete is a main source of sectional shear resistance(8)

where MA is sectional moment due to applied loads. δA is
CMOD due to applied load. δI are δrb are CMODs due toconcrete bridging stress and rebar bridging forces, respectively.
With crack face acting stresses due to applied loads, concrete
and rebar bridging stresses, these three CMODs can becalculated employing a fracture mechanics based integral
equation introduced in [5] and the crack face weight functions [6].
δs is CMOD due to bond slip which can be determinedfollowing an appropriate bond slip model [7].
Therefore, if all stresses and forces appeared in Eq. (6) to Eq.
(8) can be expressed with α, β and δ, the cracking stateparameters after every loading cycle are obtainable throughand the stress transferring capability of concrete drops
tremendously after cracking, the shearing capacity decreases
remarkably due to the propagation of punching shear cracks.
Consequetely, a localized shear failure was commonly
observed in experiments. Therefore, the final failure momentof RC slabs is predicted following some shear capacity based
failure criteria and using the obtained cracking stateparameters.
As is generally accepted, the total shear resistance for a
traditional RC slab is an integration of the vertical componentsof tensile stresses in concrete along the whole cross-section andthe dowel action of flexural reinforcement. However, for the
aimed problem, the shear resistance reduces due to follows: (1)3- 67 -

at failure moment, the shear stresses cannot be transferredtheoretical simulation successfully captured the first two stages,

effectively along moving load direction due to the existence

whereas the third stage cannot be observed in the curves. This

parallel cracks perpendicular to moving load direction; (2)can be explained as follows [10]. To facilitate understanding, aunder cyclic moving loads, the crack opening and closing

crack-tip stress intensity factor (SIF) which reflects theprocess reduce the stress transferring ability of concrete tocrack-tip stress state is employed. For the punching shear

almost zero; (3) post-moment inspections of tested specimens

cracks, the net crack-tip SIF is a resultant of SIF from external

showed that a delamination along upper rebars occurred in theapplied load and all bridging effects. The SIF amplitude due to

mid-span cross-section of specimens subjected to cyclic

external applied load increases with the length of the crack

moving loads. This is probably a result of high-fatigue shear

emanating from the tension face. At the same time, thestresses at that depth due to the existence of upper rebars and amagnitude of the negative crack-tip SIF due to crack bridging

short distance from the neutral axis. Therefore, the dowel effect(rebar forces and concrete bridging stresses) increases with theof upper rebars should be excluded in computing the punchingpropagation of the crack as well. The initial decelerated growth

shear capacity after the crack tip reaches. Therefore, the total

is caused by the effective reduction of the net crack-tip stressshear capacity should only include concrete in the assumed

intensity factor amplitude due to the rapidly increased crack

failure area as shown in Figure 4 and rebar dowel action, which

bridging related SIF. The steady state growth is caused by theandconstant value of net crack-tip SIF when the increase of SIF, respectively. In summary, the punching shear

due to applied load is balanced with the increase of negative

can be calculated following the equations from ACI
Matsui[8][1]

SIF due to bridging stress. The reason why the final accelerated

capacity of the critical RC beams is given as

V pun­2 ª¬b  1  D h º¼  1  D h  f t  4c  B  f t°° Before crack tip reaching upper rebars(9)®
° 2 ¬ªb  1  D h ¼º  1  D h  f t  2c  B  f t°
After crack tip reaching upper rebars¯

The final failure moment is predicted based on this punching
shear capacity in this study.

crack growth stage cannot be observed is that the RC beam isassumed to fail in a brittle shear failure mode according tocertain failure criterion.
However, the initial decelerated stage in the crack
depth/beam depth ratio to number of loading cycles curve forhigher load level is not as apparent as that for lower load levels.
This is because the punching shear crack is more fully
developed in the beginning several loading cycles under the

4. Numerical results and experimental verification

higher load levels, manifested with higher crack depths andIn this study, a fatigue analysis based on the proposed
method is conducted in order to investigate crack propagationand fatigue life of an RC slab (C0) tested by Civil Engineering
Research Institution for Cold Region (CERI) [9]. The method

wider crack openings. Due to the wide crack opening, thebridging effect from nonlinear concrete bridging stress maybe
small or even negligible compared with that from rebars. Thus,
the structure degradation can be regarded as mainly from theverification is done by comparing analysis results with results

lw

from experiments and some empirical equations. Four loadinglevels, i.e. 130 kN, 150 kN, 175 kN and 200 kN, are selected
for fatigue analysis.

b

4.1 Prediction of fatigue crack growth
Under the selected loading levels, the crack depth/beamAssumedfailure planesP

depth ratios are plotted with the number of loading cycles on
the semi-logarithmic scale as shown in Figure 5(a). Generally,
the fatigue growth of a bridged crack in a concrete structure
involves of three distinct stages: initial decelerated growth,steady state growth and final accelerated growth. Most of thefatigue life is spent in the second stage. It is found that the4- 68 -

σt
45°

h

45°

Crackedconcrete

Figure 4 Schematic of failure plane combined fatigue
degradation and experimental observationsdegradation of bond slip, which follows a steady state growth

were obtained through statistically fitting a huge amount ofas shown in Figure 5(a). As a result, the crack propagation

experimental data on RC slabs under cyclic moving loads,

curve seems to start from the steady state growth for higher

each empirical equation represents a set of experimental resultsload level as observed in Figure 5(a).

of RC slabs with different dimensions and loading levels. Thus,

Similar to the characteristics exhibited in the crackthese equations implicitly reflect the internal mechanisms anddepth/beam depth ratio to number of loading cycles relations,can be employed in life prediction of similar problems as

the two stages are also observed in Figure 5(b), where theemployed case in this study.

relationships between tensile depth/beam depth ratio and the

To illustrate the relationship between fatigue life and load

number of loading cycles for several fatigue loading levels arecondition, an S-N diagram is commonly employed. All the S-N

plotted on the semi-logarithmic scale.

diagrams from the proposed theoretical method and the

The evolution of CMODs with respect to the increasing

empirical equations and experimental fatigue life are plotted on

number of loading cycles for several fatigue loading levels area double-logarithmic scale as shown in Figure 6 where thedrawn on a semi-logarithmic scale Figure 5(c). It is noticed thatvertical and horizontal axes represent a load parameter

the CMOD evolution of the punching shear crack depends onnormalized with a punching shear capacity determined by [1]the load level. And, the CMOD increasing is expected to be
In summary, fatigue loading produces generally continuous
increase of crack depth, tensile depth and CMOD.
4.2 Fatigue life prediction and comparison with experimental
results and existing methods

0.8

Crack depth/Beam depth

mainly from the degradation of bond slip.

0.7
0.6

130kN
150kN
175kN
200kN

0.5
0.4
1

In this method, the final fatigue failure of RC slabs under

100

10000

1000000

1E8

1E10

Number of cycles

cyclic moving load is determined following certain brittle shear(a) α vs. number of loading cycles

failure criterion introduced in section 3.4. Substituting
parameter values into Eq. (9), the punching shear load
rebars is determined as 230.63 kN. After the crack tip reaches
the upper rebars, transverse shear cracks along the upper rebars
initiate and propagate under repeated fatigue loads. Accodingly,
the punching shear capacity reduces to 123.79 kN. Therefore,
under the selected cyclic moving loads, fatigue punching shear

Tensile depth/Beam depth

capacities for the moment of the crack tip reaching the upper

0.80
0.75
0.70

130kN
150kN
175kN
200kN

0.65
0.60
1

100

failure occurs after the crack tip reaches the upper rebars with αequaling 0.7. Comparing this critical state with the obtainedin Figure 5, the fatigue life of the RC slab subjected to different

institutions, such as Japan Society of Civil Engineers (JSCE)
and Public Work Research Institute (PWRI). All the empiricalequations were formulated as double logarithmic functions
treating a punching shear capacity of the critical RC beam in [1]as a normalizing parameter. Since these empirical equations

5- 69 -

CMODs (mm)groups, such as Matsui [1] and Abe research teams [11], and

1E8

1E10

1.00

load levels can be determined easily.
empirical life prediction equations derived by different research

1000000(b) β vs. number of loading cycles

cracking state parameters vs. number of cycles curves shown

Moreover, the fatigue life can be calculated following some

10000

Number of cycles

0.75
0.50

130kN
150kN
175kN
200kN

0.25
0.00
1

100

10000

1000000

1E8

1E10

Number of cycles(c) CMOD vs. number of loading cycles
Figure 5 Cracking state parameters vs. number of cycles

Applied load/Punching shear capacity

helpful suggestions and unpublished experimental data.

10

Matsui equation
JSCE equation
1

References
1)
PWRI equation

0.1

Proposed method
Experiment

reinforced concrete slabs. Doboku Gakkai Ronbunshu,

Abe equation

pp.133-141, 1984.

0.01
1

100

10000

Maeda, Y., & Matsui, S. : Punching shear load equation of

1000000

1E8

2)

1E10

Number of cycles

Drar, A. A. M., & Matsumoto, T. : Fatigue analysis of RC
slabs reinforced with plain bars based on the bridging

Figure 6 S-N curves from different approachstress degradation concept. Journal of Advanced Concreteand number of loading cycles, respectively. It is found that thetheoretical S-N relation agrees well with the fatigue life of theTechnology, Vol.14, No.1, pp.21-34, 2016.
3)

Zararis, P. D., & Papadakis, G. C. : Diagonal shear failure

experimental RC slab and is almost the average results of theand size effect in RC beams without web reinforcement.

four reported equations derived statistically. These good

Journal of structural engineering, Vol:127, No.7,

agreements verify the reliability of the proposed theoretical

pp.733-742, 2001.

method.

4)

Zhang, J., Stang, H., & Li, V. C. : Fatigue life prediction offiber reinforced concrete under flexural load. International

5. Conclusions

Journal of Fatigue, Vol.21, No.10, pp.1033-1049, 1999.
5)

Deng, P., & Matsumoto, T. : Estimation of the rebar force

A theoretical fatigue life prediction method for RC bridgein RC members from the analysis of the crack mouthslabs which fail in a punching shear model under cycle moving

opening displacement based on fracture mechanics.

loads has been proposed based on the nonlinear fracture

Journal of Advanced Concrete Technology, Vol.15, No.2,

mechanics for RC structures.

pp.81-93, 2017.

The method was established focusing on the propagation of

6)

Deng, P., & Matsumoto, T. : Weight function

punching shear cracks in a critical RC beam which was

determinations for shear cracks in reinforced concrete

simplified from a RC slab under cyclic moving loads

beams based on finite element method. Engineering

according to the cracking process and failure crack pattern andFracture Mechanics, Vol.177, pp.61-78, 2017.

referencing to the theoretical bases of existing researches. From

7)

this method, the fatigue crack growth of the critical punching
shear cracks was predicted. With the obtained crack growth

Maekawa, K., Okamura, H., & Pimanmas, A. : Non-linear
mechanics of reinforced concrete: CRC Press, 2003.

8)

ACI Committee 318. : Building code requirements forresults, the final failure moments or fatigue life were predictedstructural concrete (ACI 318-99) and commentary

according to certain punching shear failure criterion combining(318R-99). American Concrete Institute, Farmington Hills,

with experimentally observed failure modes. Comparisons

Michigan, pp.399, 1999.

between method predictions to results from experiment and9)

Mitamura, H., Syakushiro, K., Matsumoto, T. & Matsui,

some existing empirical life prediction equations indicate aS. : Experimantal study on fatigue durability of RC deck

reasonably good agreement, which verified the applicability

slabs with overlay retrofit. Journal of Structuraland reliability of the proposed method. In addition, aEngineering A, Vol.58, pp.1166-1177, 2012.

time-saving characteristic of this method makes it extremely10) Li, V. C., & Matsumoto, T. : Fatigue crack growthsuitable for parametric studies and analysis on the effect ofanalysis of fiber reinforced concrete with effect ofenvironment actions on the occurrence of punching shear

interfacial bond degradation. Cement and Concrete

failure.

Composites, Vol.20, No.5, pp.339-351, 1998.
11) Abe, T.: Study on punching shear load-carrying capacity

Acknowledgementof RC slab. Journal of College of Industrial Technology

We are grateful to Dr. K. Kakuma from Civil Engineering
Research Institute for Cold Region for providing us with

6
- 70 -

Nihon University, Vol.38, No.2, pp.11-21, 2005.
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