doi:10.1016/j.engstruct.2005.05.019www.elsevier.com/locate/engstruct
es
formed by nts. Finally,
Wonseok Chunga, Elisa D. Sotelinob,∗
aTrack and Civil Engineering Research Department, Korea Railroad
Research Institute, Republic of Korea bDepartment of Civil and
Environmental Engineering, 214 Patton Hall, Virginia Tech,
Blacksburg, VA 24061, USA
Received 19 February 2004; received in revised form 4 January 2005;
accepted 10 May 2005 Available online 26 August 2005
Abstract
This paper investigates finite element (FE) modeling techniques of
composite steel girder bridges focusing on the overall flexural of
the system. In particular, four three-dimensional FE bridge models
are examined. Various modeling techniques, which are e to overcome
displacement incompatibility and geometric modeling errors, are
studied and issues related to the selection of ele discussed. A
technique that uses the concept of work equivalent nodal loads to
accurately represent the applied tire pressure is also The accuracy
of each model is verified against the results acquired from
full-scale laboratory test experiments and a field test per other
researchers. Furthermore, the results are also compared with those
of a detailed finite element model that uses solid eleme the
efficiency of each model, based on a comparison of computer
resources usage, is also presented. © 2005 Elsevier Ltd. All rights
reserved.
Keywords:Finite element method; Three-dimensional analysis;
Composite bridge; Compatibility
s re na e
tly la n r
1. Introduction
With the development of high-powered computer together with
state-of-the-art finite element (FE) softwa and user-friendly
graphical interfaces, three-dimensio (3-D) FE analysis has become a
popular choice ev for straightforward bridge analysis. More
specificall the design bending moment for steel girders can
determined more accurately using FE analysis of a brid
superstructure rather than using the lateral load distribut factor
specified in the AASHTO (American Association o State Highway and
Transportation Officials) specification However, accurate finite
element models must be used reliable bridge analysis.
This study is implemented to evaluate the curren adopted 3-D FE
models used in the analysis of a popu type of bridge, composite
steel girder bridges. It is know that some FE bridge models
available in the literatu introduce geometric errors and
compatibility errors whi can result in incorrect predictions of
flexural behavio
∗ Corresponding author. Tel.: +1 540 231 3174; fax: +1 540 231
7532. E-mail address:
[email protected] (E.D. Sotelino).
0141-0296/$ - see front matter © 2005 Elsevier Ltd. All rights
reserved. doi:10.1016/j.engstruct.2005.05.019
ab. ur- the
In this study, techniques to minimize modeling errors a introduced
and implemented. These techniques include use of displacement
transformations and the proper selec of finite elements.
The simplest 3-D FE model utilizes shell elements f the deck slab
with eccentrically stiffened beam elements the girders [1–4]. The
eccentricity of the girders is take into account by using rigid
links between the centroid of t concrete slab and the centroid of
the steel girders.
Brockenbrough [5] and Tabsh and Tabatabai [6] modeled deck slabs
using four-node shell elements that includ membrane and bending
effects. Each steel girder w divided into flange and web parts.
Each flange of the gird was idealized by beam elements, and the web
was mod by the four-node shell elements. Bishara et al. [7] adopted
the same modeling technique to represent the girder, they used
three-node thin plate triangular elements to mo the slab. The
eccentricity between the concrete deck and steel girder flange was
modeled by a rigid link.
Mabsout et al. [8] used three-dimensional solid element which have
linear shape functions, to model the deck sl The steel girder
flanges and web were modeled by fo node shell elements. Imposing no
releases between
os m ck idg th
ere ce d
g ur e es om ing h of ide lin
ra nit ite idg ec b th
ng ag ry ev ed
d el hi hi or
a th , is e b
in ck t t nt f t lts
to e .) re si) in
he
de nts is an
ing
shell elements and beam elements simulated the comp behavior
between the concrete deck and steel girder. Ea and Nowak [9] used
solid elements to represent the de while beam elements were used to
represent the br girders. The beam elements were attached directly
to bottom of the solid deck. In these works, no details w provided
concerning the lack of compatibility at the interfa between the
slab solid element and the elements use model the girders.
In the published literature, various FE modelin techniques have
been used to idealize bridge superstruct However, there is a lack
of information about th compatibility at interfaces when different
element typ are adopted. The geometric errors also observed in s FE
models which overlap two shell elements by shar the same node have
not yet been fully explored. T objective of this study is, thus, to
identify the source the incompatibilities and geometric errors, and
to prov guidelines to help analysts avoid these types of mode
errors in the FE analysis of bridge superstructures.
2. Finite element models
A bridge is a hybrid structure that combines seve structural
components. Since the proper selection of fi elements is key in a
FE model, different types of fin elements have been used in an
attempt to model br superstructures. First, the modeling of the
bridge d and girders is discussed separately. This is followed a
discussion of the techniques employed to model composite behavior
of the structure.
The numerical simulations were performed usi the general-purpose
finite element software pack ABAQUS [10]. This software provides an
extensive libra of elements that can model virtually any geometry
and s eral multi-point-constraint (MPC) options that can be us to
avoid displacement incompatibility between elements.
2.1. Modeling of bridge deck
In FE modeling, the bridge deck is typically modele either by solid
or shell elements. Shell elements are wid used to idealize the
bridge deck since behavior of t structural component is governed by
flexure and in t case a mesh of shell elements is computationally m
efficient when compared to one of solid elements. In bridge
application, the finite element that accounts for transverse shear
flexibility, the Mindlin type shell element preferred for an
accurate analysis, even though the transv shear deformations are
not usually significant. It should noted that Mindlin type elements
are susceptible to lock with full integration when the thickness of
the bridge de becomes thinner. This phenomenon is due to the fact
tha shear strain energy term tends to dominate the total pote
energy in these cases. This leads to the deterioration o element
bending stiffness, thus producing over-stiff resu
ite on , e e
Fig. 1. Maximum deflection of ABAQUS shell elements.
Several ABAQUS-provided shell elements were tested evaluate their
applicability for bridge deck modeling. Th tested plate was a 2.54
m (100 in.) by 1.27 m (50 in rectangular plate under a distributed
load. All supports we clamped. The modulus of elasticity was 20.7
MPa (3000 p and Poisson’s ratio was 0.3. The exact solution for a
th plate is given by [11]
wmax = 0.00254qL4
D (1)
whereD = Eh3
12(1−ν2) . The central deflections of ABAQUS
shell elements were compared to the exact solution of t thin plate,
as shown inFig. 1. It should be noted that the mesh density of each
model presented inFig. 1 is identical. It was found that a
quadrilateral nine-node (or eight-nod shell element with reduced
integration (S9R5/S8R5) and quadrilateral eight-node thick shell
element with reduce integration (S8R) predicted the same response
up to a ra of span length to depth(L/h) of approximately 150 which
is a reasonable upper bound for bridge analyses. Thus transverse
shear deformation may be neglected in typi bridge analyses. In this
study, the shear flexible shell elem (S8R) was selected to model
the concrete bridge deck.
Some models proposed in the literature utilize sol elements to
model a concrete bridge deck. The ma drawback of solid deck models
is the computational cost predict the correct flexural behavior of
the bridge. Multipl layers are required through the thickness
direction in order model the deck with linear solid elements (e.g.,
eight-no brick elements), since the strain variation of these eleme
is constant through the thickness. An alternative option the use of
higher order solid elements, but this may entail even higher
computational cost.
2.2. Modeling of bridge girders
In this study, four different modeling techniques fo steel girders,
named G1, G2, G3, and G4, respective are investigated. The element
selection for each model technique is shown inTable 1.
W. Chung, E.D. Sotelino / Engineering Structures 28 (2006) 63–71
65
r. n r e
to in
pt he
on th
and F, nts ell re nd
he m ible
am ns.
s of
by ed
Model name Girder part Web Flanges
G1 Shell element Shell element G2 Shell element Beam element G3
Beam element Shell element G4 Beam element
Fig. 2. Bridge model 1 and girder model G1.
The G1 model is a detailed model of a steel girde The flanges and
the web are modeled by shell eleme as shown inFig. 2. It should be
noted that only the girde part of the bridge model is considered in
this section. Sh elements must be placed along the mid-surface of
struct components. Numerical tests revealed that overlapp flange
elements with web elements by sharing the same n results in
significant modeling errors because of an incorr moment of inertia
about the primary bending axis. ABAQU has the capability to input
an offset distance for sh elements from their node locations. The
moment of iner can be matched with the actual I-beam moment of iner
by adjusting the offsety, shown inFig. 2. Alternatively to the
offset option, rigid links or constraints can be used create an
offset. However, this creates additional model complexity and is
therefore not used in this study.
The next model, G2, is similar to the G1 model exce that the flange
is modeled by beam elements instead of s elements, as shown inFig.
3. As a result, this model requires less computing resources to
represent the three-dimensi nature of the girder structures. The
beam elements have same properties as a girder flange with the
centroid of
ts
g
ll
Fig. 3. Bridge model 2 and girder model G2.
flange offset from the node location by one-half of the flan
thickness.
The G3 model is proposed here to investigate the poss
incompatibility at the element connection between web a flanges
found in the previous two models. A typical flat sh element is
formulated by superimposing plate bending a membrane action. The
resulting shell element has, thus, degrees of freedom (DOFs); three
translational DOFs two in-plane rotational DOFs at each node. A
sixth DO known as the drilling DOF, is often added to these eleme
to avoid singularity. This DOF is associated with the sh normal
rotation. However, if two neighboring elements a not coplanar,
compatibility between the in-plane rotation a drilling rotation is
generally violated [12]. The G1 and G2 models share the drilling
rotation of the shell element in t web with the in-plane bending
rotation of the shell or bea element in the flanges. Thus,
displacement is not compat along the element boundary of these
models.
The G3 model places shell elements at the centroid girder flanges.
Beam elements are placed at the cent of the girder web. Rigid
links, through the constraints DOFs, are applied to ensure
composite action. This mod illustrated inFig. 4. The shape
functions of beam and sh elements should be identical to avoid
incompatibility alo the element boundary.
The G4 model is the simplest model and utilizes be elements with
the geometric properties of girder sectio This model is shown
inFig. 5. It should be noted that the G4 model is not able to
represent different material propertie web and flanges.
The performance of each girder model is evaluated working through a
numerical example. A simply support
66 W. Chung, E.D. Sotelino / Engineering Structures 28 (2006)
63–71
o ti ar
ty.
ust ions wo on igid he ips
o a
Fig. 4. Bridge model 3 and girder model G3.
Fig. 5. Bridge model 4 and girder model G4.
beam having an I-shape cross section is subjected t concentrated
load at the center span. Because an analy solution of the maximum
deflection considering she flexibility is known from the theory of
elasticity [13], a direct comparison with the FE result is
possible. The giv single girder structure is modeled by models G1,
G2, and using either linear elements or quadratic elements. The fi
element selected for the G4 model was either an Euler be element or
a shear flexible Timoshenko beam element.
The convergence trends of the FE solutions are sho in Fig. 6. The
G1 and G2 models required significant me refinement to converge to
the analytical solution, while t G3 and G4 models produced less
than 1% error compare
a cal
Fig. 6. Convergence of finite element girder models.
the analytical solution, regardless of the mesh refineme In the G1
and G2 models, the prescribed incompatibil between the drilling
rotation and the transverse rotat tends to diminish as the mesh is
refined. It is also obser that the G1 and G2 models using quadratic
eleme converged more quickly than those using linear elements
It is concluded from the previous observation that t G3 and G4
models are simple yet produce accurate res regardless of the mesh
density. Models G1 and G2, howe are able to model the local
behavior of a bridge girder b require a refined mesh due to
displacement incompatibili
2.3. Modeling of composite action
1 0 0 0 e 0
(2)
W. Chung, E.D. Sotelino / Engineering Structures 28 (2006) 63–71
67
de
d de he th ng c m e e o
of ap t i
h
e uch of d ed
Fig. 7. DOFs for solid and shell/beam elements.
wheree is the eccentricity between the solid element no and the
shell/beam element node.
In the case where shell elements are used to mo the bridge deck,
the nodes of the girder do not coinci with the nodes of the shell
elements in the deck. T shell elements in the bridge deck are
connected with prescribed girder models through an MPC. Typical
bendi elements, such as the Kirchhoff shell element (for the de and
for girder models G1 and G3) and the Bernoulli bea element (for the
girder model G2 and G4), should be avoid for the modeling of the
composite girder bridge sinc displacement incompatibility occurs at
the interface of tw bending elements [14]. These bending elements
make use a linear shape for the axial displacement and a cubic sh
for the transverse displacement. The axial displacemen the girder
is given as
ug 1 = ud
1 − e · θd 2 . (3)
This incompatibility is noticeable since the axial displac ments of
the deck and the girder (ud
1 and ug 1) are linear,
but the rotation of the deck(θd 2 ) is quadratic in the axial
direction. Even though this incompatibility error completely
disappears as the mesh is refined, many methods have b proposed to
eliminate this nonconforming error [15–17]. In the ABAQUS
implementation, the use of S8R elements f the concrete deck and B32
elements for the girder giv full compatibility between the boundary
of two differen elements.
2.4. Discretization of applied loading
AASHTO specifications [18] specify the use of tire contact area for
a more exact analysis of bridge structur The applied loading on a
bridge deck consists of press loads applied through a tire patch.
In the finite eleme modeling, this requirement imposes the need for
a fine m in the deck, so that the element is fitted with the pat
size. As a part of this research, the equivalent nodal lo algorithm
is employed in order to uncouple the patch lo
el
e
k
d
Fig. 8. Discretization error of patch load.
from the mesh size. The equivalent load of the patch loa can be
calculated by the surface integral as
Re = ∫
S NTt dS (4)
whereN is the shape function matrix andt is the surface traction.
In this formal way, one must identify the nodes an elements that
lie on the patch load. For practical purpose in this work, the
patch load is discretized as a numbe of uniformly distributed
concentrated loads which will be called “sub-point loads”. Each
sub-point load is considere as a single concentrated load. If there
areK sub-point loads applied to the tire patch on an element(p),
then equivalent nodal forces are computed as
Re = K∑
i=1
NT i pi . (5)
The saved equivalent nodal forces are assembled to loaded nodes.
The major advantage of the discretized pa load algorithm is that it
eliminates the cumbersome loa boundary search problem and numerical
integration whi the accuracy of the FE solution is retained with
the sufficien refinement of tire patch. The discretization error of
patc load is illustrated inFig. 8 when an eight-node Mindlin shell
element is loaded under distributed load represent by sub-point
loads. Different levels of discretization ar considered by
increasing the number of sub-point load The exact equivalent nodal
forces are calculated b Eq. (4). It is clear that equivalent nodal
forces for both corne nodes and interior nodes using the proposed
discretiz algorithm converge to the exact value of equivalent nod
forces as discretization level increases. It is observed th
approximately 100 sub-point loads results in less than 0.5 error
for both corner and interior nodes.
2.5. Boundary conditions
Since the main purpose of this study is to analyz bridge
superstructures, it is assumed that substructures, s as piers and
abutments, do not influence the behavior the superstructure.
Although bearings are typically locate below the beam element, many
previous models neglect
68 W. Chung, E.D. Sotelino / Engineering Structures 28 (2006)
63–71
Table 2 Material properties of Nebraska bridge test
Material properties Concrete Steel
Slab Parapet Top Bottom Bottom web Reinforcing flange flange (int)
flange (ext) bar
Young’s modulus, GPa 193.7 180.6 191.0 200.6 190 Yield stress, MPa
– – 283 268 279 323 500 Ultimate stress, MPa – – 467 445 447 415
827 Compressive strength, MPa 43.0 41.8 – – – – –
ro e
in a
lts ry t s
ia
ing
ed d
g al
al e. ts r ed in e e is ite id el el.
s rs. e st ts et. e e lid
ll ar en he
l d
this fact and assumed bearings to be located at the cent of the
beam element or at the bottom flange of th beam. In this study,
bearings are modeled by assign boundary conditions to the
zero-dimensional elements their real location. For simply supported
beams, rotatio in all directions are allowed in order to simulate
th simply supported structure. Minimum restraints are assign for
longitudinal and transverse movement while vertic restraint is
placed at the supports. Kinematic constraints also supplied to
nodes between the girders and the deck.
3. Numerical comparisons
The objective of this section is to compare the resu of finite
element analyses to those of a full-scale laborato test and an
actual field test. The comparison is restricted elastic load levels
to investigate the validity of the variou bridge models.
3.1. Full-scale laboratory test
A full-scale steel girder bridge was designed, constructe and
tested in the structural laboratory of the University Nebraska at
Lincoln [19]. The bridge is 21.4 m (70 ft) long and 7.9 m (26 ft)
wide. The superstructure consists of thr 137 cm (54 in.) deep
welded plate girders built composite with a 19 cm (7.5 in.) thick
reinforced concrete deck. The are three girders with girder spacing
of 3 m (10 ft). Th reinforced concrete deck was reinforced in both
the top a bottom of the slab. The details of the bridge
configuratio and loading can be found in the original report. The
mater properties are summarized inTable 2.
The test loading set-up consisted of 12 post-tension rods
simulating approximately two side-by-side AASHTO HS-20 design
trucks. The four rods simulating front axle are placed at 6.1 m
from the left support. The rods we spaced at 3.7 m (12 ft) and 4.6
m (15 ft) instead of th typical AAHSTO HS-20 spacing of 4.3 m (14
ft) and 4.3 m (14 ft). Two sets of loads simulating two trucks ar
placed symmetrically with respect to the center girder the
transverse direction. The tire contact area was simula using steel
plates with dimensions of 50 cm (20 in.) by 20 c (8 in.) for rear
and center wheels and 25 cm (10 in.) by 10 c (4 in.) for front
wheels. For the elastic test, 2.5 times HS2
id
l
d
truck load was applied on the rod plates. This load consist of
362.5 kN (40 kips) for the center and rear wheels an 87.5 kN (10
kips) for the front wheels. The applied loadin on the bridge deck
is discretized using the equivalent nod force algorithms as
discussed in the previous section.
In the present study, a total of five three-dimension finite
element models are implemented for the tested bridg Four models
make use of eight-node Mindlin shell elemen (ABAQUS S8R) as the
deck model with different girde models (girder models G1, G2, G3,
and G4) as discuss in the previous section. These models are
illustrated Figs. 2–5. Model 1 uses shell elements to represent th
bridge deck and the quadratic G1 model to idealiz the bridge
girders. The parapet of the tested bridge modeled by three-node
beam elements. The compos action between the deck and the girder is
modeled by rig links. The second model (Model 2) is the same as Mod
1, but the girders are modeled by the quadratic G2 mod In Model 3,
the girder model G3 is used for modeling the girders. Model G4 is
the simplest model, which use three-node beam elements (G4) to
represent bridge girde This model is known as the eccentric beam
model. Th last model is denoted as “Solid Model”, and uses the mo
detailed girder model, G1, while quadratic solid elemen (ABAQUS
C3D20) are used to model the deck and parap Full composite action
and displacement compatibility ar achieved by imposing kinematic
constraints between th solid elements and shell elements. The FE
mesh of the So Model is shown inFig. 9.
Figs. 10 and 11 present the bottom flange deflections at the
interior girder and exterior girders, respectively. A finite
element models generally produce deflections simil to the measured
deflections. The maximum error betwe predicted deflection and
measured deflection is 5% for t Solid Model and 6% for Model 4. The
predicted bottom flange strains are compared to the measured
strains Fig. 12. The maximum error is 5% for the interior girder at
mid-span in Model 1 and 9% for the exterior girder a the
quarter-span in the Solid Model. Model 3 and Mode 4 predict values
that are closer to the measured strains both cases.
The lateral load distributions of each bridge mode are compared
through the AASHTO specified loa distribution factor (LDF).Fig.
13shows the LDF from the
W. Chung, E.D. Sotelino / Engineering Structures 28 (2006) 63–71
69
Fig. 9. Finite element model of Nebraska bridge.
s. tly ar
Fig. 10. Deflection of interior girder.
Fig. 11. Deflection of exterior girders.
AASHTO Standard specifications [18], AASHTO LRFD specifications
[20], experiments, and finite element model The LDFs calculated by
finite element models are sligh larger (up to 6%) than the measured
LDF. However, it is cle that the LDF values from the
AASHTO-Standard and t ASSHTO-LRFD are more conservative than both
measu
Fig. 12. Strains at the bottom flanges.
Fig. 13. Load distribution factors.
and predicted LDF values. All finite element models a capable of
predicting the load distribution mechanism of composite
bridge.
The amount of computational resources used by various FE models is
compared inFig. 14. The total number of DOFs required for the Solid
Model is three times larg than the number required for Model 4.
This indicates th the eccentric beam model (Model 4) is as accurate
as other models yet it is the simplest model. The eccentric
be
70 W. Chung, E.D. Sotelino / Engineering Structures 28 (2006)
63–71
t
re an d
ed
l 4) ong tely g re, s a ge
A &
Fig. 14. Total number of DOFs.
model is thus further verified with a field test in the nex
section.
3.2. Michigan field test
The eccentric beam model (Model 4) is verified with the results
from a field test conducted at the University Michigan [21]. The
tested bridge is a simple span located o Stanley Road over I-75 in
Flint, Michigan. The span length 38.4 m (126 ft). There are seven
girders with girder spacin of 2.2 m (7.25 ft) and an overhanging
width of 74.7 cm (2.45 ft). The slab thickness is 20 cm (8 in.).
Strain gaug were installed at the bottom flanges of the girders.
All strain were measured along the centerline of the bridge span. T
test load was the Michigan three-unit, 11-axle truck. Th load test
was performed with the truck at crawl speed produce the maximum
static strain at the steel girders.
The predicted strains and load distribution factors a compared to
those obtained from the test results. As c be seen inFig. 15, good
agreement between measured an calculated values is observed in all
girders. The maximu error in the finite element model is within 6%
of the measured strain value at girder 4. The model is also able
predict the lateral load distribution accurately. Therefore, can be
concluded that the eccentric beam model (Model used in this study,
is capable of accurately predicting th actual behavior of steel
girder bridges.
4. Summary and conclusions
This paper discusses two key issues in the thre dimensional finite
element modeling of composite girde bridges: element compatibility
and geometric error. Sinc the proper selection of finite elements
is key to avoi compatibility errors, different types of finite
elements wer used to model bridge superstructures. In addition,
techniqu to reduce geometric errors and to discretize the appli
pressure load are also presented.
Based on comparisons between the results obtained us several finite
element models and available experimen results or analytical
solutions, the following conclusion
e
(b) Load distribution factor.
Fig. 15. Results of Michigan field tests.
can be drawn. First, girder models, which utilize sh elements for
their girder modeling (girder model G1 a G2), require a higher
level of mesh refinement to conve due to the displacement
incompatibility between the drilli DOF of the web element and the
rotational DOF of t flange element. In general, for the same level
of me density, quadratic elements are more accurate than lin
elements. Secondly, the eccentric beam model (Mode has been
identified as the most economical model am all studied models,
since this model is capable of accura predicting the flexural
behavior of girder bridges, includin deflection, strain, and
lateral load distribution. Furthermo the finite element model for
slab on girder bridges provide rational tool for the understanding
of the behavior of brid superstructures.
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Three-dimensional finite element modeling of composite girder
bridges
Introduction