Engineering Structures 181 (2019) 617–628
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Engineering Structures
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The shear-lag effect of composite box girder bridges with corrugated steel
webs and trusses
T
Yiyan Chenb, Jucan Donga,b, , Tianhua Xub,c, Yufeng Xiaob, Ruijuan Jiangb,c, Xinmin Nieb
⁎
a
School of Civil Engineering, Hunan University, Changsha, China
Shenzhen Municipal Design & Research Institute Co., Ltd., Shenzhen, China
c
School of Civil Engineering, Shandong University, Jinan, China
b
ARTICLE INFO
ABSTRACT
Keywords:
Shear-lag effect
Composite box girders
Corrugated steel webs
Trusses
Energy variational principle
The composite box girder bridge with corrugated steel webs and trusses is a recently proposed enhanced
composite box girder bridge structure. The shear-lag effect of this kind of structure is studied in detail in this
research. Two 8.744 m-long test beams with and without concrete filled in the bottom steel tubes were constructed and tested. The non-uniform distribution of cross-sectional stress in the top concrete slab was captured.
The filling of concrete inside the bottom steel tubes is shown not to have a great influence on the shear-lag effect.
Numerical parametric studies were carried out to study the influence of various factors on the shear-lag effect of
this kind of structure. The numerical results indicate that the magnitude of the shear-lag effect increases with the
width-to-span ratio and the suspension ratio. Analytical equations were derived to calculate the shear-lag
coefficient for composite box girders with corrugated steel webs and trusses based on the energy variational
principle. The experimental and numerical validation indicates that the proposed equations can be well applied
in engineering practice.
1. Introduction
Recently, Chen et al. [1] proposed a new type of advanced bridge
structure — the Composite Box Girder Bridge with Corrugated Steel
Web and Trusses (CBGB-CSWT, Fig. 1), which is mainly composed of a
top concrete slab, corrugated steel webs and two bottom steel tubes
connected by trusses. This new type of bridge structure integrates the
advantages in the mechanical behaviors of concrete-filled steel tubes
and corrugated steel webs, which greatly reduces the bridge self-weight
and facilitates the bridge construction. The positive bending moments
are sustained by the top concrete slab and the bottom steel tubes, while
the negative bending moments are carried by the bottom concrete-filled
steel tubes and the prestressing tendons in the top concrete slab. In this
way, all the materials in the bridge work effectively, and the ductility of
bridges can be improved. Similar to other kinds of box girder structures,
CBGB-CSWTs also experience the shear-lag effect. The shear-lag effect
refers to the non-uniform longitudinal normal stress distribution within
the flanges induced by the shearing interaction between the webs and
the flanges (e.g. [2]). Whether or not the shear-lag effect is properly
considered in design has a great influence on the safety of a CBGBCSWT. Therefore, the shear-lag effect of this new kind of composite
structure is worth being studied.
⁎
The idea of a CBGB-CSWT mainly originates from traditional
Composite Box Girder Bridges with Corrugated Steel Webs (CBGBCSWs), including the ones with a concrete bottom slab or a single
bottom concrete-filled steel tube. For traditional CBGB-CSWs, comprehensive studies have been carried out on their flexural [3–6], torsional
[7,8], shearing [9–11], dynamic [12,13] behaviors. However, only
limited studies focus on their shear-lag effect. Chen et al. [14] proposed
a shear-lag warping displacement function to evaluate the shear-lag
effect of a single-box multi-cell girder with corrugated steel webs. Using
the energy variational method, the basic differential equations of twinand triple-cell box girders with corrugated steel webs were obtained.
Hu et al. [15] investigated the shear-lag effect of curved CBGB-CSWs,
and showed that the radius of curvature of the girder, the width-to-span
ratio, the spacing of the webs and the transverse loading position are
important influencing parameters. Zhang et al. [16] incorporated the
influence of the shear-lag effect in the derivation of analytical equations
for the natural frequencies formulas of CBGB-CSWs, which shows that
the calculated natural frequencies become smaller when the shear-lag
effect is considered. Ji et al. [17] considered the shear lag and shear
deformation in the energy-based calculation of the vertical deflection of
CBGB-CSWs. Jiang et al. [18] proposed a new definition of effective
flange width coefficient that can be used as an effective indicator to
Corresponding author at: School of Civil Engineering, Hunan University, Changsha, China.
E-mail address: dongjican@szmedi.com.cn (J. Dong).
https://doi.org/10.1016/j.engstruct.2018.12.048
Received 13 August 2018; Received in revised form 6 December 2018; Accepted 14 December 2018
Available online 24 December 2018
0141-0296/ © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/BY-NC-ND/4.0/).
Engineering Structures 181 (2019) 617–628
Y. Chen et al.
determined according to the scale rules of modeling tests [1].
Load Case 1 (LC 1): Two layers of concrete blocks were placed on
the top surface of the test specimen. The height of the first layer is
47 cm, which simulates the self-weight of the reference viaduct. The
second layer has a height of 29 cm, which simulates the weight of the
pavements and the facilities on the viaduct.
Load Case 2 (LC 2): A two-point concentrated loadings were applied
on the top surface of the test beam. Each loading point was 2450 mm
away from the nearest end of the test beam.
Load Case 3 (LC 3): A one-point concentrated loading was applied
on the top surface at the mid-span to simulate the live load.
LC 2 and LC 3 are applied to simulate the live load of the referenced
viaduct.
Fig. 3 shows the overview of the test setup. The specimens were
loaded with a 1000 kN hydraulic jack. Before the test, a pre-loading
force equal to 10% of the estimated ultimate flexural capacity was first
applied to the test beam. The pre-loading force was kept for 3 min and
then the test beam was unloaded. A force control process was adopted
for the test. During the test, when the specimen is at the elastic stage,
the increased load is 10 kN at each load step, which was applied with a
loading speed of 0.67 kN/s. At the elasto-plastic or plastic stage, the
increased load at a load step was reduced to 5 kN, and the loading rate
was 0.17 kN/s. At each load step, the reading was taken manually after
the data became stable.
The strain of the test specimen was measured at four cross sections
(A–D) shown in Fig. 4(a). The arrangement of strain gauges at the top
concrete slab, the corrugated steel webs and the bottom steel tubes is
demonstrated in Fig. 4(b). The strain of concrete was measured with
BX120-100AA strain gauges with a grid length of 100 mm and an
electric resistance of 120 Ω. The strain of steel in the webs or the steel
tubes was measured with BX120-3CA strain gauges. The sensing unit of
a BX120-3CA strain gauge consists of three measuring grids arranged
along 0°, 45° and 90° directions, respectively. The deflection of the
specimen was measured with YHD-200 linear variable differential
transducers (LVDT). The range of measurement is −100 mm to
100 mm.
Top concrete slab
Corrugated steel webs
Bottom trusses
Bottom steel tubes
Concrete
Fig. 1. Schematical diaphragm of a composite box girder bridge with corrugated steel web and trusses (CBGB-CSWT).
help evaluate the shear-lag effect of a CBGB-CSW. In summary, the
above-mentioned studies only provide a shallow insight on the shearlag effect in bridges adopting corrugated steel webs. Experimental and
detailed parametric studies have yet to be conducted. Moreover, as the
CBGB-CSWT is recently proposed, its shear-lag effect has not yet been
investigated. The influence of its structural members (such as the
bottom concrete-filled steel tubes) and geometric parameters on the
shear-lag effect should be characterized.
The aim of this research is to comprehensively study the shear-lag
effect of CBGB-CSWTs. First, modeling tests on the shear-lag effect of
this kind of structure will be presented. Both the specimens with and
without concrete filled in the bottom steel tubes were tested.
Afterwards, a comprehensive numerical parametric research will be
carried out to highlight the important factors that influence the shearlag effect of this kind of structure. The difference between the shear-lag
effects in this kind of structure and CBGB-CSWs will be indicated.
Finally, analytical equations will be developed to calculate the shearlag coefficients for CBGB-CSWTs to facilitate the calculation of shear lag
effects in real bridge design.
2. Experimental study
2.1. Test specimens
2.3. Experimental results and analysis
The two test specimens in this research were constructed with reference to the design of the left sub-bridge of the Maluanshan Park
Viaduct in Shenzhen, Guangdong Province, China [1]. The linear dimension scaled factor between the test specimens and the referenced
viaduct (Sl) is 1/5. In the first specimen (SP1), the bottom steel tube was
filled with concrete. On the contrary, the other specimen (SP2) was
constructed with hollow bottom steel tubes.
Fig. 2 shows the detailed design of the test specimens. The net
length of the test beam is 8.984 m. The depth of the girder and the
diaphragms is 0.56 m and 0.6 m respectively. The width of the top
concrete slab is 2.08 m. The trusses connecting the steel tubes are
comprised of transverse and diagonal braces. Each of these braces was
made of a steel channel. The braces were all welded to the bottom steel
tubes. The mechanical properties of the test beams are shown in
Table 1. All the values listed in Table 1 are average values obtained
from laboratory tests. The elastic modulus of concrete were obtained
from the compression tests of three 300 mm × 150 mm × 150 mm
prismatic specimens, and the compressive strengths were measured
from the compression tests of three 150 mm × 150 mm × 150 mm
cubic specimens. Tension tests were carried out for three steel coupons
to determine the elastic modulus and the tensile strength of steel.
For the experimental results, the shear-lag coefficient is calculated
by dividing the measured cross-sectional normal stress by the mean
cross-sectional normal stress on the top surface. Let x denote the longitudinal coordinate with the origin set at the centerline of support at
one end, and L represent the calculated span of the test specimen.
Fig. 5(a) and (b) compare the measured shear-lag coefficients on the top
surface of the top slab in SP1 and SP2 under the distributed loading in
LC 1. The shear lag effect is not obvious at both the mid-span cross
section and the x = L/3 cross section sections of two test beams under
the distributed loading. The maximum shear-lag coefficients of the test
beam SP1 and SP2 are 1.029 and 1.009, respectively. Fig. 5(c)–(f) are
the shear-lag coefficients distribution on the top surface of the top slab
at the mid-span and x = L/3 cross sections under the concentrated
loadings in LC 2 and LC 3. According to the figures, positive shear-lag
effect can be observed at the cross sections next to the loading position
(i.e. the x = L/3 cross sections in LC2 and the mid-span cross section in
LC3). On these cross sections, the shear-lag coefficient reaches the
largest value at the intersection between the top concrete slab and the
corrugated steel webs. The maximum value of the shear-lag coefficient
along the entire test specimens for SP1 and SP2 are 1.126 and 1.108,
respectively. The relative difference of these two values is only 2%. On
the cross sections relatively far away from the loading position (i.e. the
mid-span cross section in LC2 and the L/3 cross sections in LC3), a
nearly uniform distribution of the shear-lag coefficient can be observed,
indicating that the magnitude of the shear-lag effect is small at these
cross sections.
In Fig. 5, the distribution of the shear-lag coefficient of SP1 nearly
2.2. Loading set up and instrumentations
In the modeling tests, three types of loading were applied to the test
specimen, which includes a distributed loading, a two-point concentrated loading and a one-point concentrated loading. The tests were
carried out under three different load cases as follows, which are
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Top concrete slab
9000
8504
120 120 8
600
600
560
8 120 120
Corrugated steel webs
Bottom steel tube
Centerline of support
Centerline of support
(a)
2080
960
560
560
100
560
Bottom steel tube
Φ146x6
31 °
44
(b)
74
86
86
74
320
(c)
Fig. 2. Design of test specimens (Unit: mm): (a) side view; (b) typical cross-section; (c) a corrugation of web.
overlaps with that of SP2. The relative difference of the maximum values of the shear-lag coefficient is within 2%. Therefore, it can be inferred that the filling of concrete inside bottom steel tubes does not
greatly influence the shear-lag effect of a CBGB-CSWT. This conclusion
will be further verified by the numerical investigations in the following
sections of the paper.
each other. The slip between the webs and the concrete slab was also
ignored. At one end of the finite element model, the y and z displacements of the nodes at the bottom of the diaphragm were set to 0. For the
nodes at the same location at the other end of the finite element model,
the y displacement is restraint.
The concrete in the top slab was modeled using the recommended
stress-strain relationship in [20]. The stress-strain relationship of concrete under compression can be determined by Eqs. (1)–(4):
3. Finite element analysis
= (1
3.1. Finite element model
dc =
Numerical analysis was carried out to further investigate the characteristics of the shear-lag effect of a CBGB-CSWT. The typical cross
section of the finite element models is shown in Fig. 4(b). The finite
element model was developed using ABAQUS [19]. An example of the
finite element models is shown in Fig. 6. The concrete parts were
modeled using C3D8R solid elements, which is a kind of 8-node linear
brick element with reduced integration (i.e. 1 integration point). The
bottom steel tube, the corrugated steel webs, the K-braces and the
trusses were simulated by S4R shell elements, which is a kind of 4-node
linear shell element with reduced integration. Truss elements were used
to model the steel reinforcement bars. The bars were directly embedded
into the concrete without considering the slip between each other. The
corrugated steel webs were connected to the top concrete with the
“TIE” constraint in ABAQUS. This constraint is used for making the
degrees of freedom equal for a pair of surfaces when they are close to
c =
n=
(1)
d c ) Ec
1
1
cn
Xc
1 + Xcn
n
c
c (x
1)2 + Xc
1
Xc > 1
(2)
fck
Ec c, r
(3)
Ec c, r
Ec c, r fck
(4)
where σ and ε are the stress and strain of concrete, respectively; Ec is the
elastic modulus of concrete; αc is a model coefficient influencing the
shape of the stress-strain curve; fck is the characteristic strength of
concrete calculated by fck = 0.67 fcu; and fcu is the cube strength of
concrete; εc,r is the strain of concrete corresponding to the peak of the
stress-strain curve; dc is a damage coefficient; Xc = ε/εc,r is the normalized compressive strain. εc,r and αc can be adopted from Table 2
Table 1
Material properties of test specimens.
Material
Concrete
Steel
Component
Top slab
Inside bottom steel tubes
Corrugated steel web
Bottom steel tubes
Elastic modulus (N/m2)
4
3.96 × 10
3.68 × 104
2.03 × 105
2.47 × 105
Compressive strength (MPa)
Yielding strength (MPa)
Ultimate tensile strength (MPa)
53
49.5
–
–
–
–
316
332
–
–
419
426
Note: All the values in Table 1 are average values obtained from laboratory tests.
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[20].
The constitutive relationship proposed by Han et al. [21] was
adopted for the concrete inside the bottom steel tubes. In this model,
the equivalent stress-strain model of the core concrete in a concretefilled steel tube under compression can be written as:
y=
x 2 (x
2x
0 (x
1)
x
(x > 1)
1) 0 + x
(5)
where
x = ε/ε0;
y = σ/σ0;
σ0 = fck;
ε0 = [(1300 + 12.5 ×
fck + 800 × θ0.2] × 10−6;
η0 = 1.6 + 1.5/x; 0 = f ck0.1 /(1.2 1 + ) ;
θ = Asfy/Acfck; As and Ac are the cross-sectional areas of the steel tube
and the concrete core, respectively; fy is the yield strength of steel (unit:
N/mm2).
In this research, the material properties listed in Table 1 were used
for the above mentioned constitutive models of concrete. For the top
concrete slab, Ec = 3.96 × 104 MPa and fcu = 53 MPa. For the concrete
inside bottom steel tubes, Ec = 3.96 × 104 MPa and fcu = 53 MPa.
The steel bars and webs are modeled using the idealized elastoplastic model in ABAQUS [19]. The elastic modulus of steel are
2.06 × 105 MPa, which is determined based on the measured value in
Table 1. The Poisson’s ratio of steel is set to be 0.30.
3.2. Model verification
As a very first step of the numerical research, two numerical models
for the test specimens SP1 and SP2 were developed to verify the numerical method. The longitudinal normal strain on the top surface of
the concrete slab at some typical cross sections obtained from numerical simulation is compared to the experimental results. Fig. 7 shows
such a comparison at the mid-span and x = L/3 cross sections for the
test specimens under the concentrated load applied at the mid-span (LC
3). It is corresponding to a test load of 33.6 kN. According to the figure,
the variations of the longitudinal normal strain in these two cases agree
well with each other. The numerical and experimental values of the
longitudinal normal strain are similar at each strain gauge location.
Fig. 3. Overview of test set up.
8744
4372
4372
P
2450
P
320
1280
320
1600
2450
320
A
BC
D
A
BC
D
Longitudinal concrete strain gauge
P: Test loading corresponding to LC2
(a)
2080
30
250
200 160 200
200
200
200 160 200
250 30
Longitudinal concrete strain gauge
Longitudinal steel strain gauge
(b)
Fig. 4. Loading set up and instrumentation of the specimen (Unit: mm): (a) side view; (b) typical cross-section.
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1.2
SP1
SP2
1.1
Shear lag coefficient
Shear lag coefficient
1.2
1.0
0.9
web positions
0.8
0
500
1000
1500
Transverse position (mm)
SP1
SP2
1.1
1.0
0.9
web positions
0.8
2000
0
(a)
1.2
SP1
SP2
1.1
1.0
0.9
SP1
SP2
1.1
1.0
0.9
web positions
0.8
0
500
1000
1500
Transverse position (mm)
0.8
2000
web positions
0
(c)
1.2
SP1
SP2
1.1
1.0
0.9
web positions
0.8
0
500
1000
1500
Transverse position (mm)
500
1000
1500
Transverse position (mm)
2000
(d)
Shear lag coefficient
Shear lag coefficient
1.2
2000
(b)
Shear lag coefficient
Shear lag coefficient
1.2
500
1000
1500
Transverse position (mm)
1.1
1.0
0.9
web positions
0.8
2000
(e)
SP1
SP2
0
500
1000
1500
Transverse position (mm)
2000
(f)
Fig. 5. Shear-lag coefficient on top surface of top slab: (a) mid-span section (LC 1); (b) x = L/3 section (LC 1); (c) mid-span section (LC 2); (d) x = L/3 section (LC 2);
(e) mid-span section (LC 3); (f) x = L/3 section (LC 3).
Moreover, as shown in Fig. 8, the simulated load-deflection curve
agrees well with the experimental curve. Therefore, the finite element
models in this research can well simulate the test specimens and hence
can be used to further investigate the shear-lag effect of this kind of
bridge structure.
3.3.1. Influence of the filling of concrete inside the bottom steel tubes
To study the influence of the filling of concrete inside the bottom
steel tubes, a series of finite element models of CBGB-CSWTs with the
same span but different widths were developed. The calculated span of
the finite element models is 8744 mm. The widths of the finite element
models vary from 874 mm to 8744 mm. Given the same width of the
models, both the cases with and without concrete filled inside bottom
steel tubes were considered. The maximum shear-lag coefficient at the
mid-span cross section (λmax,1/2) is used as an indicator to show the
magnitude of shear-lag effect for each finite element model. Herein, the
maximum shear-lag coefficient in a cross section is defined as the
maximum value of the ratio between the numerical simulated normal
stress and that calculated following the elementary beam theory.
Fig. 10. shows the λmax,1/2 values for the abovementioned finite
3.3. Parameteric analysis
Detailed numerical research has been carried out to investigate the
influence of important parameters on the shear-lag effect of CBGBCSWTs, which includes the filling of concrete inside the bottom steel
tubes, the width-to-span ratio (b/L), the suspension ratio (b1/b2) and
the width-to-depth ratio (b/h). The definition of the geometry parameters b, b1, b2 and h are shown in Fig. 9.
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Y. Chen et al.
Concrete slab
Corrugated steel webs
Bottom steel tube
End diaphragm
(a)
Vertical K brace
Corrugated steel webs
Bottom truss structure
Bottom steel tube
End diaphragm
(b)
Fig. 6. Finite element model of a CBGB-CSWT: (a) whole model; (b) model with hidden concrete slab.
element models. Given the same dimensions and loading condition, the
λmax,1/2 value in the case with concrete-filled bottom steel tubes is
larger than that in the case with hollow bottom steel tubes, but the
difference between the two results is small. The largest relative difference is only 4.5%, which is observed for the two models with a widthto-span ratio of 1.0 subjected to a uniform distributed loading. Therefore, the filling of concrete inside the bottom steel tubes does not
greatly influence the shear-lag effect of a CBGB-CSWT. This conclusion
is coincident with that drawn from the experimental research.
3.3.3. Influence of the suspension ratio (b1/b2)
As the filling of concrete is shown not to greatly influence the shearlag effect of a CBGB-CSWT, the parametric study is carried out based on
the models with concrete-filled bottom steel tubes. In this research, the
influence of the suspension ratio is studied by varying b1 from 240 to
960 mm while keeping L = 8744 mm and b2 = 480 mm unchanged in
the finite element models.
The λmax,1/2 values corresponding to difference b1/b2 values under
two loading conditions are shown in Fig. 11. The suspension ratios b1/
b2 vary from 0.75 to 2.00. λmax,1/2 increases from 1.028 to 1.164
(13.2%) when the model is subjected to a concentrated loading at the
mid-span, and from 1.009 to 1.064 (5.4%) when the model is subjected
to a uniform distributed loading. Therefore, the influence of the suspension ratio on the shear-lag effect of a CBGB-CSWT is not as large as
that of the width-to-span ratio.
3.3.2. Influence of the width-to-span ratio (b/L)
Fig. 10 also shows the influence of the width-to-span ratio. The
width-to-span ratios in the cases in Fig. 10 vary from 0.10 to 1.00. For
both the cases with and without concrete filled inside the bottom steel
tubes, the λmax,1/2 values increases dramatically with the width-to-span
ratio. When b/L = 0.10, λmax,1/2 is close to 1.00, indicating that the
longitudinal normal stress distributes uniformly when the width of the
model is small. When b/L increases to 1.00, λmax,1/2 increases to about
2.00 under a concentrated loading at the mid-span and about 1.35
under a uniform distributed loading. In other words, the maximum
longitudinal normal stress is 100% and 35% larger than the average
value. Therefore, the width-to-span ratio has a great influence on the
shear-lag effect of a CBGB-CSWT.
3.3.4. Influence of the width-to-depth ratio (b/h)
Finite element models with different depths were developed. The
depths of these models are 297–693 mm, and the calculated span and
the width are 8744 mm and 2080 mm, respectively. The corresponding
width-to-depth ratios are 3.00–7.00. Two loading cases are considered,
including a concentrated loading at the mid-span cross section and a
uniformly distributed loading along the whole girder. Fig. 12 shows the
λmax,1/2 values in the simulated cases. The curves in the figure are
nearly horizontal, indicating that width-to-depth ratio almost has no
Table 2
Coefficients in the stress-strain relationship for concrete in the top slab.
fc,k (MPa)
εc,r (1 0 −6)
αc
20
1470
0.74
25
1560
1.06
30
1640
1.36
35
1720
1.65
40
1790
1.94
45
1850
2.21
622
50
1920
2.48
55
1980
2.74
60
2030
3.00
65
2080
3.25
70
2130
3.50
75
2190
3.75
80
2240
3.99
Engineering Structures 181 (2019) 617–628
Y. Chen et al.
Transverse postion (mm)
Transverse postion (mm)
-10
0
500
1000
1500
2000
-10
0
1000
2000
web positions
-20
Strain (µε)
Strain (µε)
-20
-30
-30
web positions
-40
-40
FEM
FEM
Experiment
-50
Experiment
-50
(a)
(b)
Transverse postion (mm)
Transverse postion (mm)
-10
0
1000
2000
500
-10
1000
1500
2000
FEM
Experiment
-20
web positions
Strain (µε)
Strain (µε)
-20
-30
-30
web positions
-40
-40
FEM
-50
Experiment
-50
(c)
(d)
t2
20
b
b2
tw
b1
x
y
t
tu
tw b2
t1
b1
h
Load (kN)
30
h2 h1
Fig. 7. Longitudinal normal strain of the top slab (LC 3, load = 33.6 kN): (a) mid-span section of SP1; (b) x = L/3 section of SP1; (c) mid-span section of SP2; (d)
x = L/3 section of SP2.
10
0
0.0
FEM
Experiment
0.5
1.0
1.5
2.0
Deflection (mm)
2.5
d
3.0
z
d
Fig. 9. Geometry parameters in a typical cross section of a CBGB-CSWT.
of one of the investigated CBGB-CSWT models and the corresponding
equivalent cross section of the CBGB-CSW model are shown in
Figs. 4(b) and 13, respectively.
The calculated span length of all the finite element models for the
comparison is 8744 mm. The widths of the top concrete slabs in these
models are 874–12,242 mm, which are corresponding to the width-tospan ratios of 0.1–1.4. The element types and material properties used
in the finite element models of these two kinds of structures were kept
the same with those in previous sections. Simply supported boundary
conditions were adopted for all the models. The comparison is also
carried out under two loading conditions, which are a concentrated
loading at the mid-span and a uniformly distributed loading. Fig. 14
shows a finite element model for a CBGB-CSW.
The simulated λmax,1/2 values for all the numerical models of CBGBCSWs and CBGBs-CSWTs are shown in Fig. 15. Under both loading
conditions, the magnitude of the shear-lag effect increases with the
Fig. 8. Load-deflection curve at x = L/2 under a concentrated loading at midspan (LC 3).
influence on the shear-lag effect.
3.4. Comparison to composite box girder bridges with corrugated steel webs
The main difference between a CBGB-CSW and a CBGB-CSWT is that
in the former structure, a concrete bottom slab is adopted instead of two
bottom steel tubes connected with trusses. In this research, the characteristics of the shear-lag effect in these two kinds of structures are
compared based on finite element simulation.
In this research, an equivalent cross section of a CBGB-CSW is first
developed. To be more specific, given a cross section of a CBGB-CSWT,
the bottom steel tubes are replaced by a bottom concrete slab while the
flexural stiffness of the cross section remains the same. The cross section
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Y. Chen et al.
2.1
with concrete-filled bottom steel tube
with hollow bottom steel tube
100
79
1.2
0.1
0.2
0.3
0.4 0.5 0.6 0.7 0.8
Width-to-span ratio (b/L)
0.9
2.1
1.0
λmax,1/2
1.5
1.2
0.1
0.2
0.3
0.4 0.5 0.6 0.7 0.8
Width-to-span ratio (b/L)
0.9
4. Calculation of the shear-lag coefficient of a CBGB-CSWT
1.0
4.1. Governing equations
(b)
The following assumptions have to be made to analytically calculate
the shear-lag coefficient of a CBGB-CSWT.
Fig. 10. λmax,1/2-b/L curves of numerical models under: (a) a concentrated
loading at mid-span; (b) a uniform distributed loading.
1.2
(1) The flexural strain energy and the shearing strain energy of the top
concrete slab and the bottom steel tubes are considered, while the
transverse deformation and the out-of-plane deformation of the
concrete slab are ignored.
(2) The strain energy of the corrugated steel webs is not considered due
to the accordion effect.
(3) The box girder is at the elastic stage.
(4) The concrete-filled bottom steel tubes are modeled as a kind of
composite material, and the stress in a cross section of the tubes is
uniformly distributed.
Concentrated loading at midspan
Uniformly distributed loading
λmax,1/2
1.1
1.0
0.9
0.50
0.75
1.00
1.25
1.50
Suspension ratio (b1/b2)
1.75
In this research, the longitudinal displacement of the top concrete
slab is modeled using a cubic function:
2.00
u i (x , y , z ) = z
Fig. 11. λmax,1/2-b1/b2 curves of numerical models.
1.2
λmax,1/2
1.0
3
4
5
6
Width-to-depth ratio (b/h)
dw
+ 1
dx
y3
U (x )
bi3
(6)
where ui is the longitudinal displacement of the i-th part of the top
concrete slab of a typical cross section shown in Fig. 9; x, y and z are the
longitudinal, transverse and vertical coordinates, respectively; w = w
(x) is the vertical displacement; bi is the width of the i-th part of the top
concrete slab; and U(x) is the maximum difference of the shear rotation
angle. Eq. (6) has been adopted by other researchers in their studies on
the shear-lag effect and shown to provide accurate results (e.g.
[22,23]). The longitudinal displacement of the concrete-filled bottom
steel tubes, usc, is assumed as
Concentrated loading at midspan
Uniformly distributed loading
1.1
0.9
Equivalent bottom
concrete slab
79
width-to-span ratio for both kinds of structures. For the models with
smaller width-to-span ratios, the shear-lag effect in a CBGB-CSWT is
similar to or slightly stronger to that in a CBGB-CSW. On the contrary,
for the models with larger width-to-span ratios (i.e. b/L > 0.7 under a
concentrated loading at the mid-span and b/L > 0.9 under a uniformly
distributed loading), the shear-lag effect in a CBGB-CSW is stronger.
The difference between the magnitudes of the shear-lag effect of these
two kinds of structure seems increases with the width-to-span ratio.
Considering the difference between the observations in these two kinds
of structure, the equations to calculate the shear-lag coefficient of a
CBGB-CSW cannot be directly applied to a CBGB-CSWT.
with concrete-filled bottom steel tube
with hollow bottom steel tube
1.8
960
1118
Fig. 13. The equivalent section of corrugated steel web PC composite beam
(unit: mm).
(a)
0.9
560
Tendons
1.5
0.9
2080
960
95
λmax,1/2
1.8
560
usc (x , y , z ) = h2
7
dw
dx
(7)
where h2 is the distance from the center of the tube to the y axis (Fig. 9).
The strain energy for each part of a CBGB-CSWT. The strain energy
of the top concrete slab can be expressed by
Fig. 12. λmax,1/2-b/h curves of CBGB-CSWT.
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Engineering Structures 181 (2019) 617–628
Y. Chen et al.
Top concrete slab
Corrugated steel webs
Bottom concrete slab
Fig. 14. Finite element model of a CBGB-CSW.
3.0
2.5
λmax,1/2
1
V¯ss =
2
CBGB-CSWT
CBGB-CSW
1.0
0.4
0.6
0.8
1.0
Width-to-span ratio (b/L)
1.2
1.4
=
E I
+ sc2 sc
=
λmax,1/2
(EI)
+ 2c
1.0
1
EI
i 2 c i
0.2
0.4
0.6
0.8
1.0
Width-to-span ratio (b/L)
1.2
(EI)c
1.4
=
(
E I
dx + s2 d
2
w
EI
dx + 2
w dx
9
U 2dx +
14
3
w
4
U dx + (EI)c
M (x ) + EIw
(EI)c
9
(
9G
}
U + 14 U 2 + 5E bc 2 U 2 dx
c i
2
w dx
(EI )c
9I1 Gc
9I G
+ 102 b2c
10 b12
2
)
3
w
4
U dx
U 2dx
1
i 2
V¯si =
{w
2
3
w
2
2(Ec xi2 + Gc i2 ) ti dxdy
9
U + 14 U 22 +
9Gc
5Ec bi2
2Esc sc2 Adx+
)
2Gsc sc2 r 2dx =
9
+ 14 U
U dx +
(
9I1 Gc
9I G
+ 52 b2c
5 b12
2
w dx + (EI )c
(
)
3
U
4
(EI )c
3
w
4
9I1 Gc
9I G
+ 52 b2c
5 b12
2
w dx
) U Udx
9
+ 14 U
) U Udx
U
x2
x1
Let δΠ = 0 for any given values of δw” and δU, the following
equations can be obtained:
}
U 2 dx
(EI )sc
2
(
9
U
14
3
(EI )c U
4
3
w
4
w w dx
(12)
M (x ) + EIw
(8)
where Ec and Ac are the Young’s modulus and the cross-sectional area of
the top concrete slab, respectively. The strain energy of the concretefilled bottom steel tubes can be written as
1
V¯sc =
2
3
w
2
2
2
M (x ) w dx + EI
Fig. 15. λmax,1/2 of CBGB-CSWTs and CBGB-CSWs under: (a) a concentrated
loading at mid-span; (b) a uniformly distributed loading.
=
{w
=
(b)
i
(10)
(11)
1.5
V¯s =
w 2dx
where EI = EcI1 + EcI2 + EsId+ (EI)sc is the equivalent flexural stiffness
of the whole cross section; and (EI)c = 2EcI1 + 2EcI2 is the flexural
stiffness of the top concrete slab.
The derivative of Π can be written as:
2.0
0.5
M (x ) w
CBGB-CSWT
CBGB-CSW
2.5
Es Id
2
2
1
EI
i 2 c i
M (x ) w dx +
(a)
3.0
2Gs s2 r 2dx =
where EsId = EsAd (cosα) h2 is the inertial moment of the truss
members along the longitudinal direction of the girder; Ad is the crosssectional area of the truss members along the longitudinal direction of
the girder; and α is the angle between the truss members and the
longitudinal direction of the girder.
With the consideration of the strain energy of each part and the
potential energy of the external loading, the total potential energy of a
CBGB-CSWT, Π, can be expressed by
1.5
0.2
)
2Es x2 Adx+
3
2.0
0.5
(
w 2dx
(EI )c
(EI )c
(9)
3
w
4
3
(EI )c U = 0
4
9
U
14
3
9
w +
U
4
14
(13)
9I1 Gc
9I G
+ 2 c2 U = 0
5 b12
5 b2
+
U
x2
=0
(14)
(15)
x1
Eqs. (13) and (14) are the governing equation with two unknown
variables w and U, and Eq. (15) is one of the boundary conditions that w
and U satisfy. U can then be obtained by solving these equations:
where εsc and γsc are the normal and shear strains of the bottom steel
tubes, respectively. (EI)sc is the equivalent flexural stiffness of the
bottom steel tubes. When the bottom steel tubes are filled with concrete, (EI)sc = (EA)sc h22, while for hollow bottom steel tubes,
(EI)sc = (EsAs) h22. In these equations, (EA)sc = EsAs + mEcAc is the
equivalent axial stiffness of the concrete-filled steel tubes; m = 0.1
[24]; Ec is the Young’s modulus of concrete filled in the steel tubes; Es is
the Young’s modulus of steel; As is the cross-sectional area of the steel
tubes, respectively. The strain energy of the truss members along the
longitudinal direction of the girder is
(16)
U (x ) = m (C1 shkx + C2 chkx + U )
where k 2 =
1
G
G
EI I1 2c + I2 2c
5
b1
b2
;m =
1
1
EI (EI )c
(EI )2c
14
16
3
EI
7
1
2
; U* is a particular solution
3
(EI )c
8
of the governing differential equations related to the distribution of the
shear force; and C1 and C2 are two coefficients determined by the
boundary conditions. Then the longitudinal displacement of the top
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Engineering Structures 181 (2019) 617–628
Y. Chen et al.
concrete slab can be calculated using Eq. (6).
For the intersection point between the webs and the top concrete
slab at a typical cross section shown in Fig. 9,
u (x , y = b2) =
h1
dw
dx
1
M (x )
EI
(17)
3
(EI ) 0 U
4
(18)
Therefore, the normal stress at such an intersection point can be
calculated by
u (x , y )
=
x
x = Ec
Ec h1 w = Ec h1
1
M (x )
EI
3
(EI ) 0 U
4
On the other hand, the cross-sectional normal stress calculated
based on the elementary beam theory is
0 =
=
x
=
3
(EI )c U
4
M (x )
Ec h1 EI
0
=1
=1
3m
l
(EI )c thk
2 kl
2
(26)
1
(l
2
2x )
1
chkl 1
shkx +
chkx
k
k shkl
(27)
The bending moment at the mid-span cross section is ql /8, thus the
shear-lag coefficient at the mid-span cross section in this case is
3 (EI )c
U
4 M (x )
=1
3 (EI )c
U
4 M (x )
l
3 (EI )c P shk 2
l
m
shk
4 Pl/4 k shkl
2
2
Therefore, the shear-lag coefficient can be calculated by
1
3 (EI )c
U =1
4 M (x )
q
U=m 2
k
(20)
Ec h1 EI M (x )
l /2 = 1
When a simply supported CBGB-CSWT is subjected to a uniformly
distributed loading with a magnitude of q (Fig. 16(b)),
(19)
M (x )
Ec h1 w = Ec h1
EI
(25)
where l2 = l – l1 is the length of the CB segment. When the concentrated
loading is applied at the mid-span, l2 = l1 = l/2, and the bending moment at the mid-span is Pl/4. Then the corresponding shear-lag coefficient at the mid-span cross section can be obtained:
where h1 is the distance from the centerline of the top concrete slab to
the neutral axis of the cross section. According to Eq. (13),
w =
l
shkl1·cthkl·chkx + 1
l
P
U2 = m 2 shkl1·shkx
k
=1
(21)
q
3 (EI )c
m
4 ql2 / 8 k 2
=1
m
kl
6 2 2 (EI)c 1
(1
l
chk 2 +
l
chk 2 +
chkl
2chk
chkl 1
l
shk 2
shkl
)
1
l
2
(28)
4.2. Calculation of the shear-lag coefficient for simply supported girders
4.3. Experimental and numerical validation
For a simply supported CBGB-CSWT, the boundary conditions can
be written as
To validate the proposed analytical calculation method for the
shear-lag coefficient, the calculation results are compared to experimental and numerical results. The experimental and numerical results
of test beams are from the research presented in the previous sections.
Finite element simulation is also carried out for a real bridge to provide
results for the validation. The finite element model is developed according to the design of the Maluanshan Park Viaduct. Figs. 17 and 18
show the typical cross section and the finite element model of the
viaduct, respectively. The net length of the real bridge model is
44.92 m. The types of elements and the material properties used to
model the structure elements are the same with those for the finite
element model of the test specimens. Simply supported boundary
conditions are assigned to the real bridge model.
Fig. 19 compares the values of λmax obtained using the analytical
equation and from the experimental and numerical study. These two
kinds of results are shown with the horizontal and vertical axes, respectively. The investigated cross section is located at the mid-span.
Two loading cases are considered, which are a concentrated loading
applied at the mid-span and a uniformly distributed loading acting
along the whole girder, respectively. In the figures, the symbols representing the investigated cases are all located next to the straight line
with a slope of 1.0, indicating that in all these cases the two kinds of
results agree well with each other. The largest relative error in all these
investigated cases is smaller than 4%, thus the analytical equation is
able to be applied in practical engineering, which may help efficiently
estimate the shear-lag effect.
First consider the case with a concentrated loading P applied at the
location x = l1, which is shown in Fig. 16(a). Let UAC represent the U
values at the AC segment, and UCB represent that at the CB segment. At
the loading location C, the value of U should be unique, i.e.
(23)
UAC | x = l1 = UCB | x = l1
Based on these boundary conditions, the U values at the AC and CB
segments can be obtained:
P shk (l l1)
U1 = m 2
chkx
k
shkl
l1
A
l2
l
P
(24)
l2
B
C
x
l
z
(a)
x
q
x
10400
4800
500
1250
1200
2800
1200
1250
2800
z
200
l
300
2800
300
(22)
U1 |x= 0 = 0 U2 |x= l = 0
(b)
Φ720mmx20mm
Fig. 16. Simply supported CBGB-CSWTs under: (a) a concentrated loading; (b)
a uniformly distributed loading.
Fig. 17. Typical cross-section of a real bridge girder (unit: mm).
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Engineering Structures 181 (2019) 617–628
Y. Chen et al.
Concrete slab
Corrugated steel webs
Bottom steel tube
End diaphragm
Fig. 18. Finite element model of the real bridge.
λmax,1/2(Theory)
2.1
1.9
shear-lag effect, a small width-to-span ratio should be adopted. The
magnitude of the shear-lag effect of a CBGB-CSWT also increases
with the suspension ratio, while the width-to-depth ratio does not
show a great influence.
(4) The shear-lag coefficient in a simply supported CBGB-CSWT can be
accurately estimated by the proposed analytical equation in this
paper, which can help consider the shear-lag effect in engineering
practice.
SP1
Investigated real bridge
Specimens with different b/L
1.7
1.5
1.3
λmax,1/2(Theory)=λmax,1/2(FEM)
1.1
0.9
0.9
Acknowledgements
1.1
1.3
1.5
1.7
λmax,1/2(FEM)
1.9
This research is funded by the National Natural Science Foundation
of China (Project No. 51578323.), Postdoctoral Science Foundation of
China (Project No. 2018M633156), Guangdong Provincial Department
of Science and Technology (Project No. 2012-02-025) and Science and
Technology Innovation Committee of Shenzhen (Project No.
JCYJ2014090216222).
2.1
(a)
λmax,1/2(Theory)
2.1
1.9
SP1
Investigated real bridge
Specimens with different b/L
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1.3
λmax,1/2(Theory)=λmax,1/2(FEM)
1.1
0.9
0.9
1.1
1.3
1.5
1.7
λmax,1/2(FEM)
1.9
2.1
(b)
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5. Summary and conclusions
In this paper, experimental and numerical studies have been carried
out to investigate the shear-lag effect of composite box girders with
corrugated steel webs and trusses in detail. Analytical equations have
also been developed to estimate the shear-lag effect of this kind of
bridge in practical engineering. The main conclusions of this paper are
as follows.
(1) As a new type of bridge structure, composite box girders with
corrugated steel webs and trusses experience the shear-lag effect.
The maximum cross-sectional normal stress in the top concrete slab
occurs at the intersection with the corrugated steel webs.
(2) Hollow or concrete-filled bottom steel tubes can be adopted in a
CBGB-CSWT for different purposes. The filling of concrete inside
the bottom steel tubes does not greatly influence the shear-lag effect
of a CBGB-CSWT. The difference between the shear-lag coefficients
in the two cases is only 2% for the test specimens in this research.
(3) The width-to-span ratio is the most important influencing parameter of the shear-lag effect of a CBGB-CSWT. To avoid a severe
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