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: Kongms@uos.ac.kr

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.

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