Proceedings of The 12th Symposium on Decks of Highway Bridges JSCE

Technical Paper
Non-Linear FEA of a Steel-Concrete Composite Deck under Increased Fatigue
Loading and Damage Evaluation of the Concrete Elements by Stress InvariantsDaniel Apenyo*, Toshio Matsumura**, Takuji Kumano**, Hitoshi Wada***, Masashi Yamamoto*,
Chikako Fujiyama*****MSc. of Eng., Japan Bridge Association (1-6-11 Nishi-Shinbashi, Minato-ku, Tokyo 105-0003)**Dr. of Eng., Japan Bridge Association (1-6-11 Nishi-Shinbashi, Minato-ku, Tokyo 105-0003)***BS. Civil Eng., Japan Bridge Association (1-6-11 Nishi-Shinbashi, Minato-ku, Tokyo 105-0003)****Dr. of Eng., Associate Professor, Institute of Urban Innovation, Yokohama National University (79-5 Tokiwadai,Hodogaya-ku, Yokohama-shi, 240-8501)
Strength and fatigue resistance of a Steel-concrete deck composite structure havebeen verified by wheel running tests however, fatigue damage process due to full-scale
bridge loads isn’t clear. The purpose of this research is to conduct a nonlinear finite
element analysis on the composite structure under fatigue loading (8 times T-Load) andevaluate by stress invariants the damage in concrete elements. A cross section ofelements beneath the wheel track between two adjacent ribs was evaluated using theDrucker-Prager criteria to confirm stress state of each element, identify criticalelements and classify the damage pattern of a steel-concrete deck composite.
Keywords: Nonlinear Finite Element Analysis (FEA), Fatigue Analysis, DruckerPrager Criteria, Steel-Concrete Composite Deck
1. BACKGROUND
Steel-concrete deck is a reinforced concrete slab with a steel
plate at the bottom. This plate that works as tension
reinforcement is connected to concrete by headed studs weldedon the plate. Strength and fatigue resistance of the composite
structure have already been verified by wheel running tests
however, the fatigue damage process due to full scale bridge
loading isn’t clear.
Fatigue problems on both steel decks and reinforced
concrete decks have been investigated both experimentally andanalytically however, the number of investigations of steelconcrete composite decks is small1). In a study on the damage
analysis of headed stud on the steel-concrete composite deck bynumerical analysis, the failure process under fatigue loading
considering a heavy load of 800KN (4 times T-Load) was
conducted2). The maximum mid-point deflection (8mm)
obtained by fatigue analysis was 1.75 times larger than thatobtained by static analysis. Also, in the fatigue analysis, thisdeflection did not increase with the increase in the number ofloading cycles up to 100 million. However due to the bottom
steel plate, damage inside the concrete remains invisible thatmakes it hard to obtain practical damage information. Therefore,

the purpose of this research is to conduct a nonlinear finite
element analysis to a steel-concrete composite deck underincreased fatigue loading of 1600KN (8 times T-Load) andevaluate by stress invariants the damage in the concrete elements.
1.1 Failure criteria and Stress Invariants
According to Hawkins and Mitchell3), the failure mode ofcomposite structures is greatly influenced by the mechanicalproperties of shear connectors. However, in formulating failure
criteria for concrete materials under combined states of stress,
agreement must be reached on a proper definition of failure.
Criteria such as yielding, initiation of cracking, load-carrying
capacity and extent of deformation have been used to define
failure. A failure criterion of isotropic materials based upon stateof stress must be an invariant function of state of stress that is
independent of choice of coordinates system in which the stressin defined4). This criterion can be applied to un-cracked concrete,
however the cracked concrete is well known as a highly
anisotropic material.
Considering a 3D element subjected to pressure loading, thecomponents of stress tensor 𝜎 are given in Figure 1. In theprincipal direction, the stress state at any point inside a concretematerial can be defined by stress tensor 𝜎 in terms of three- 195 -Fig. 1 Components of stress tensor in an element.
principal invariants I1, I2 and I3. The stress tensor can also be
decomposed into two components (Equation 1) i.e., a purelyhydrostatic stress 𝜎 and a deviatoric stress 𝑆 part which
represents state of pure shear. The invariants of stress deviator
are expressed as J1, J2 and J3.
𝜎 =𝑆 +𝜎 𝛿(1)1.2 Non-Linear Finite Element Analysis
The numerical software used in the study is COM3 based on
the Maekawa Concrete Model in which fatigue damage ofconcrete is comprehensively enhanced by the embedded
constitutive laws5) to consider; a) decrease of stiffness andplasticity accumulation by continuous fracturing of concrete incompression, b) decrease of tension stiffness by bond fatigueand c) decrease of shear transfer normal to crack by continuous
deterioration of rough crack surface.
2. METHODOLOGY
2.1 Target Bridge
The target bridge (Figure 2) is a twin girder highway bridge
supporting a steel-concrete composite deck of width 6m andaverage span 49m. The 260mm thick concrete slab withembedded reinforcing bars is connected to 8mm thick bottom
steel plate by headed studs of diameter 16mm and withreinforcing ribs.
2.2 Modeling and Material properties
The analysis considers a half cross section of target bridge(Figure 2), modelled 20m in length and boundary conditions set
as in Figure 3. The girder is constrained at the lower flange ends
as a simply supported beam. For the half model of the bridge
considering symmetry, the nodes at the centerline areconstrained in the direction perpendicular to the bridge axis.
Material strengths for concrete (compression strength, f’c = 30
MPa and tension strength, ft = 2.22 MPa) and steel (main girder,yield strength, fy = 345 MPa, cross girder, bottom plate, studs andribs, yield strength, fy = 245 MPa and Reinforcement bar(SD345), yield strength, fy = 345 MPa) were applied.
The interface between the different materials to represent
adhesion and friction are reproduced by bond elements withparameters as show in Table 1.

Fig. 2 Bridge cross section (dimension in mm).

Fig. 3 Analysis model showing boundary conditions.
Table 1 Bond element properties
Parameter

Value

Shear stiffness in closure mode

76923 (N/mm2)

Normal stiffness in closure mode

200000 (N/mm2)

Normal/Shear stiffness in Open mode

15 (N/mm2)

Initial Bond (Normal/Shear)

3 (N/mm2)

Friction

0.5

Table 2 Fatigue loading cases
AnalysisLoading

Number of cycles

Case 1800 KN (4 x T-Load)

10,000,000

Case 21600 KN (8 x T-Load)

1,000,000,000

2.3 Fatigue loading
The fatigue loading cases are shown in Table 2. T-Load isdefined as a Truck Load specified by the Japanese Standard6). In
this study, excessive loads (not expected in reality on the actualbridge) are intentionally set to investigate failure process by
nonlinear analysis. It is also assumed that the loads are
symmetrically applied in the whole bridge model.
2.4 Bridge mid-point vertical displacement
The bridge mid-point vertical displacement was evaluatedaccording to load cases in Table 2. According to Figure 4, Case
2 deflection history shows that the slab mid-point verticaldisplacement increased with increase in the number of cycles
whereas in Case 1, the maximum displacement remained
constant regardless. In the first loading cycle, Case 1 produced amaximum deflection of 7mm as compared to 16mm for Case 2- 196 -Fig. 4 Bridge mid-point displacement history.

4xT-Load

8xT-Load

Fig. 5 Bridge deformed shape (scale factor = 50) with load at the center of slab (first cycle loading).

Fig. 6 Stud line S02 location.

Fig. 7 Cross section showing focus elements at stud line S02.

attributed to difference in the loading conditions. Figure 5 shows
bridge deformation (scale factor = 50) with load at center of slab.
3. ELEMENT DAMAGE EVALUATION
3.1 Focus area and Load consideration.
The area of focus is a cross section beneath the wheel track
along stud line S02 as indicated in Figure 6. A detailed crosssection of focus elements for each layer of concrete, locatedbetween adjacent ribs is shown in Figure 7. Additionally,
damage evaluation of cross section based on element location i.e.
near the rib (L1) and beneath the stud head (L4) have been
considered as indicated in Figure 7.
According to the bridge mid-point displacement results in
Section 2.4, only damage in concrete by load Case 2 (Table 2)

described in Section 2.3 was evaluated for invariant stress states.
3.2 Drucker-Prager yield criterion
The Drucker-Prager yield criterion is a pressure independent
model often referred to as the smooth version of Mohr-Coulomb
yield surface. This has been widely used in the geotechnical andconcrete engineering field to determine whether a material has
failed or undergone plastic yielding when subjected to loading,
based on the first invariant stress (I1) and the second invariant of
the deviatory part of stress (√J2 ) relationship. The damage
evaluation is derived from the Drucker-Prager failure boundary
surface governed by equation 2 below.

- 197 -𝐽 = 𝛼𝐼 + 𝛽𝐼 =𝜎+𝜎+𝜎(2)(2a)𝐽 =1⁄6𝜎−𝜎+ 𝜎−𝜎+ (𝜎 − 𝜎 )+ 𝜏+𝜏+𝜏(2b)Where I1 is the first invariant of Cauchy stress given byEquation 2a, J2 is the second invariant of the deviatory part ofCauchy stress derived from Equation 2b and 𝛼 , 𝛽 areparameters determined from experiments. Also the I1 - √J2graph can be used to confirm element confinement (Figure 8).

Fig. 8 Confinement indicator.
3.3 Tension and compression capacities of concrete
The first invariant of stress, I1 is an indicator of both tension
and compression in an element. Based on the tension strength, ft
and compression strength, f’c of concrete i.e. 2.22 MPa and 30
MPa respectively, the stress states given by Equations 3 and 4 to
identify elements under high stress values have been defined.
For compression side,
For tension side,> 1.0(3)< −0.074(4)3.4 Capacity of Shear
The second invariant of the deviatory stress, √ J2
represents shear effect. Based on the theory of shear transfer
capacity along a crack plane under monotonic loading5), themaximum shear, 𝜏is given by Equation 5.
𝜏= 3.83𝑓′

(𝑀𝑃𝑎)

(5)

In this study, the authors defined shear capacity as Equation
6 in order to apply to the general Drucker-Prager criteria.
𝐽𝜏
=𝑓′𝑓′> 0.4(6)
Fig. 9 Drucker-Prager relationship.

4. RESULTS AND DISCUSSIONS
4.1 Failure mode classification
Figure 9 shows the I1 – √ J2 relationship for selectedelements (half of focus cross section) in each layer depicting thestress state at selected loading cycles (1, 10,000 and
1,000,000,000). However, an evaluation of all cross-section
elements was conducted and failure modes explained in Section
3 were classified as shown in Figures 10-a, 10-b and 10-c.

In Figure 10-a, high compression stresses exceeding 30 MPa
occurred at the first layer on elements located directly above therib, beneath the stud head, between the two studs and also inthird layer near the bottom of the stud, attributed to the global
deformation of the deck slab. Additionally, for regions beneaththe stud head and at the bottom of stud, local deformation of boththe stud and bottom steel plate causes confinement to theelements thus a high compression.

- 198 -a) Compression sideb) Tension sidec) Boundary surface and capacity of sheard) Combined damage distribution map
Fig 10 Element failure mode classification.
Figure 10-b, shows elements in tension failure attributed toglobal deformation of the deck slab. Generally, layer 1 element
at center, layer 3 elements near the rib and element at slab center
experienced high-tension stress exceeding 2.22 MPa.
Figure 10-c, shear failure occurs at elements in the region
below the stud head and at the bottom of stud. In this region, as
compressive stress increases due to element confinement, shear
failure occurs. Failure considering the Drucker-Prager yieldsurface was confirmed at elements near bottom of the rib and at
the centroid of the slab.
The combined damage distribution (Figure 10-d) shows that,elements in the forward direction of moving load are more
susceptible to damage as compared to those on the opposite side.
4.2 Invariant stress envelopeThe stress envelop in Figure 9 for the 3 layers and Table 3
shows the invariant stress range for each layer during the entire
loading history. Layer 1 is characterized by an elongation on I1
axis and wide distribution on the √ J2 axis with highcompression stress values due to element confinement. Layer 2
has a shorter distribution implying low stress values due toproximity to neutral axis, while layer 3 has an elongated but thin
distribution on both axes. In layer 3, the elongated distribution
on the I1 axis is due to element 5390 near bottom of the stud
being subjected to high compressive stress due to confinement.
4.3 Comparison of elements stress state by location
Figure 11 shows the non-dimensional Drucker-Prager
relationship comparing element stress state according to location,

Table 3 Invariant stress distribution.
Layer 1

Layer 2

Layer 3I1 (MPa)-1.0～42.2-2.1～19.0-3.3～42.4

𝐽 (MPa)0～11.60～6.900～13.60

i.e. near the rib (L1) and beneath the stud (L4).
Near the rib (L1: Elements 4478 and 5243), all elements
experience compressive stress less than 50% of 30 MPa due tominimal confinement effect. However, due to global
deformation of slab, element 5243 located at the bottom exhibits
failure on tension side as stress path is outside failure boundary
surface and also the maximum tension stress exceeds 2.22 MPa.
Beneath the stud head (L4: Elements 5027 and 5390), allelements experience high compression due to the local bending
behaviour of the stud and the global deformation of the bottom
plate that causes element confinement. In element 5027,
compression stress exceeds 30.0 MPa by about 18 % in the firstloading cycle and further exceeds to about 49% as the loading
cycle increases. After shear failure occurs, compressive stressdecreases. Element 5390, compression stress only exceeds 30.0
MPa by 37% in the first loading cycle but reduces after shear
failure has occurred.
5. CONCLUSION
The damage of concrete element due to fatigue loading (8 xT-Load) assessed by stress invariants showed that;
1. Most damaged elements are located in layer 3 while layer
1 elements experience high compression stresses in excessof 30 MPa. However, from the bridge centroid, the- 199 -Fig. 11 Drucker-Prager relationship (non-dimensional) according to element locations L1 and L4.
damage distribution shows elements in the forward
direction of moving load are more susceptible to damage
as compared to elements on the opposite side.
2. At the rib, for element above, high compressive stress wasobserved while near the bottom, element failed in tension
and failure confirmed by Drucker-Prager yield criterion.
3. Elements beneath the stud head and at bottom experienced
high compressive stresses and shear failure was confirmed.
4. At the slab centroid, layer 1 is in high compression
however, failure is governed by high tension stress at bothlayer 1 and 3 elements. Also, failure of elements in layer 2and 3 were also confirmed by the Drucker-Prager criterion.
The results of this research based on stress invariants alone
are inadequate to predict damage. Therefore, consideration forstrain evaluation, stud deformation, etc. are recommended to
fully comprehend damage in steel-concrete composite decks.
ACKNOWLEDGEMENT
We are grateful to Japan Bridge Association for the funding
and Yokohama National University for the non-linear analytical
tools and work environment for the research.

REFERENCES
1) C. Fujiyama and K. Maekawa: Fatigue Damage Assessment
for Steel-Concrete Composite Deck, Society for Social
Management Systems Internet Journal, 6(1), March 2010.
2) C. Fujiyama, H. Hoshina, H. Wada, T. Kumano and T.
Matsumura: Fatigue Damage Analysis of Steel-concrete
Composite Bridge deck by FEM, MECHCOMP3,International Conference on Mechanics of Composites,
2017, Italy.
3) M. N. Hawkins and D. Mitchell: Seismic response ofcomposite shear connectors, Journal of StructuralEngineering, 110(9), pp.2120-2136, 1984.
4) Wai-Fah Chen and Atef F. Saleeb: Constitutive Equations
for Engineering Materials, Vol. 1, pp. 251-347, Elsevier,Netherlands, 1994.
5) K. Maekawa, A. Pimanmas and H. Okamura: Nonlinear
Mechanics of Reinforce Concrete, pp. 125-175, Spon Press,London, 2003.
6) Japan Road Association: Specifications for highway bridges
part 1, p. 94, 2017 (in Japanese).
（Received July 8, 2022）（Accepted September 9, 2022）- 200 -