dynamic analysis of the multip le-arch bowstring …c).pdfdynamic analysis of the multiple-arch...
TRANSCRIPT
Steel Structures 6 (2006) 227-236 www.kssc.or.kr
Dynamic Analysis of the Multiple-Arch Bowstring Bridge and
Conventional Arch Subjected to Moving Loads
Min-Sik Kong1, Sung-Soon Yhim2,*, Suk-Ho Son3 and Dong-Yong Kim4
1Department of Civil Engineering, University of Seoul, Seoul 130-743, Korea2Department of Civil Engineering, University of Seoul, Seoul 130-743, Korea
3Dongsung Engineering, Ltd., 15-2, Sukchon-Dong, Songpa-gu, Seoul 138-842, Korea4Dongsung Engineering, Ltd., 15-2, Sukchon-Dong, Songpa-gu, Seoul 138-842, Korea
Abstract
Multiple-arch bowstring bridge (MABB) is a structural type of arch in which arch ribs and stiffened girders are connectedwith two internal arches. In this study, the static and dynamic behavior of MABB was analyzed in comparison with those ofconventional arches for the investigation of the structural effect of MABB on moving loads. For the purpose of surveying theeffect of internal arches on the dynamic behavior of structure, natural frequency and natural vibration mode were investigatedand the static and dynamic behavior were analyzed by the method of idealizing train loads as travelling loads consisting of agroup of concentrated loads. From the results, the following conclusions are known. First, it is concluded that in MABB,decreasing the section of stiffened girders is possible as compared with conventional arches because the increase of stiffnessby internal arches is larger than that of mass by internal arches. Second, it is concluded that MABB have the advantage of betterstability of dynamic behavior because the dynamic behavior of the MABB on moving loads is usually investigated in a morestable way than that of conventional arches
Keywords: MABB, Internal Arch, Moving Loads, Natural Vibration Mode, Natural Frequency, Dynamic Behavior
1. Introduction
Arch bridge is a structure that distributes load applied
from the girder to arch ribs through hangers, and transfers
such loads to the supports. Arch bridges come in various
structural types - Tied Arch, Langer Arch, Rose Arch,
Nielsen Arch, etc. - depending on how girders are connected
with arch ribs. Tied-arch bridge allows for horizontal
displacement on the supports by the tied bar. Langer-arch
bridge is a type that allows only axial forces to occur to
arch rib. Rose-arch bridge is a type that allows axial
forces and moments to occur to arch rib. Nielsen arch
bridge is a structure type in which a cable hanger is
arranged out of perpendicular. This study presents the
multiple-arch bowstring bridge (MABB), which connects
arch ribs and stiffened girders with two internal arches in
contrast to above-mentioned arch types. Also, this study
investigates the structural effect of MABB by analyzing
structural behavior in comparison with conventional arch.
Generally, arch bridges are constructed as middle-span
bridges due to their structural characteristics, and those
bridges that exceed the middle span have the
disadvantage of vibration induced by live loads such as
seismic load, wind load, and vehicle moving load.
Especially, because the railroad bridge bears a large
vehicle load compared with its own weight, the dynamic
stability is influenced by vehicle loads and traveling
property. Traveling stability and ride comfort are seriously
affected by the train and structure interaction. Sufficient
investigations on the matter are executed from the
perspective of design. Developed countries in the field of
railroad restrict natural frequency, vertical acceleration of
the upper flange, dynamic displacement, vertical acceleration
of vehicle, and other factors to ensure stability and
efficiency.
Therefore, this study investigates natural frequencies
and natural vibration modes of MABB and conventional
arch. The static behavior of MABB and conventional
arch is analyzed by the method of substituting train loads
for traveling loads composed of a group of concentrated
loads. Also, the dynamic behavior is investigated according
to traveling velocities.
2. Analysis Model and Load Case
For the purpose of comparison on the behavior of
MABB and conventional arch (C. Arch), other members
except internal arch are modeled on conditions similar to
what were used in this study. The section property is
presented in Table 1 and the geometric conditions are
*Corresponding authorTel: +82-2-2210-2953E-mail: [email protected]
228 Min-Sik Kong et al.
shown in Fig. 1. In the analysis model, a span is 72 m
long and hangers are arranged every 6m on the stiffened
girder. The span–rise ratio is about 1/6. Also, the train
load of the analysis model is EL-18 shown in Fig. 2 and
the load is composed of four passenger cars considering
span and analysis time.
3. Static Analysis
Considering the geometric condition of the internal
arch that connects stiffened girder and hangers as Fig. 1b,
static behavior is analyzed in the two load cases. In Case
1, the train was loaded in a half span, whereas in case 2,
the train was loaded in full span.
Especially, this study investigates the vertical displacement
and moment of stiffened girder, moment, and axial force
of the arch rib considering the fact that structural type of
analysis model is an arch bridge.
3.1. Displacement
According to load cases, the maximum vertical
displacements and deformed shapes are shown in Table 2
and Fig. 4, respectively. In MABB, vertical displacement
Table 1. Parameters for the MABB and conventional arch
ElementModulus of
elasticity(kN/m2)
Crosssectionalarea (m2)
Moment of inertia(m4)
Girder
20.58E+07
0.123 0.022
Arch Rib 0.12 0.019
Internal Arch 0.035 0.792E-04
Hanger 0.021 0.213E-03
Figure 1. Geometry for the analysis model.
Figure 2. EL-18.
Figure 3. Load configuration of EL-18.
Table 2. Vertical displacement of MABB and c. arch
LoadThe maximum vertical displacement [m]
MABB C. Arch Ratio[%]
A half span –0.0112 –0.0483 76.812
Full span –0.0156 –0.0244 36.066
Where, R = | (MABB-C. Arch)/C. Arch × 100 |
Figure 4. Deformed shapes of MABB and C. Arch byEL-18.
Dynamic Analysis of the Multiple-Arch Bowstring Bridge and Conventional Arch Subjected to Moving Loads 229
of the case 1 occurred less than that in case 2. On the
contrary, the conventional arch has larger vertical
displacements in case 1. Deformed shapes of case 1 are
shown in Fig. 4a and Fig. 4b and deformed shapes of case
2 are shown in Fig. 4c and 4d. Because MABB connects
stiffened girder and hangers through internal arches, the
upward vertical displacement of the stiffened girder did
not occur in MABB as Fig. 4b. In the middle of the
stiffened girder, the vertical displacement of MABB
occurred less than that with conventional arch as shown
in Fig. 4d. Because of internal arches, the deformed shape
of MABB is flat in the middle of stiffened girder.
3.2. Member forces
According to the load cases as seen in Fig. 3, axial
forces of the arch rib of MABB and the conventional arch
are shown in Fig. 5, and moments of the stiffened girder
and arch rib are shown in Fig. 6. Table 3 shows the
maximum member forces. The maximum member forces
of MABB are less than those of the conventional arch.
Axial force occurred less about 4.53% and, moments of
arch rib and stiffened girder occurred less about 85.50%,
67.79%, respectively. Moments of stiffened girder and
arch rib of case 1 were less than those of case 2. There
was, however, a slight increase in the axial force of the
arch rib in case 1.
Because the internal arch partially converts moments of
stiffened girder and arch rib into axial force of the arch
rib, it is concluded that displacements of stiffened girder
and arch rib decrease and axial force of arch rib slightly
increases as in the case of EL-18 applied to a half span.
The difference of deformed shapes as seen in Figs. 4a and
4b occurred for the same reason.
4. Dynamic Analysis
4.1. Free vibration
The free vibration is the vibration characteristic that a
structure has in the absence of externally applied forces.
This means the minimum or smallest value of potential
energy. The vibration characteristic indicates low-level
points of potential energy and vibration shape at these
points. This problem is called eigenvalue problem, and
low-level points and vibration shape is defined as
eigenvalue and eigenvector mathematically.
Natural frequency and natural vibration mode of a
structure are determined by the analysis of the eigenvalue
problem. Natural frequency presents the basis that can
estimate the possibility of resonance according to
frequency of excitation and is the representative value that
Figure 5. The maximum axial force of arch rib by EL-18.
Figure 6. The maximum moment of arch rib and stiffenedgirder by EL-18.
Table 3. The maximum member forces by EL-18
Content
Max. axial force
Max. moment
Arch rib Arch ribStiffened
girder
Case 1
MABB 1594.77 164.366 526.11
C. Arch 1300.12 1157.27 1633.26
Ratio [%] 22.66 85.80 67.79
Case 2
MABB 2206.25 187.263 467.00
C. Arch 2311.02 395.90 651.07
Ratio [%] 4.53 52.70 28.27
Where, R = | (MABB-C. Arch)/C. Arch × 100 |
230 Min-Sik Kong et al.
can estimate the structural stability. Because the natural
vibration mode is the vibration shape of structure, this is an
important physical value that can estimate the tendency of
behavior change of a structure. Also, the natural vibration
mode presents the possibility of displacement occurrence.
The first natural frequency and vibration mode are called
basic natural frequency and basic natural vibration mode. If
external force is not applied in a direction that is
completely opposite of the basic natural vibration mode,
structures are deformed according to the direction of basic
natural vibration mode. Namely, basic natural vibration
refers to the minimum energy of structure. If acquired
dynamic response is almost the same with the basic natural
vibration mode, resonance occurs. When structure has
damping and vibration occurred in the mixed state of
natural vibration modes, only the dynamic magnification
factor increases and resonance almost never happens.
When only the dynamic response similar to natural
vibration mode occurs, there is also an increase of
amplitude in resonance. So, this study carries out free
vibration analysis for the purpose of investigating the
change of natural frequency and vibration mode of MABB
due to internal arches.
In the result, natural vibration modes are shown in Fig.
7, with natural frequencies of MABB appearing to be
generally larger than those of the conventional arch. The
first natural vibration mode of MABB is symmetric but
that of conventional arch is asymmetric. With the natural
vibration modes of MABB, vibration mode shapes of
internal arch are generally asymmetric and were almost
similar to the first mode shape of conventional arch.
From the fact that the natural frequency of MABB is
larger than that of conventional arch, it is concluded that
the increase of stiffness is more than the increase of mass
due to internal arches.
4.2. Dynamic behavior
Moving load is one of the dynamic loads subjected to
bridges. Applying moving load requires that the location
of load varies with time, so that velocity is dominant.
Analytical model of a vehicle can be composed of
moving load, moving mass, and multi-degree of freedom
spring-mass system. Friction, road roughness, impact,
and other factors can be added to the analytical model for
the purpose of describing the interaction with bridge
vibration. But this study applies the following method to
analyze dynamic behavior of bridge prior to vehicle
vibration.
Train load is idealized traveling load composed of a
group of concentrated loads. This traveling load is
subsequently applied to bridges with uniform velocity.
Numerical analysis method is classified into direct
method and indirect method. The former is to integrate
directly the dynamic equilibrium equations and the latter
is to calculate the solution of each independent equation
transformed from the dynamic equilibrium equations by
the modal matrix. In the field of direct method, various
analytical methods are presented, and in which the
Newmark method and Wilson method are used most
frequently. The reason is that these methods always show
numerically stable solution for the special coefficient
value. The indirect method, called mode superposition
method, is the only method that acquires each independent
equation through the mode, eigenvector. Therefore, for
the application of this method, the calculation of eigenvalue
must be preceded. Because the solution of the equation is
acquired from a correct solution determined already, this
method has the advantage of reduction of operation time.
But this mode superposition method only can be applied
to a linear system due to the use of the superposition
method.
This study used the Newmark method, which can be
applied to both linear and nonlinear systems, and traveling
load was loaded as the velocity varied from 60 km/h to
300 km/h. Considering the first natural vibration mode
and the location of maximum displacement that occurred
by its own weight, dynamic responses to vertical displacement,
and acceleration were analyzed at the middle and the
quarter span. Also, both the moment of the arch rib and
stiffened girder and axial force of arch rib were analyzed.
Figure 7. Natural frequencies and natural vibration modes of MABB and conventional arch.
Dynamic Analysis of the Multiple-Arch Bowstring Bridge and Conventional Arch Subjected to Moving Loads 231
Figure 8. Displacement at the middle of span.
Figure 9. Acceleration at the middle of span.
232 Min-Sik Kong et al.
Figure 10. Displacement at a quarter of span.
Figure 11. Acceleration at a quarter of span.
Dynamic Analysis of the Multiple-Arch Bowstring Bridge and Conventional Arch Subjected to Moving Loads 233
Figure 12. Axial force for arch rib.
Figure 13. Moment for arch rib.
234 Min-Sik Kong et al.
4.2.1. Vertical displacement at the middle and the
quarter span
As a result of the dynamic response of vertical
displacement at the middle span, vertical displacement of
90 km/h and 140 km/h occurred largely as shown in Fig.
8. The vertical acceleration of the middle span is shown
in Fig. 9. On the basis of Fig. 8 and Fig. 9, it is concluded
that the vibration of conventional arch is larger than that
of MABB in dynamic response of vertical displacement
at the middle span.
In contrast with conventional bridges, the first vibration
mode of arch bridges is asymmetric as Fig. 7b. According
to this characteristic of arches, this study analyzes the
vertical displacement and acceleration at the quarter span.
As a result of the analysis, vertical displacement of 80km/
h and 110km/h occurred largely as shown in Fig. 11, and
the vertical acceleration is shown in Fig. 12. From Fig.
10, it is known that the vibration of conventional arch is
generally larger than that of MABB. So for the same
reason that the difference of displacement shapes
appeared as shown in Fig. 4a and Fig. 4b, it is concluded
that dynamic responses to the vertical displacement at a
quarter span appeared differently.
4.2.2. Axial force and moment of arch rib
The result of the dynamic response to axial force and
moment of arch rib is presented in Figs. 12 and 13. As a
whole, the dynamic response to the axial force of the arch
rib of the MABB is similar to that found in conventional
arch in contrast with vertical displacement. It can be
known that vibrations of conventional arch appear larger,
and the length of negative moment of MABB appears
shorter than that of the conventional arch. It is concluded
that the relatively long occurrence of length of negative
moment reflects the arch’s characteristics, which is that
the sign of end moment of arch rib happens reversely in
case of loads subjected to a half span as shown in Fig. 6b.
Namely, because the internal arches convert partially the
moment of arch ribs into the axial force of arch rib as
shown in Fig. 6a, it is concluded that MABB has a shorter
length of negative moment than the conventional arch.
4.3. Dynamic magnification factor
Dynamic magnification factor (DMF) is the value that
is used to estimate how large the maximum dynamic
response is with regards to the maximum static response
when a moving load is subjected. Namely, DMF is
defined as the ratio of the maximum static response to the
maximum dynamic response. When the moving loads are
loaded according to each specified velocities, DMF is the
ratio of the maximum value from dynamic analysis to the
maximum value from static analysis for a specified behavior,
Figure 14. DMF for conventional arch and MABB.
Dynamic Analysis of the Multiple-Arch Bowstring Bridge and Conventional Arch Subjected to Moving Loads 235
for example vertical displacement, member forces, reactions
for the design of shoes, etc. So, DMF is a factor in the
analytical result obtained on the assumption that dynamic
load is replaced as equivalent static load to correspond
with the maximum value. As earlier mentioned, this study
analyzed the DMF of vertical displacements at the middle
and the quarter span and of axial force and moment of
arch rib as shown in Fig. 14.
As shown in Fig. 14, the DMF of MABB is smaller
than those of the conventional arch in case of vertical
displacement and moment of arch rib. But on the
contrary, the DMF of axial force of arch rib is larger in
the MABB than in the conventional arch. It is estimated
that the result showing the DMF of MABB to be larger
with regards to the axial force of arch rib is induced by
the fact that the internal arch rib converts moment of
stiffened girder and arch rib into axial force of the arch
rib.
Looking into the DMF of vertical displacement and of
the moment of the arch rib, the DMF variation of MABB
appeared to be smaller than that of conventional arch. It
is shown that the DMF of vertical displacement almost
never changes when the velocity is below 250 km/h.
From this fact, it is concluded that the internal arch rib
that connects the stiffened girder and arch rib decreases
the dynamic variation of the stiffened girder.
5. Conclusion
This study analyzed the static and dynamic behavior of
conventional arch and MABB, which connects the
stiffened girder and arch rib for moving loads. Also, free
vibration analysis is carried out to compare natural
frequencies and vibration modes that dominate the dynamic
response of a structure. The free vibration analysis is an
important analytical method physically and because this
is an eigenvalue problem mathematically and is the only
method that can be used to estimate the tendency of
structural deformation. Moving loads are described as
moving concentrated loads, moving masses, or moving
vehicle loads. But this study focused on the analysis of
dynamic behavior of bridges rather than on vehicular
vibration. So, train load was idealized traveling load
composed of a group of concentrated loads. Traveling
load was subsequently applied to bridges with uniform
velocity. The findings of this study are as follows.
1) The basic natural vibration mode of MABB is
symmetric but that of the conventional arch is
asymmetric. Also, natural frequencies of MABB are
larger than those of the conventional arch. Therefore,
based on frequency analysis, it is concluded that the
stiffness increase is larger than the mass increase of a
structure.
2) When the train load is applied to a half span, vertical
displacement of the middle span decreases about 76.8%
than that of the conventional arch. When the train load is
applied to full span, a vertical displacement of middle
span decreases by about 36.0% than that of the
conventional arch. The maximum vertical displacement
of MABB occurred in case of the train load applied to full
span, whereas with the conventional arch, displacement
occurred when train load is applied to a half span.
In case of train load applied to full span, the maximum
moment of arch rib and the maximum axial force of arch
rib decreases about 85.80% and 4.53%, respectively. In
case of train load applied to a half span, the maximum
moment of arch rib decreases about 52.70%, but the
maximum axial force of the arch rib increases about
22.66% than the conventional arch. But the maximum
axial force of the arch rib occurs when the train load is
applied to full span. It is estimated that the increase of
axial force of the arch rib is induced by the fact that the
internal arch rib converts moment of stiffened girder and
arch rib into the axial force of arch rib.
3) As a result of analyzing the dynamic behavior
according to each velocity, the DMF of vertical
displacement of stiffened girder, vertical acceleration and
moment of arch rib appeared smaller in the MABB than
in the conventional arch. But the DMF of the axial force
of the arch rib is slightly larger than that of the
conventional arch. From this result, the following
conclusions are made. Because the internal arch converts
the moment of stiffened girder and arch rib into the axial
force of the arch rib, the dynamic vibration variation of
stiffened girder decreases considerably and the dynamic
variation of axial force increases slightly.
From the results previously described, this study
presents the following conclusions. First, it is concluded
that MABB can decrease the size of the cross-section of
the stiffened girder because its increase in stiffness is
larger than the mass increase of structure due to internal
arch rib that connects the stiffened girder and arch rib.
Second, it is estimated that MABB has the advantage of
ensuring the dynamic stability for the moving load
because the dynamic behavior of MABB is more stable
than that of the conventional arch.
References
Gwak Jong-Won, Ha Sang-Gil, Kim Seong-Il, Jang Seung-
Pil (1998), Vibration of steel composite railway bridges
under high-speed train, Journal of Korean Society of Steel
Construction, Vol.10, No.4, pp. 577-587.
Kim Sang Hyo, Park Hong Seok, Heo Jin Yeong (1999),
Study on dynamic responses of bridges using high-speed
railway vehicle models, Journal of the Computational
Structural Engineering Institute of Korea, Vol.12, No.4,
pp. 629-638.
Choi Chang Geun, Song Byeong Gwan, Yang Sin Chu
(2000), A model for simplified three-dimensional analysis
of high-speed train vehicle (TGV)–bridge interactions,
Journal of the Computational Structural Engineering
Institute of Korea, Vol.13, No.2, pp. 165-178.
Choi Seong Rak, Lee Yong Seon, Kim Sang Hyo, Kim
Byeong Seok (2002), Verified 20-car model of high-
236 Min-Sik Kong et al.
speed train for dynamic response analysis of railway
bridges, Journal of the Computational Structural Engineering
Institute of Korea, Vol.15, No.4, pp. 693-702.
Hwang Sun-geun, Development of core technology for
performance enhancement of railway system-track: civil
development of the design specifications for improving
dynamic characteristics of railway bridges. Research
paper. Korea Railroad Research Institute.
Kong Jun (2005), Dynamic analysis of cable-stayed bridges
under moving vehicle loads. Master’s Thesis. University
of Seoul Graduate School.
Kim Sung-il (2000), Bridge-train interaction analysis of
high-speed railway bridges. Doctorate Thesis. Seoul
National University Graduate School.
Franco BRAGA et al. (2003), Bowstring bridges for high-
speed railway transportation, IABSE SYMPOSIUM.