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Dynamic response of an under-deck cable-stayed Timber-Concrete Composite bridge under a moving load

Authors:

Abstract

Timber-Concrete Composite bridges have the potential to achieve significant levels of structural efficiency through the synergistic use of Engineering Wood Products (EWPs) and reinforced concrete. With the implementation of post-tensioned under-deck tendons, the range of application of TCC bridges can be extended to medium spans. However, little work has been done to date to study the dynamic response of these newly proposed bridges. In this paper, a set of FE models representing 60-m span structures are analysed to gain understanding on the dynamic response of post-tensioned under-deck TCC bridges. Two models with Euler and Timoshenko beam idealizations are considered in order to evaluate the significance of shear deformations on deflection, structural stresses and connector shear forces. Besides, an analytical model is formulated and compared against the numerical predictions. The results show that timber shear deformations should be considered in the design of post-tensioned under-deck TCC bridges. The dynamic characteristics of the bridge models were studied. The dynamic amplification caused by a moving point load on key response parameters such as deflection, stresses and connector shear forces is discussed. Also, a sensitivity study on the speed of moving load is conducted to investigate its influence on the bridge dynamic response.
Dynamic response of an under-deck cable-stayed Timber-Concrete Composite
bridge under a moving load
Zhan Lyu , Christian Málaga-Chuquitaype, and Ana M. Ruiz-Teran
ABSTRACT: Timber-Concrete Composite bridges have the potential to achieve significant levels of structural
efficiency through the synergistic use of Engineering Wood Products (EWPs) and reinforced concrete. With the
implementation of post-tensioned under-deck tendons, the range of application of TCC bridges can be extended to
medium spans. However, little work has been done to date to study the dynamic response of these newly proposed
bridges. In this paper, a set of FE models representing 60-m span structures are analysed to gain understanding on the
dynamic response of post-tensioned under-deck TCC bridges. Two models with Euler and Timoshenko beam
idealizations are considered in order to evaluate the significance of shear deformations on deflection, structural stresses
and connector shear forces. Besides, an analytical model is formulated and compared against the numerical predictions.
The results show that timber shear deformations should be considered in the design of post-tensioned under-deck TCC
bridges. The dynamic characteristics of the bridge models were studied. The dynamic amplification caused by a moving
point load on key response parameters such as deflection, stresses and connector shear forces is discussed. Also, a
sensitivity study on the speed of moving load is conducted to investigate its influence on the bridge dynamic response.
KEYWORDS: Timber-concrete composite, post-tensioning, dynamic response, dynamic amplification factors, moving
load
1 INTRODUCTION
123
The high strength, light weight and low carbon
characteristics of Engineered Wood Products (EWPs)
make them ideal for bridge construction [1]. These
structural materials have been employed in a wide range
of bridge typologies such as beam bridges, arch bridges,
truss bridges, suspension bridges and cable-stayed
bridges.
By combining EWPs with reinforced concrete, a new
system, known as timber-concrete composite (TCC)
system, is created. TCC structures are associated with
high material efficiency, low construction cost and long
service life [2]. However, the relatively large flexibility
of TCC bridges hampers their applicability for larger
spans. An effective and innovative solution is the
implementation of under-deck post-tensioned tendons
[3]. The static and dynamic response of concrete and
steel-concrete composite bridges using under-deck
tendon systems have been conducted in the last two
decades [4-6]. However, there is a dearth of studies on
the structural response of TCC bridges with under-deck
post-tensioned tendons, especially related to their
Zhan Lyu, Imperial College London, UK,
z.lyu15@imperial.ac.uk
Christian Málaga-Chuquitaype, Imperial College London, UK,
c.malaga@imperial.ac.uk
Ana M. Ruiz-Teran, Imperial College London, UK,
a.ruiz-teran@imperial.ac.uk
dynamic behaviour under dynamic action such as heavy
moving vehicles.
This paper presents a series of numerical studies carried
out in order to evaluate the dynamic responses of a post-
tensioned under-deck TCC bridge under moving point
loads. Focus is placed on simply-support bridge with a
span of 60 m. The Euler and Timoshenko beam models
implemented in Abaqus [7] are considered. Besides, an
analytical formulation is put forward for the estimation
of displacements and compared against the numerical
predictions. These comparisons reveal that timber shear
deformations have a significant impact in the response of
the newly proposed bridges. An eigenvalue analysis is
performed to determine the vibration properties of the
bridge models being considered. The results of a
sensitivity study on the speed of the moving point load
show that the dynamic amplification factors (DAFs)
significantly increase with increasing speeds in the range
of 45 m/s to 55 m/s, reaching a local maximum plateau
for speeds larger than 55 m/s.
2 BRIDGE CONFIGURATION
Figure 1 shows views of a post-tensioned under-deck
TCC bridge where l and l
p
refer to the span length and
the side sub-span, respectively, b to the section width
and h to the section height. The deck of the simply
supported TCC bridge is made up of one concrete slab
and two timber beams. 16 mm studs are used for
connecting the concrete slab to the timber beams. Four
struts, which are pinned to the deck, are used at sections
A and B to transfer the deviation force to the main
beams, dividing the total span into three sub-spans. The
under-deck tendons are anchored to the edge of the
beams at the diaphragms.
In this study, a span of 60 m is chosen with the side sub-
span length equal to quarter of the total span, while the
section width and height are 14 m and 2.4 m,
respectively. The thickness of the concrete slab varies
from 0.3 m at the middle of the cross section and at the
two edges, increasing up to 0.5 m over the timber beams.
Each timber beam has a width of 2 m and a depth of 1.9
m. The spacing of the connectors is 0.1 m in the area
close to the support section over the abutments and 0.2
m elsewhere. The under-deck tendons are made of 112
pre-stressed strands consisting of 7 wires with an area of
140 mm
2
each. An eccentricity of l/10 was adopted for
the tendons following the recommendations by Ruiz-
Teran and Aparicio [8]. General and specific aspects of
the design of post-tensioned under-deck bridges and
their structural advantages can be found in Lyu [3].
z (
eccentricity
)
l
p
diaphragm
centroid
l
timber
Section A
concrete
post-tensioned tendons
l
p
Section B
diaphragm
(a) Elevation view
struts
w
t
h
c1
h
t
h
c2
timber beam
h
tendons
w
t
b
concrete slab
(b) Mid-span section
connector
concrete deck
timber beam
(c)
Connection system
Figure 1: Post-tensioned under-deck TCC bridge
3 BRIDGE MODELS
3.1 ANALYTICAL MODEL
Post-tensioned under-deck TCC bridges can be idealized
as shown in Figure 2. In this analytical model, two
springs are used to account for the effect of the under-
deck cable system. The equivalent stiffness of the
springs is calculated by means of the Flexibility Method.
This analytical model neglects the contribution of the
stay cables to the axial compression in the deck, which
would be relevant in the response only if the geometrical
nonlinearities are important. Previous research has
shown that these nonlinearities can be neglected when
analysing the response under live loads [9].
Section A
K K
Section B
Figure 2: Analytical model for post-tensioned under-deck TCC
bridges
Figure 3 depicts the typical deflected shape of post-
tensioned under-deck TCC bridges under symmetrical
loads such as a concentrated load at mid-span or uniform
load. The vertical deflection at the top tip of the strut,
δ
v
induces the vertical component of the deviation force in
the under-deck cable system, F
v
, as follows:
vv
KF
δ
=
(1)
where K is the equivalent stiffness of the springs shown
in Figure 2. The horizontal and vertical deflections at the
bottom tip of the strut,
x
and
y
, respectively, are also
indicated in Figure 3.
z
l
Deviation force
D
y
F
v
initial shape
α
T
i
l
p
x
α
T
h
δ
v
deflected shape
Figure 3: Typical deflection shape of post-tensioned under-
deck TCC bridges under symmetrical load
By neglecting the elongation of the struts and the
horizontal deformation of the deck, it can be shown that:
22
zll
z
D
z
pp
yv
x
==
δ
(2)
where z is the eccentricity of the post-tensioned tendons
at mid-span and D is the horizontal projection of the
strut. The axial forces of the horizontal and inclined stay
cables, T
h
and T
i
, and their relationships are given in
equations (3)-(5).
22
2zll
EA
EAT
p
x
hh
==
ε
(3)
(
)
p
xy
ii
l
EA
EAT )2cos()2sin(
α
α
ε
==
(4)
TTT
ih
=
=
(5)
where E represent Young’s modulus of steel, A is the
cross section area of the stay cables, Ɛ
h
the axial strains
of the horizontal cable, Ɛ
i
the axial strains of the inclined
stay cable, and
α
the angle between the cables and the
strut. The relationship between the vertical component of
the deviation force and the axial force of the stay cables
is:
α
α
sincos2
=
TF
v
(6)
Combining equations (1)-(6), the equivalent stiffness of
the equivalent vertical springs (K in Figure 2) can be
obtained as
( )
3222
2
2
ppp
llzll
EAz
K+
=
(7)
The calculation of the timber-concrete composite beam
follows the linear-elastic method presented in EN 1995-1
[10], also known as the gamma method. It should be
noted that the analytical formulation described above
does not include shear deformations at this stage. A more
comprehensive analytical model able to incorporate
shear deflections is currently being developed. Also, this
analytical model is available only for the static solution
under symmetrical loads. The result of the analytical
model in this paper will be compared with an Abaqus
model in the following sections of this paper.
3.2 ABAQUS MODELS
Two models, Model A and Model B, have been
developed in Abaqus to investigate the structural
response of the post-tensioned under-deck TCC bridges.
Table 1 summarizes the element types employed. These
include shell elements, Euler and Timoshenko beam
elements, truss elements and spring elements. The
stiffnesses of the springs, K
p
, representing the stud
timber-concrete connectors are estimated from:
23
2
5.1
d
K
p
ρ
=
(8)
where
ρ
and d refer to the density and diameter of each
stud, respectively. K
p
is expressed in [N/mm] with
ρ
expressed in [kg/m
3
].
A Timoshenko beam idealization is employed in Model
B to account for timber’s shear deformation, while
Model A employs a Bernoulli beam approximation. The
comparison between models A and B allows the
evaluation of the influence of timber’s shear deformation
on key structural response parameters.
Table 1: Element types for components of numerical models
Components Model A Model B
Concrete slab shell shell
Timber beams Euler beam Timoshenko beam
Struts Euler beam Euler beam
Tendons truss truss
Connectors spring spring
3.3 MODEL VALIDATION AND COMPARISON
Figure 4 presents the comparison of response estimations
obtained from the three alternative models described
above when subjected to a concentrated load at mid-
span. The deflection at mid-span,
δ
, concrete
compression stress at the mid-span section,
σ
cc
, timber
tensile stress at the mid-span section,
σ
tt
, timber shear
stress at Section A/B,
σ
ts
, tendon stress,
σ
T
, and
maximum connector shear force, Q, were compared in
this study. The graph in Figure 4 presents the estimations
from models A and B normalized by the corresponding
analytical predictions, where a value of 1 (red dash line)
corresponds to a full equivalence between numerical and
analytical estimations. The red numbers on top of the
bars are the differences between results obtained from
Model A and Model B, and reflect the influence of
accounting for timber’s shear deformations.
Figure 4 shows that the predictions of Model A are very
close to those of the analytical model, with typical
differences of less than 2.0%. Therefore, the simpler
analytical formulation is well qualified to simulate the
structural performance of post-tensioned under-deck
TCC bridges where shear deformations are not expected
to be significant. However, for the bridge aspect ratio
being considered herein (60 m corresponding to h/l =
1/25), shear deformations significantly affect connector
shear force (10.2%), the vertical deflection (8.1%),
timber shear stress (6.8%) and tendon stress (5.4%), in
that order, and may need to be adequately considered.
The following sections focus on Model B estimations
which account for the shear related effects.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
Q
σ
T
σ
ts
σ
tt
σ
cc
10.2%
5.4%
6.8%
3.2%
2.7%
Model B / Analytical
Model A / Analytical
8.1%
δ
Figure 4: Model validation and comparison
4 DYNAMIC RESPONSE
4.1 MODAL ANALYSES
An eigenvalue analysis was performed employing
Abaqus Model B. Table 2 reports the first 10 vibration
modes of the post-tensioned under-deck TCC bridge. As
expected, the national mode is the first vertical mode
(V1), with a frequency of 1.42 Hz. The first 10 modes
include vertical (V), torsional (To), transversal (Tr) and
lateral (L) modes (shown in Figure 5).
(a) Vertical mode (mode1: 1.42 Hz)
(b) Torsional mode (mode2: 2.33 Hz)
(c) Transversal mode (mode5: 6.22 Hz)
(d) Lateral mode (mode6: 6.54 Hz)
Figure 5: Examples of four typical vibration modes
Table 2: Details of the first 10 vibration modes
Mode Frequency (Hz) Name Description
1 1.42 V1 1st vertical mode
2 2.33 To1 1st torsional mode
3 4.04 V2 2nd vertical mode
4 5.23 To2 2nd torsional mode
5 6.22 Tr1 1st transversal mode
6 6.54 L1 1st lateral mode
7 7.08 V3 3th vertical mode
8 7.86 Tr2 2nd transversal mode
9 9.04 To3 3rd torsional mode
10 9.29 V3+Tr1 3rd vertical mode +
1st transversal mode
4.2 CONCENTRIC LOAD CASE
A concentric load of 400 kN and speed of 60 m/s, was
applied to Model B moving from one side of the bridge
to the other in order to evaluate its dynamic response.
Figure 6 shows the frequency-domain amplitude for the
acceleration of the concrete slab at the mid-point of the
mid-span section as a function of the frequency of the
response. The first three vertical (V) and Transversal
(Tr) modes contribute the most to the acceleration, with
the first vertical mode contributing the most and its
amplitude after Fourier Transform reaching
approximately 0.3 m/s. As expected, torsional (To)
modes are negligible for this concentric load case.
0 2 4 6 8 10 12
0.0
0.1
0.2
0.3
L1
Tr3
V3+Tr1
To3
Tr2
V3
Tr1
To2
V2
To1
Amplitude (m/s)
Frequency (Hz)
F
v = 60 m/s
400 kN
V1
Figure 6: Frequency-domain response for the acceleration at
the mid-point of the mid-span section
Figure 7 to Figure 12 depict the modal contribution to
deflection, stresses and connector shear forces. The
vertical axis of each figure represents the corresponding
response amplitude normalized by the maximum
amplitude over the whole frequency range. Therefore,
the mode contributing the most to each response has an
ordinate of 1.0.
0 2 4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
1.2
V3+Tr1
To3
L1
Tr1
To2
V2
To1 V3
R / R
max
Frequency (Hz)
V1
Figure 7: Frequency content for vertical deflection at mid-
span section over timber beams
0 2 4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
1.2
V3+Tr1
L1
Tr1
V2
To1 V3
R / R
max
Frequency (Hz)
V1
Figure 8: Frequency content for concrete compression stress
at mid-span section
0 2 4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
1.2
L1
Tr1
V2
To1 V3+Tr1
V3
R / R
max
Frequency (Hz)
V1
Figure 9: Frequency content for timber tensile stress at mid-
span section
0 2 4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Tr1 L1
To1 V3+Tr1
V3
V2
R / R
max
Frequency (Hz)
V1
Figure 10: Frequency content for timber shear stress at
Section A/B
0 2 4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
1.2
V3+Tr1
L1
Tr1
To1
V3
V2
R / R
max
Frequency (Hz)
V1
Figure 11: Frequency content for tendon stress
0 2 4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Tr1 L1
To1
V3+Tr1
V3
V2
R / R
max
Frequency (Hz)
V1
Figure 12: Frequency content for maximum connector shear
force
It is clear from these figures that the first vertical mode
(V1) dominates the vertical deflection at mid-span of the
timber beams, as well as the concrete compression,
timber tensile stresses at mid-span and tendon stresses.
With regards to timber shear stress at section A/B and
the maximum connector shear force, the second vertical
mode (V2) was found to contribute the most.
4.3 SENSITIVITY STUDY ON THE SPEED OF
MOVING LOAD
A series of moving loads with speeds varying from 5 m/s
to 60 m/s were applied to Model B to investigate the
influence of speed on the dynamic amplification factor
(DAF) of key response parameters. The parameters
examined include vertical deflection at mid-span section,
concrete compression stress at mid-span section, timber
tensile stress at mid-span section, timber shear stress at
section A/B, tendon stress and maximum connector
shear force.
Figure 13 reports the DAFs obtained as a function of
different load speeds. It can be appreciated from Figure
13 that the maximum connector shear force takes the
largest DAF values, reaching 1.76 when the speed of
moving load is 60 m/s. All the DAFs, except the one for
maximum connector shear force, stay at relatively low
levers (less than 1.25) for load speeds of less than 45
m/s. Afterwards, the values of DAFs significantly
increase for load speeds in the range of 45 m/s to 55 m/s.
Most of the DAFs reach a local maximum plateau at
speeds larger than 55 m/s.
0 10 20 30 40 50 60 70
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Dynamic amplification factor (DAF)
Speed (m/s)
Deflection
Concrete compression stress
Timber tensile stress
Timber shear stress
Tendon stress
Connector shear force
Figure 13: Relationship between DAF and speed of moving
load
5 CONCLUTION
The dynamic behaviour of a post-tensioned under-deck
TCC bridge under the action of a moving load has been
studied in this paper. Two FE models and an analytical
formulation have been developed. It was demonstrated
that for the bridge under considerations (60 m span), the
consideration of shear deformations in the timber beams
can lead to up to 10.2% greater deformations and
stresses. Therefore, it is recommended that appropriate
consideration of shear related effects be made for post-
tensioned under-deck TCC bridges of similar aspect
ratios. Modal analyses have been conducted and load
speed sensitivity studies have been carried out on the
most advanced FE model. The following conclusions can
be offered from the analyses presented:
(1) Timber shear deformations have an important
influence on the response of the post-tensioned
under-deck TCC bridge analyzed (slenderness of
1/25). These shear deformations affect significantly
the connector shear force (increasing it by 10.2%),
the vertical deflection (8.1%), timber shear stress
(6.8%) and tendon stress (5.4%).
(2) When under a concentric moving load, the dynamic
mid-span deflection, concrete compression stress,
timber tensile stress and tendon stress are dominated
by the first mode of vibration (1.42 Hz), whereas the
second vertical mode becomes more important for
timber shear stress and maximum connector shear
forces.
(3) Peak dynamic amplification factors (DAFs) for
maximum connector shear force, tendon stress and
deflections are more than 1.5, in the order of 1.76,
1.58, and 1.52, respectively.
(4) The DAFs significantly increase when the moving
load reached speeds in the range of 45 m/s to 55 m/s
due to resonant effects, reaching a local maximum
plateau at speeds between 55 m/s and 60 m/s.
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In the past quarter century, a number of bridges have been built that do not fit into the conventional types of cable-stayed bridges. These are under-deck cable-stayed bridges and combined cable-stayed bridges. In this paper we define the first of these two types and set out its mechanisms of response. We then establish and analyze the parameters that determine the permanent response and the response to live load of these bridges. Lastly, we draw conclusions relating to their behaviour and define design criteria for them with the aim of making cable-staying systems highly efficient and allowing the design of much lighter and slimmer structures.Key words: unconventional cable-stayed bridges, under-deck cable-stayed bridges, combined cable-stayed bridges.
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This paper deals with the application of under-deck cable-staying systems and combined cable-staying systems to prestressed concrete road bridges with multiple spans of medium length. Schemes using under-deck cable-staying systems are not suitable for continuous bridges, as they are not efficient under traffic live load and only allow for the compensation of permanent load. However, combined cable-staying systems are very efficient for continuous bridges and enable the design of very slender decks (1/100th of span) where the amount of materials used is halved in comparison with conventional schemes without stay cables. In this paper, the substantial advantages provided by combined cable-staying systems for continuous bridges (such as high structural efficiency, varied construction possibilities, both economic and aesthetical benefits, and landscape integration) are set out. Finally, design criteria are included.
Vertically laminated timber bridges
  • G Freedman
G. Freedman. Vertically laminated timber bridges. Proceedings of the Institution of Civil Engineers -Construction Materials, 162: 4, 181-190, 2009.
Stress-Laminated Timber Bridge Decks: Non-linear Effects in Ultimate and Serviceability Limit States
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  • P Jacobsson
  • R Kliger
M. Ekevad, P. Jacobsson, and R. Kliger. Stress-Laminated Timber Bridge Decks: Non-linear Effects in Ultimate and Serviceability Limit States. Proceedings of International Conference on Timber Bridges 2013-Las Vegas, Nevada USA, 2013.
Eurocode 5: Design of timber structures -Part 1-1: General, common rules and rules for buildings
European Committee for Standardization. Eurocode 5: Design of timber structures -Part 1-1: General, common rules and rules for buildings. CEN, Brussels, 1995.