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This is to certify that the

thesis entitled

Three-dimensional Elastic Seismic Response
of Deck-type Arch Bridges

presented by

Robert Joseph Millies

has been accepted towards fulfillment
of the requirements for

M. S. degree in Civil Engineering

 

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7/W

Major professor

Date 4/ 2/92

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THREE-DIMENSIONAL ELASTIC SEISMIC RESPONSE OF
DECK-TYPE ARCH BRIDGES
BY
Robert Joseph Millies

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Civil and Environmental Engineering

1992

ABSTRACT

THREE-DIMENSIONAL ELASTIC SEISMIC RESPONSE OF DECK-TYPE ARCH
BRIDGES

By

Robert Joseph Millies

A study of the lateral stiffness of the end towers of a
bridge was conducted using computer modelling. The bridge
model is symmetric, consisting of geometrically nonlinear
curved beam elements for parabolic ribs with.cross-bracing and
a trussed deck. Three span lengths of 200, 600 and 1000 feet
were studied. The 1940 El Centro ground motion in 3 direc-
tions was used.

The tower lateral stiffness essentially affected the out-
of—plane response, only. Ftu' short spans, this response
contributed significantly to the rib stresses. Accordingly,
an increase in the tower stiffness increases the rib stress in
the 200 ft. span. For the longer spans, the tower stiffness
had small affects on the rib stresses because the lateral
responses were small. However, the displacements at the
towers were significantly decreased by a tower stiffness
increase. Lateral tower stiffness of the order of 50%
relative to that of the rib system seems appropriate for

design purposes.

To my dad (Robert H.), mom (Maureen), sisters,
”The Real Men of Pittsford”, friends 8
my future wife (whomever she may be)
for their incalculable support towards

my every endeavor, thanks.

iii

ACKNOWLEDGEMENTS

At this time I would like to acknowledge with sincere
appreciation the opportunity given by Dr. Robert K. L. Wen,
Professor of Civil Engineering, Michigan State University, to
conduct this study. His role as thesis advisor, as well as,
engineering scholar, mentor, general counselor and friend is
appreciated and will not be forgotten. Without Dr. Wen, this
study would not have been possible.

I would also like to thank the National Science Founda-
tion for the financial support. I would like to thank my
committee members, the Civil and Environmental Engineering
Department and Case Computing Center' at. Michigan State
University. Finally, I would like to thank God for this

opportunity and for the family and friends He has given me.

iv

TABLE OF CONTENTS

Chapter Page
LIST OF TABLES vii
LIST OF FIGURES viii

I INTRODUCTION........................................1
II MODELLING AND PARAMETERS............................5
2.1 Bridge System Considered.......................5
2.1.1 Description of Bridge Model.............5
2.1.2 Assumptions of the System Studied.......7
2.1.3 Discussion of Analysis..................8
2.2 Parameters.....................................10
2.2.1 Parameters Associated with the
In-Plane Behavior.......................11
2.2.2 Additional Parameters Associated with
the Out-of-Plane Behavior...............11
III RESULTS.............................................l4
3.1 Determination of Inter-Rib Bracing Parameters..14
3.2 Role of Tower Lateral Stiffness................16
3.2.1 General Comments on the Lateral
Stiffness in Tower......................16
3.2.2 Effect of a on the Sequencing of

In-Plane and Out—of-Plane Normal Modes..20

TABLE or CONTENTS (Continued)

Chapter Page
3.2.3 Effect of a on Maximum Stress Ratio.....21

3.2.4 Effect of a on Deck Lateral (z-)
Displacement............................23

3.2.5 Effect of a on Lateral Load

Distribution............................25

3.2.6 Effect of a on Seismic and Wind
Stresses................................27
IV SUMMARY 8 CONCLUSION....... ......... ................30
4.1 Summary........................................30
4.2 Conclusion.....................................35
TABLES..............................................38
FIGURES.............................................42
LIST OF REFERENCES..................................73

APPENDIX - GLOSSARY OF SYMBOLS......................75

vi

LIST OF TABLES

Table Page
1 Parameters for In-Plane and Additional Parameters
for Out-of-Plane Studies...........................38
2 Distribution of Stresses vs. Loading in Curved
Beams(L = 600’, a =1.0)............................39
3 Natural Frequencies(cps)...........................40

4 Comparison of Wind and Seismic Stresses............41

vii

LIST OF FIGURES

Figure Page
1 Sketch of Bridge Model........................42
2 Cross-Section of Twin-Rib System and Computer

Model.........................................43
3 Ground Acceleration, Velocity and Displacement
of 1940 El Centro Earthquake..................44
4 Z-Displacement of Quarter Point as a Function
of Nonlinear and Linearized Analysis .......... 45
S Z-Displacement of Quarter Point as a Function
of Linear or Nonlinear Analysis...............46
6 Z-Displacement of Quarter Point as Function of
Linear and Linearized Analysis................47
7 First Out-of-Plane Natural Frequency as a
Function of Ifi/Iw.............................48
8 First Out-of-Plane Natural Frequency..........49
9 Rib Force Due to Static z-Load................50
10 Response Spectrum as a Function of Period.....51
11 Distribution of Stress in a Rib (L=200 ft.)...52
12 Distribution of Stress in a rib (L=600 ft.)...53
13 Distribution of SR (L = 200 ft.)..............54

14 Distribution of SR (L = 600 ft.)..... ....... ..55

viii

LIST OF FIGURES (continued)

Figure

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

Page
Distribution of SR (L = 1000 ft.).............56
SR for Different Combination of Earthquake
Components (L = 200 ft.)......................57
SR for Different Combination of Earthquake
Components (L = 600 ft.)... ..... ..... ......... 58

Maximum SR as a Function of a (L = 600 ft.)...59

Maximum SR as a Function of a (L 200 ft.)...60
Maximum SR as a Function of a (L = 1000 ft.)..61
Linear and Linearized Analysis................62
Deck z-Displacement Along Span Length

(L = 600 ft.).................................63
Deck z-Displacement as a Function of a

(L = 200 ft.).................................64
Deck z-Displacement Along Span Length

(L = 1000 ft.)................................65
Maximum D, as a Function of a (L = 200 ft.) ...66

Maximum Tower D, as a Function of a
(L=6OO ft.)...OOOOOOOODOOOOOOOOOOO0000......67

Maximum Tower D, as a Function of a

(L - 1000 ft.)................................68
Linear and Linearized Analysis (L = 200 ft.,

600 ft., 1000 ft.)............................69

Maximum a, as a Function of a..................70

ix

LIST OF FIGURES (continued)

Figure Page

30 Linear and Linearized Analysis (L = 200 ft.,

600 ft., 1000 ft.)............................71
31 Effect of a on SWR (L = 200 ft., 600 ft.,

1000 ft.) ..................................... 72

931193;;
Introductio

According to the theory of plate tectonics, the earth’s
surface consists of a number of plates which are constantly in
motion. As these plates move against each other, strain
energy is accumulated in them until, eventually, the material
fails, sending off shock waves. This is the cause of most
earthquakes. The San Fernando earthquake that occurred in
1971 caused much.damage tolbridgesm Since then, the engineer-
ing field has given much time and effort to the study of
earthquakes and how to build stronger, safer structures to
withstand earthquakes. This study is a step towards a better
understanding of the effects of earthquakes on a particular
type of structure, steel arch bridges.

Thakkar and Arya [9] had reported the linear response of
single arch ribs to in-plane seismic motions. Using a simple
model of lumped flexibility and lumped mass, Raithel and.Fran-
ciosi [8] illustrated the elastic vertical vibrations of
arches caused by horizontal seismic excitations. The decrease
of the natural frequencies due to compression in the arch was
also illustrated.

The linear behavior of deck arch bridges subjected to

2
realistic ground motion in three dimensional space has been
discussed by Dusseau and Wen [2,3]. Lee [S] has presented a
method of analysis for the geometrically nonlinear and
inelastic responses of arch bridges. A general computer
program was developed and illustrative applications were
given.

The above computer program was modified to include a more
effective type of nonlinear curved beam elements [13] and used
to produce application oriented data for in-plane response
[12].

The objective of this thesis is to study (using the
program) the seismic response from a designer's point of view,
chiefly in the consideration of the lateral stiffness needed
for the end towers of the bridge. In the course of the work,
an appreciable amount of effort was expended in adapting the
modified program for design studies in a three-dimensional
setting.

The program 'was modified to include the automatic
generation of the nodal coordinates for the cross beams
between the ribs. The time step-size (used in the dynamic
analysis) input was modified so as to be a function of the
fundamental period. A Lagrange interpolation function was
implemented to determine the ratio of the two principal
moments of inertia of the rib cross section as a function of
span length. This ratio then allowed for the computation of

the other geometrical properties.

3

This study concentrates on the role of the end towers
which are the "bents" at the bridge ends rising from the arch
support level to support the deck, see Figure 1. They are
analogous to the towers of suspension bridges, in that their
actions under gravity loads are a relatively simple one of
transmitting vertical loads down to the foundation through
compression of the vertical members.

When the bridge is subjected to lateral (normal to the
planes of the arches) loads, such as wind or seismic loads,
the designer needs to decide on a scheme for the transmission
of the loads to the foundation. Assuming that a horizontal
shear connection is provided between the deck and the arch
ribs (usually at the crown), the towers and the ribs partake
in the lateral load transference. Such action obviously
depends on the lateral stiffness of the tower relative to that
of the arches.

In the case of static loads (wind load is usually treated
as such) this relative stiffness would play the major role in
determining the proportioning of the given total design
lateral loads to the ribs and the end towers. It is well-
known that the seismic design load on a structure depends on
its natural frequencies. In the case of seismic loading
(due to lateral ground motion), the stiffness would have the
additional effect of actually defining the loading itself as
the stiffness affects the out-of-plane natural frequencies of

the structure.

4
The main purpose of this thesis is to elucidate the
preceding observation using computer analysis. The computer
model and the parameters of the systems studied are described
in Chapter 2. The results of the study are presented in

Chapter 3. A summary and conclusions are given in Chapter 4.

Chapter II
Modelling and Parameters

This chapter discusses the modelling system and the
parameters. In the first section, the bridge system studied
is discussed. In the second section, a discussion of the
parameters of the system is given.

2,; Bridge Systeg Considered

Herein, the components of the bridge and their action are
described. The assumptions of the system studied are given,
followed by a summary of the analysis.

3.1.; Qescription of Bridge Model

A real arch bridge containing a multitude of components
is a system with a large number of degrees of freedom (d.-
o.f.). It was necessary, due to resource limitations, to use
a.model that had as few d.o.f. as feasible on the one hand and
yet could still be expected to capture the essential feature
of the prototype on the other; This is particularly true here
since the analysis is nonlinear and consequently requires more
computing resources.

In an arch bridge, the major structural components are
the arch ribs which are the main object of this study. Thus,
in formulating the bridge model, the subsystem of the ribs is

given greater precision than the deck and the end towers. For

6
the latter, the model would provide only the interaction
between them and the ribs. The modelling for the deck and
tower is not intended to reflect the internal behavior of the
deck subsystem or the tower subsystem at their component
level.

The model used is shown in Figure 1. It consists of a
twin-rib system: a rib in front and a rib in back. The arch
is parabolic in shape. The ribs consist of curved beam
elements. The ribs are braced together to form a unit by
"cross beams" and cross-bracing (X-bracing). The cross beams
which are perpendicular to the planes of the ribs are modelled
by straight beam elements. The X-bracing run in diagonal
directions between the cross beams and are modelled as truss
elements.

The main feature of the 3-D system when compared with the
2-D in-plane system, is the out-of-plane torsional bending
behavior. To strengthen the torsional response, the rib
bracing could have X-bracing along the top faces of the ribs
and another X-bracing along the bottom faces” This would form
a "torsional box" as shown in Figure 2a. Since it is cumber-
some to include two layers of X-bracing in the model, a model
is used as shown in Figure 2b. To simulate the relatively
large torsional stiffness.of the "torsional.box" in Figure 2a,
the value:of the torsional stiffness of the individual ribs in
Figure 2b could be increased.

The deck system which includes "stringers”, "X-bracing"

7
and "cross beams," is represented entirely by truss elements.
It is connected to the end towers and to the crown of the
arch. The in-plane bending stiffness of the deck stringers
are neglected. Similarly, the deck in the model does not
contribute significantly to the torsional stiffness of the
system. However, the deck as a horizontal truss can resist
substantial bending in the horizontal plane.

The deck and ribs are connected by "columns” modelled as
truss elements. Truss elements are also used for X-bracing .
for the deck.

Each end tower is represented.by a vertical bent.consist-
ing of four truss elements; two vertical elements and two
diagonal elements (the stiffness of which essentially defines
the lateral stiffness of the tower).

Lastly, the support conditions are as follows. For the
rib ends, all d.o.f. are restrained except rotation about the
z-axis (i.e. rotation in the plane of the arch). The towers
are hinged at their lower ends and ("pin-”) connected to the
deck at the upper ends. As mentioned previously, the deck is
also directly hinged to the ribs at the crown.

Mia—W

The system studied is symmetric about two vertical
planes: one containing the crown and parallel to the end
towers; the other is in between the two planes of the two
ribs. The bridge, as a whole, was modeled. The mass of the

deck and that of the arch is distributed unifomly along their

8
horizontal projection. All rib elements have the same cross-
section. Likewise, all elements within a group (e.g. X-beams,
or rib X-bracing) have the same cross-sectional properties.
s s of Ana 3

The computer program used for the study and the method of
analysis on which it was based are essentially those as
described by C. M. Lee [5]. There, the truss elements and
straight beam elements were linearly elastic. For the curved
beam elements both geometric nonlinearity and plasticity were
allowed. The masses were lumped at the nodes. Rayleigh
damping was assumed. Two coordinate systems were used for the
"Global Stiffness Matrix”: Curvilinear coordinate system for
the arch nodes and the common Cartesian coordinate system for
the deck nodes.

The computation of the seismic response would consist of
two steps: 1. A linearly elastic. static analysis was made
for the dead load; and 2. The results were used as the
initial conditions to compute the seismic response. For the
latter analysis, the Newmark B-method (B = 1/4) of numerical
integration was used for the time domain, and the Newton-
Raphson method was used to deal with the nonlinearity content
of the problem.

The computer program used in this study is a modification
of the one described above. The modifications include the
following:

1. A new nonlinear curved beam element [13] is

9
used to replace the old one. The new element makes
it possible to compute the resistance much more
accurately.
2. The dual coordinate systems described previous-
ly is replaced by a single Cartesian system (X, Y
and 2 denote global coordinate system; x, y and 2
denote member coordinate system). Although this
would incur more computing, it is more than compen-
sated by the avoidance of confusion to the user of
the program.
3. Straight beams with.general global geometry are
allowed. (Previously, in order to save computing
time, straight beams were fixed as horizontal or
vertical members.)
4. The two separate inputs for computer solutions
(dead load - static, and then seismic - dynamic)
are combined into one computer run.
5. The eigensolution values needed for the specifi-
cation of the Rayleigh damping is also incorporated
directly into the analysis. (Previously, this was
done separately.)
6. As a consequence of 5. above, the size of the
time increment used.in the numerical integration is
made a fraction of the computed fundamental period
(instead of a preset constant).

7. The stiffness of the rib system is calculated

10
using F = kz. A uniformly distributed load (F) in
the z-direction is applied to the deck and ribs and
the corresponding crown displacement (z) is mea-
sured. The stiffness (k) is equal to the total
load applied divided by the displacement. This
value of the rib lateral stiffness multiplied by a
ratio of end tower lateral stiffness to rib lateral
stiffness will give the lateral stiffness of the
end towers. The lateral stiffness of the end tower
is defined as the load applied at the top of the
end tower in the z-direction to cause a unit dis-
placement thereat.
8. A number of convenient measures are added for
engineering studies (such as this one). They
include the automatic generation of input data such
as mass and stiffness quantities based on the
parameters defined in the next section as well as
the ”post processing” of data such as searching for
the maximum of the response values.
242__2£I£E!§!£!
The following section discusses the parameters chosen to
define the system being considered. Estimations of their
ranges, ”central values" and those used in the study are also

presented.

11
oo w the n- v r

There are ten parameters that are associated with the
bridge in-plane behavior and are given in Table 1. The first
column lists the parameters; the second column, their ranges;
the third, the values used in this study.

The parameters associated with the in-plane behavior are
as follows: H/L is the rise (H) to span length (L) ratio;
L/ry is the ”slenderness ratio", representing the span length
over the radius of gyration about the rib’s local y-axis (the
local y-axis of curved beam cross section is its principal
axis normal to the plane of the arch and the local x-axis is
the other principal axis); G is equal to MgL’l (EIfl) (in which
M is the total mass per foot of span length, L, g is the
gravitational acceleration, E is the Young's modulus and Iyr is
the moment of inertia about y); M,/M is the mass of the rib
over the total mass; N is the number of panels; 5 is the
critical damping coefficient for the first two modes; and the
ratio cylr, in which c’ is one half of the depth of the rib
cross-section; and x, and y, are the accelerations in the X-
and Y-directions, respectively.

. d t at e s oci ed w t Out-
- eh

The additional parameters associated with out-of-plane
behavior are also listed in Table 1. They include: W, the
‘width; I,/Ifl, the moment of inertia about x-axis of the rib to

the moment of inertia about the y-axis of the rib; c,/r,, in

12

which c, is the distance in the rib local x-direction from the
neutral axis to the extreme fibers (or one half of the width)
and r, is the radius of gyration about the x-axis of the rib;
A,/A,, in which A, is the cross-sectional (x-sectional) area of
the rib X—bracing and.A,is the x-sectional area of the rib;
A,/A,, in which A, is the x-sectional area of the cross beams;
Iw/Ifl in which Iyb is the moment of inertia about the y-axis of
the cross beam; IfilI» in which I, is the moment of inertia
about the x-axis for the cross beam; K,/Iyb in which R, is the
torsional constant of the cross beam; a is the ratio of the
lateral stiffness of the end tower to the lateral stiffness of
the braced system of the two ribs; A,/A, in which A, is the x-
sectional area of the deck stringers; and z, is the ground
acceleration in the global Z-direction.

The columns (between the deck and the ribs) and the X-
bracing of the deck.are:represented.by essentially rigid truss
members. These elements are not formally regarded as parame-
ters of the model.

By a review of the properties of several existing bridges
[7] [14], estimates of the range and/or representative values
of the parameters are listed in column two of Table 1. For
the present study, the values of the parameters used are given
in column three of the same table.

The ground motions of the 1940 El Centro earthquake are
shown in Figure 3. The variables x,, y, and z, are used to

represent, respectively, the north-south, vertical and east-

13
west components of the El Centro ground accelerations. The
magnitude of the ground accelerations were not scaled through-
out this study.

The program has the capability of using a linear,
geometrically nonlinear or "linearized” model for the dynamic
analysis. The ”linearized” model employs the nonlinear model
for the initial dead load solution and the subsequent response
to seismic motion would be calculated based on a linear
analysis using the tangent stiffness of the structure under
dead load. A comparison of the program results using linear,
nonlinear or linearized analysis was made and is shown in
Figures 4 - 6. These figures show the time history displace-
ments of an arch quarter point. From the figures, it is seen
that there is a relatively large difference between the linear
and nonlinear case and the linear and linearized case; but
there is not a noticeable difference between the nonlinear and
linearized case. Therefore, the method of linearized analysis

was chosen for use in this study.

£§A22£B_Ill
BEEELIE

This chapter presents and discusses the results obtained
for the study. In the first section of this chapter, the
inter-rib bracing parameters are determined. In the second
section, the roles of the end towers are investigated.

3,; Determination of Inter-Rib Bracing gerametezs

The inter-rib>bracing consists of cross beam members and
cross-bracing (X-bracing) truss members placed between the
ribs as shown in Figure 1. Since this study focuses on rib
response, the deck was made essentially rigid. Because of the
limited role of the inter-rib bracing members, and with a view
to minimizing the number of parameters, a study was made to
fix the numerical values of the parameters representing the
inter-rib bracing.

With the proper bracing, the two ribs would act as a
unit. An ”optimal" or "efficient" amount of bracing material
to enable this action needs to be determined. For the beam,
this was interpreted to mean that an additional increase
beyond that amount would not have a major effect on the
fundamental out-of-plane frequency. For the x-bracing, this
was interpreted to mean that an additional increase would have

little effect on the magnitudes of the member forces caused by

14

15
a lateral load.

The cross-sectional properties of the beams are area
(A,) , moment of inertia about y- and x-axis (1,, and I“) , and
torsion constant (K,) . The X-bracing as truss members has
only the area property (A,) to define their cross-sectional
properties. The parameters used for this study are A,/A, (the
area of the beam to that of rib), belIy, (the beam moment of
inertia about the y-axis to the rib moment of inertia about
the y-axis), leIyb (the beam moment of inertia about the x-
axis to the beam moment of inertia about the y-axis) , K,/I,,
(the torsional constant to the moment of inertia about the y-
axis for the beam) and A,/A, (the X—bracing area to the rib
area). Values for the above parameters were determined as
follows.

First, as indicated by an existing bridge, KO/be - 1,]be
= 1.0 seemed reasonable. After some initial trial-and-error,
it was decided to use:

be/Iyr = 0.05
A,/A, = 0.10

A,/A, = 0.04
In support of the above choices Figure 7 shows the effect

of be/ Iyr on the fundamental out-of-plane natural frequency.
It may be seen that the frequency began levelling off,
approximately, at belly, = 0.05. Figure 8 shows the fundamen-
tal out-of-plane natural frequency is not influenced by a

variation of A,/A,. Figure 9 shows the variation of the rib X-

16
bracing member forces as a factor of A,/A,. It is seen that
the member force increases rapidly with small values of A,/A,.
However, the increase levelled off after it reached 0.04.
. ow te if ness

This section is concerned with the effects of the end
towers, its lateral stiffness in particular. The first part
presents some general comments. In part two, the sequencing
of the in-plane and out-of—plane modes is considered. In part
three, the influence of the tower lateral stiffness on the
stress ratio in the arch rib is discussed. In part four, the
influence of the tower lateral stiffness on the deck lateral
displacement is discussed, In part five, the influence of the
tower lateral stiffness on the lateral load proportioning
between the arch ribs and the towers is discussed. And in
part six, static wind stresses are compared with dynamic
seismic stresses.

e e o es e

Let the ratio of the lateral stiffness of the tower to
the lateral stiffness of the braced.arch ribs be denoted by a.
The lateral stiffness of the rib and tower is defined in
Chapter 2.1.3. The parameter a influences two aspects of the
bridge response: 1.) the local tower displacements, and 2.)
the overall bridge response to lateral ground motion as it
affects the out-of-plane natural frequencies.

The total dynamic response, R, may be expressed as R 8 R31-

* AF. Rfl.is the static response; it depends on the stiffness

17

of the structure. AF denotes "amplification factor" which
depends mainly on the natural frequency. As the structural
stiffness is increased, Rsr is decreased, but the corresponding
change in the natural frequency may increase or decrease AF.
The nature of dependence of AF on the natural period of
vibration is illustrated in Figure 10, which is the accelera-
tion response spectrum given in the AASHTO Seismic Design
Specifications for Bridges.

In order to aid in the understanding of the dynamic
stress distribution in the arch ribs due to three dimensional
excitations, the stresses under static loads for lengths of
200 ft. and 600 ft. are discussed here. For these span
lengths, the stresses for three loading cases are presented in
Figure 11 and 12, respectively. The cases are static loadings
of 0.39 applied through the nodal masses separately in the X-,
Y- and z-directions.

Under x-loading for span length = 200 and 600 ft., the
maximum static stress occurs in the area of the quarter points
of the arch (1/4 of the span length, from the end points) and
is a minimum at the crown of the arch and.the supports (ends).
The x-loading produces the largest stress when compared to y-
and z-loadings.

Under y-loading for span length = 200 and 600 ft., the
maximum stress occurs at the middle of the arch and minimum at
the supports. There is a difference on the order of 10

between the x-loading and y-loading.

18

Under the z-loading for span length = 200 and 600 ft.,
the distribution of stresses resembles an inversion of that
due to the x-loading. The maximum stress occurs at the
supports and crown of arch, while the minimum occurs at the
quarter points. The maximum stresses due to z-loading are
approximately one-third the stresses due to x-loading and
approximately three times the stresses due to y-loading.

The distribution of the stresses due to y-loading differs
from the known stress distribution due to a uniformly distrib-
uted vertical load on the horizontal projection of the arch
(having a uniform cross-section)in which the maximum stress
occurs at the supports. And the only type of stress involved
is a compressive stress. But in this model, the loading is
not uniform; the vertical load from the deck is transferred
through the columns to the rib and is applied as concentrated
loads. These concentrated loads not only produce compressive
forces but also bending moments. ‘The moment due to bending is
the largest at the crown and therefore causes large bending
stress to occur, despite the relatively small compressive
stress involved.

Table 2 shows the distribution of stresses for individual
members. The stress is chosen from the I-node only of each
curved beam member of the rib and is calculated using:

stress = |P/A| + |M,/S,| + |M,/S,| . . . . . . . . . . . (1)
where: I | denotes absolute value,

P = compressive force

19

A a cross-sectional area

My, M, = the bending moment about y-axis

and x-axis, respectively

S = the section modulus.
The stresses listed occurred in the specified member under the
given static loading, i.e., x-, y- or z-loading of 0.39 each.
The member number of the front rib is listed, and the number
next to it in parenthesis is the member in the back rib,
indicating symmetry between the front and back ribs. The SAM,
SYM and SXM headings pertain to the ratio of stress due to
axial force or bending moments about the y- and x-axis,

respectively, divided by the total stress:

SAM = IP/Allzw
SYN = IMy/SyIIXmI
SXM = IMx/le/zml

where Zuuzrepresents the total stress.

SAM, SYM and SXM should always sum to one. For example,
member 2, y-only (y-loading only) has a stress at the I-node
of 3.01 ksi and that 57% of it comes from axial stress and 43%
from bending about the y-axis.

The z-loading case is the only case that has significant
values from each of SAM, SYM and SXM. The x- and y-loading
cases have contributions to the total stress from SAM, SYM but
not SXM. The stresses due to the x-loading case is governed by
bending moment stresses at internal nodes, but at the sup-

ports, the stresses are due to axial force. .As noted earlier,

20
the support has a moment release about z-axis only.
. e t o e Be one n- -
o - ane Norm Modes
The parameter a influences the eigen-solutions of the
arch bridge. As a is varied for a given length, the sequenc-
ing of the in-plane and out-of—plane normal modes changes. In
'Table 3, these modes are listed for lengths of 200 ft., 600
ft., and 1000 ft. The letters "in" mean in-plane and the
letters "out” mean out-of-plane, the number beside ”in" or
”out" is the frequency of that mode, in Hertz. The T; and T“,
values represent the period of the fundamental in-plane (a
full wave-length in the x-Y plane) and fundamental out-of-
plane (a half wave-length in the x-z plane) modes, respec-
tively. As can be seen, as a increases, the in-plane modes
rise up in ranking and become more dominant. For the length
= 200 ft. case, as a increases from 0.5 to 9.0, the difference
between the fundamental periods of the in-plane and the out-
of-plane increases. This is due to the fact that as 0
increases the stiffness in the lateral direction increases
while the stiffness in the X-Y plane remains essentially
unchanged. The increase in lateral stiffness moves the out-
of-plane modes down in the "ranking” (the larger the value of
the natural period, the higher the ranking). The same
situation occurs for lengths of 600 ft. and 1000 ft. The
fundamental mode for length = 600 ft. when a is 0.0 is out-of-

plane and becomes in-plane when a is increased to 0.5. For

21
the length = 1000 ft., the out-of-plane remains the fundamen-
tal mode for all values of a, but other than that the in-plane
modes still rise in rank as (1 increases. Therefore, the
selection of a for a given length can influence the type of
modes of vibration that will be dominant in dynamic response.

W

The study of the maximum stress ratio (SR) that occurs
within the ribs during an earthquake loading with all three
(X,Y,2) components is described in this section for varying a
and lengths.

For length.= 200 ft., the SR in the ribs varies from 2.54
to 3.86, see Figure 13. For length = 600 ft., the SR in the
ribs varies from 2.5 to 3.7, see Figure 14, and for length =
1000 ft., the SR varies from 1.96 to 2.76, see Figure 15. In
all three cases, the largest value of SR is near the ends.‘ It
is also observed that the shorter span is influenced more by
the variation of a than the other span lengths.

For lengths = 200 ft. and 600 ft., the contribution of
each component of the earthquake loading to SR is shown in
Figures 16 and 17, respectively. The SR from the Z component
is small for the length - 600 ft. case, and contributes very
little to the stress when all three ground motions, X, Y and
z, are considered.

This is not the case for length = 200 ft. In this case
the response to the z component of the earthquake loading is

:much larger. An explanation for the difference may be found

22
in Figure 10. It may be seen that for the 600 ft. span, with
Tu,(a = 0.1) = 3.42 s, the spectral response is relatively low
(approximately 0.22). For the 200 ft. span, with.Tw,(a - 0.1)
= 1.11 s, the spectral response is relatively high (approxi-
mately 0.45).

The maximum SR as a function of a, for length = 600 ft.,
is shown in Figure 18. The largest maximum SR that occurs is
approximately 3.90 when a = 1.0. But is only about 5% more
than the minimum. For length = 200 ft, see Figure 19, the
largest maximum SR occurs at a larger a, specifically 3.0, and
it is about 18% more than the minimum. Opposite to the length
= 200 ft. case is the length = 1000 ft. case, see Figure 20,
its largest maximum SR occurs at a lesser a than the length 8
600 ft. case, specifically 0.1. The maximum SR for length =
1000 ft. is about 5% more than the minimum.

One may also note that the value of the largest maximum
SR decreases as the length increases, from approximately 4.3
to 3.9 to 2.9. This is due mainly to the variance of values
of natural periods, and hence the AF values, for the different
lengths.

Lastly, the linear and the linearized analyses results
were compared, as shown in Figure 21. When a is close to
zero, the largest difference between the two types of analyses
occur when the length = 200 ft., and the difference between

the two becomes smaller as the length increases.

23
. e t o as ater - D s as e t

The effects of the variable a on the z-displacement (DJ
of the deck is illustrated in Figure 22 for the case span
length = 600 ft. The displacement D, is directly influenced
by the stiffnesses of the tower and the arch.ribs. It is seen
that the deck end displacement decreases as (1 increases.
Conversely, the deck.middle point displacement increases as 0
increases. It may be worth noting again that at the mid-span,
the deck and the rib crown meet, and thus haVe equal displace-
ments. Also, the displacement shape of the deck is concave
for a = 0 and 0.1, while it is convex for a larger than 0.5.
As the value of a changes from 0 to 0.5, there is about a 50%
reduction in the displacement at the end of the deck, while
there is about a 15% increase in the displacement at the
middle of the deck.

For the case of length = 200 ft. see Figure 23. The
displacements vary much less from the end to the middle than
for the case of the 600 ft. span, but the shapes still
indicate a single curvature, either concave or convex. There

is a large difference between a - 0 and 0.1, approximately a

50% reduction in D,. This is probably due to the change in
the frequency response via natural periods of vibration.

In Figure 24, the behavior of the deck's displacement for
the case of length = 1000 ft. is shown to be quite similar to
'the length.= 600 ft. case for approximately a 50% reduction in
the displacement at the ends of the deck. At the middle of

24

the span, a general decrease of D, occurs with increasing the
value of a past 0.5. One fact is worth noting. .In Figure 24,
D, increased at all points when a was increased from 0 to 0.1.
This is contrary to the behavior of other lengths for the same
a range. It was probably due to a change in the frequency
response of the structure. The natural period of the struc-
ture changed with an increase in a. An explanation can be
given, similar to above, using Figure 10. After a was
increased beyond 0.1, the D, started to decrease at the ends,
indicating that the increase in the static effect of the tower
stiffness becomes dominate.

A comparison of the D, for the three different lengths
shows that the largest D, occurs for length = 200 ft. when a
= 0, the other two lengths, 600 ft.and 1000 ft., are very
similar. At the middle of the span, the largest D, occurs
when the length = 600 ft. and a = 1.

The maximum value of the ratio of the maximum tower Z-
displacement to the maximum arch crown z-displacement (D,) is
shown in Figures 25 - 27 as a function of a for the three span
lengths. Since the rib D, does not vary much, the variation
of D, would be due mainly to variation in the tower displace-
ment. For lengths == 600 ft. and 1000 ft. (Figs. 26 8 27), the
curves sharply drop as a is increased from 0.0 to 2.0. The
graph for the length = 200 ft. case (Figure 25) does not show
as sharp a drop but does show a considerable decrease in the

D, with (2. These figures also suggest that for an a value as

25

little as 0.5, the D, decreases approximately 62% for lengths
= 600 ft. and 1000 ft. For the length = 200 ft. case, the
decrease in D, of approximately 50% is associated with a value
of 3.8 (derived from interpolation from figure 25) for a. ‘Ehe
large difference between the lengths.= 200 ft. and 600 ft (and
1000 ft.) is probably due to the fact that with the same
width, the shorter bridge has a larger "depth-to-span ratio”
with respect to the lateral loading than the longer bridge.
Also, since the other properties of the deck were constant for
each length, thetdeck became stiffer for the span length.= 200
ft. case and therefore forced the tower and ribs to displace
more uniformly. This uniformity demand by the deck caused the
decrease in D, to be smoother than the other lengths.

The linear and linearized analyses for the maximum D, as
a function of a were compared and is shown in figure 28. It
is shown that there is very little difference between the two
types of analyses.

5 B feet 0 on La er 0 d t bu

The total lateral load applied to the arch bridge, is
transferred to the supports by the ribs and the towers.
Therefore, the parameter a (tower stiffness to rib stiffness)
influences the amount of force in the z-direction, z-force,
that is taken by the towers and ribs. For a = 0, all the
lateral load will be taken by the ribs, since the towers do
not have any lateral stiffness for resisting lateral loads.

Figure 29 shows, for the three different span lengths, the

26
ratio of the z-force through a single tower to the z-force
through the rib system (or) as a function of a.

For small values of 0: (approximately 0.1) , the a, is
similar (0.1) as would be expected, but as (1 increases a, does
not respond linearly as one might think. An increase in the
tower stiffness does not result in a corresponding increase in
the tower Z-force. The magnitude of a? is less as the length
is increased. For length = 1000 ft. case, 0% levels off with
increasing a starting at approximately a = 1.0. For the
length = 600 ft., a, levels off at approximately 3 = 2.0. For
the length = 200 ft. case, the response of aw‘to a is not the
same as the other lengths, it does not have a distinct
leveling-off point, but the rate of increase does decrease.
For the 200 ft. bridge, an optimal value for u might be
selected as 3.0.

For each length, when a is increased the trend of
increasing a, is similar but the magnitude of up is different.
The magnitude of or decreases as the length of the bridge
increases. This suggests that of the total lateral load that
is created from the earthquake's displacement, the amount
transferred to the towers decreases with length relative to
the load transferred through the ribs.

In Figure 30, the values of or using linear and linear-
ized are graphed as a function of a. The difference between

the two types of analyses is small.

27
as o o e m o d I n at s

The wind stresses and seismic stresses are compared.here
to relate the two considerations in design. Given a location
for a bridge, the governing design lateral load is generally
based on either a wind load or an earthquake, depending on
which is more severe.

In this section, the wind load stresses are compared to
seismic stresses. The wind load applied to the bridge was
calculated according to the AASHTO bridge specifications [14] .
The specifications state that for an arch bridge use a wind
load of 75 psf applied to the windward side of the bridge.
The windward side is the XY—plane of the bridge. The surface
area was estimated as twice the surface area of the arch, The
lumped wind load at each node was then.calculated, in terms of
gravity (g), to be 0.1259 times the nodal mass.

The stresses due to wind and seismic input (z-direction
only) are shown in Table 4. The members with the maximum
stress due to wind loading are located nearest the supports.
According to Table 2 for member 1, the stresses are composed
of 65% axial and 35% bending about the x-axis. The members
with the maximum stress due to seismic loading are located
near the crown of the arch. According to Table 2, these
stresses are composed of 70% bending about the y-axis, 15%
bending about the x-axis and 15% axial.

In Table 4, the symbol "1/4" represents the stress in the

arch at the quarter point (which happens to be not the

28

maximum). It is observed that for larger spans, the wind
stress governs; for shorter spans, the seismic stress governs.
Under the wind loading, the stresses tend to decrease as a
increases. Under the seismic loading, the opposite occurs
when the length = 200 ft. as 0 increases, the stress increas-
es. For lengths = 600 ft. and 1000 ft., the seismic stresses
do not change much with a and the direction of change is
erratic.

The ratio SWR, the seismic stress to the wind stress
ratio, varies with length and a as shown in Figure 31. For
length = 1000 ft., it is seen that wind stresses are larger
than seismic stresses. The SWR values range from 0.19 at a =
0.1, to 0.25 at a = 3.0. For length - 600 ft., the wind
stress is larger at a = 0.1, but when a = 0.5 or larger, the
seismic stress is larger. For length = 600 ft., SWR ranges
from 0.74 at a = 0.1, to 1.77 at a = 8.0. For length = 200
ft., the seismic stress.dominates for all a'sc The SWR.ranges
from 2.65 at a = 0.1 to 17.59 at a = 8.0.

Figure 31 can be used as an aid in designing arch
bridges. It pertains to the El Centro earthquake with a
magnitude of 7 measured on the Richter scale. It is, in all
probability, the largest earthquake that will occur in the
United States. The spectral response of El Centro is very
similar to Figure 10. For application of these results to a
particular site with a different seismicity, the designer can

scale accordingly. Considering this, the seismic response

29
coefficienct, cg, is calculated (according to AASHTO) in the
following manner:
c, = l.2*Ac*S/T,2’3 (2)
where Ac = the ground acceleration coefficient; S = the dimen-
sionless for the soil profile characteristics of the site; and
I; = the period of the bridge.
For El Centro, Ac = 0.4 and S = 1, therefore, scaling may

proceed as:

317,,

(ash/(145),“, ...............(3)

(AS),,/0.4 (4)

or , SF“,
The seismic stresses may then be multiplied by SE“,(scaling
factor) to obtain the stresses that may occur at the site due
to a design earthquake. The design should then be checked to

determine if the wind or seismic load governs.

.EMT

ha e v

W

W

This study is concerned with seismic effects on deck-type
arch bridges in three dimensional space. The focus of the
study was on the influence of the lateral stiffness of the end
towers on the responses of the bridge. The responses investi-
gated were:

1. the sequencing of the in-plane and out-of-plane

modes

2. the stress ratio in the arch ribs

3. the deck and tower lateral displacements

4. the lateral load proportioning between the arch

ribs and the end towers

5. the seismic stresses compared to the wind stres-

ses.

The study was carried out using a computer program that
had been developed previously and was modified to include
several measures to improve the accuracy of the analysis as
well as to allow ease in the handling and gathering of the
output results. The modifications include the following: a.)

a new nonlinear curved beam element is used to replace the old

30

31

one; b.) the dual coordinate systems is replaced by a single
Cartesian system; c.) straight beams with general global
geometry'are.allowed; d.) the two separate inputs for computer
solutions (dead load - static, and then seismic - dynamic) are
combined into one computer run; e.) the eigensolution needed
for the specification of the Rayleigh damping is incorporated
directly into the analysis (previously, this was done sepa-
rately); f.) as a consequence of e.) above, the size of the
time increment used in the numerical integration is made a
fraction of the computed fundamental period (instead of a
preset constant).; and, g.) a number of convenient measures
are added for engineering studies (such as this one), they
include the automatic generation of input data such as mass
and stiffness quantities based on the input (dimensionless)
parameters as well as the ”post processing” of output data
such as searching for the maximum of the response values.

The bridge model consists of two ribs as the main
elements of the arch. The end towers are the "bents" at the
ends of the bridge that rise from the arch support level to
support the deck (Figure 1).

The major parameter in this study is a, the ratio of the
lateral stiffness of the end tower to that of the rib system.
The stiffness of the rib system is calculated using F - kz.
A uniformly distributed load (F) in the z-direction is applied
to the deck and ribs and the corresponding crown displacement

(z) is computed. The stiffness (k) is computed as the total

32
load, F, divided by the displacement. The lateral stiffness
of the end tower is defined as the load applied at the top of
the end tower in the z-direction to cause a unit displacement
thereat.

Other parameters included in this study: span length;
I’ll” (in which Iyb is the moment of inertia about the y-axis
of the cross beam andIyr is the moment of inertia about the y—
axis for the rib); A,,/Ar (in which A, is the cross sectional
area of the cross beams and.A,is.the cross sectional area of
the rib); and, A,/A, (in which A, is the cross sectional area
of the rib cross-bracing). The ground motion input used in
this study is that of the 1940 El Centro earthquake, with all
three orthogonal components.

Three bridge span lengths are considered in this study,
200, 600 and 1000 ft. The study began with the determination
of the parameter values for the bracing system between the
ribs. Using eigenanalysis, the study of the inter-rib bracing
parameter, Iflllfl, focused on the behavior of the out-of-plane
natural frequency. I,,,,/Iyr - 0.05 was considered to be the
”optimal" value to use. Similarly, the parameter A,/A, was set
equal to 0.1. The value of A,/A, was set equal to 0.04 because
further increase in its value would not significantly change
the forces in the cross-bracing members.

The parameter a influences the eigensolutions of the arch
bridge. As a is varied for a given span length, the sequenc-

ing of the in-plane and out-of-plane normal modes changes

33

(Table 3). An increase in end tower lateral stiffness (i.e.
an increase in a) moves the in-plane modes up in ranking and
the out-of-plane modes down in ranking. For relatively large
values of a, the fundamental.mode is an in-plane mode for span
length = 200 ft. and 600 ft.; for span length = 1000 ft., it
would remain to be an out-of-plane mode. For small values of
a, the fundamental mode is an in-plane mode for span length -
200 ft. only; for span lengths = 600 ft. and 1000 ft., it is
an out-of-plane mode.

The maximum stress ratio, SR, varies with span length and
a. For span length.= 200 ft., the SR varies approximately 18%
from minimum to maximum. For span length = 600 ft. and 1000
ft. , the SR varies approximately 5%. Therefore, for medium to
long spans (i.e. 600 ft. - 1000 ft.), the SR does not vary
significantly with a (Figs. 13, 14 and 15).

The effects of a on the z-displacement, D,, of the deck
is studied. For span length = 200 ft., a reduction of 50% for
D, at the ends and middle of the deck occurs when a is
increased from 0 to 0.1. For a span length = 600 ft., a 50%
reduction of D, occurs at the ends of the deck when a increas-
es from 0 to 0.5. For the same range of a, D, increased about
15% at the middle of the deck (Figure 22). For span length =
1000 ft. , a 50% reduction of D, at the ends of the deck occurs
when (2 increases from 0 to 0.5. The D, at the middle of the
span increased approximately 18% when a changed from 0 to 0.1

and then decreased when a increased beyond 0.1. Among all

34
cases, the largest D, at the end of the deck and the middle of
the deck occurred when span length = 200 ft. and a -= 0.
Therefore, in order to have a 50% reduction of the displace-
ments at zero tower lateral stiffness, the shortest span, 200
ft., needs the smallest value of a.

The maximum value of the ratio of the maximum tower z-
displacement to the maximum arch crown z-displacement, D,, is
a function of a. For the span lengths = 600 ft. and 1000 ft.,
I%.drops rapidly as a is increased. The span length = 200 ft.
has a smoother decrease of D, as a is increased. A possible
reason for the sharp decrease of D, for increasing span
lengths is due to the fact that with the same width, the
shorter bridge has a larger ”depth-to-span ratio" with respect
to the lateral loading than the longer bridge. The larger
"depth-to-span ratio" represents a stiffer bridge; the larger
stiffness does not allow the change in displacement to occur
rapidly.

The ratio of force in the z-direction (z-force) trans-
ferred by’a single tower to the z-force transferred by the rib
system, a,, is influenced by a. For each span length case ,
a, is linearly related to a for small values of a. But as a
increases, a, does not remain linearly related; instead, for
span lengths - 600 ft. and 1000 ft., or levels-off quickly,
while a, for span length = 200 ft. does not level-off as
quickly. Also, the magnitude of a, decreases as span length

increases; in other words, the amount of force transferred. to

35
the towers decreases with increasing span length (Figure 29).

For the three span lengths, a comparison of stresses due
to wind load and seismic loads was made. It is shown that for
the span length = 200 ft., the seismic stress dominates for
all values of a. For span length = 1000 ft., the wind stress
is larger than the seismic stress. But for span length = 600
ft. and small values of a (less than or equal to 0.1), the
wind stress is larger than the seismic stress; for larger a,
the seismic stress becomes larger than the wind stress. It is
observed that for longer spans, the wind stress would govern;
seismic stress governs for short spans.

4.; Copclusiog

A study of the seismic behavior of the deck-type arch
bridges in the three dimensional space has been.conducted, In
the study, emphasis has been placed on the role of the end
towers of the bridge systems when the structure is subjected
to a ground motion that has a lateral component. It is found
that the behavior is governed mainly by the parameter a, the
ratio of the tower lateral stiffness to that of the rib
system.

As a varies, the lateral or out-of-plane natural frequen-
cies change, and the seismic response changes accordingly; It
is found that such changes had a larger effect for bridges
with a short span length (200 ft.) than the longer ones (600
ft. or 1000 ft.) . When compared .with the usual strength

requirement, or in terms of stresses, the results obtained

36
indicated for shorter spans, the lateral design load would be
based on the seismic load rather than the wind load. For
longer spans, the wind load would be the governing lateral
design load. .

For displacement, generally, the larger the a, for any
span length, the less the displacement of the deck and ribs.

Based on the results obtained, there seems to be not a
need for a large value of a; a value of 0.5 or 1.0 may be
sufficient.

The seismic stresses presented here could be scaled. The
scaling factor may be based on the seismic response coeffi-
cient, C" According to AASHTO Bridge Seismic Specification,
C, can be calculated as C, = 1.21||rAc1IIIS/T2’3
where Ac = the ground acceleration coefficient; S -= the
dimensionless coefficient for the soil profile characteristics
of the site; and T = the period of the bridge. If one takes
A.= 0.4 and S - 1 for the El Centro earthquake and the dynamic
analysis used, the scaling factor, SF“, would simply be the
seismic response coefficient of the new site to the seismic
response coefficient corresponding to the El Centro earth-
quake, i.e. SF... = (AS)*/(AS)EC,,, or, SF,, 8 (AS),-./0.4.
The seismic stresses presented herein may then be multiplied
by SF“, to obtain the stresses that may occur at the site.

It is recognized that the data presented herein covered
only a very small part of the range of parameters. In a

specific design situation, it would be necessary to obtain a

37

computer solution. based on 'the jparameter ‘values of the
structure being planned. But the results presented here
should provide insight into the behavior of such bridges and
aid the engineer in the preliminary stages of the design,
particularly with reference to the design of the end tower.

Future research along the line of the study here may
consider such ”secondary” parameters, for example, I,/Iw, the
moment of inertia about the x-axis of the rib>to the moment of
inertia about y-axis of the rib. Also, different earthquake
motions (real and artificial) might be used in the future to
develop a more general description of the bridge behavior.
Cost studies could be implemented to study the cost-benefit

relation for lateral stiffness of the end towers.

Table 1

38

for Out-of-Plane Studies.

  

Parameters for

 

 
 

In-Plane Behavior

Parameters for In-Plane and Additional Parameters

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

! L/ry 100, 300 200
G 2.63 - 10.5 10.5
M,/M 0.344 - 0.760 0.265
N 6 - 24 8
£ 2%, 5% 2%
cylr, 1.00 - 1.55 1.27
L 200 - 1000 ft. 200, 600, 1000
x, 0 - 0.50g 0.319
23 0 - 0.509 [J4 0.239
Additional
Parameters for
Out-of-Plane
Behavior
W 30 - 60 30
I IE/I” 0.32 - 0.11 0.32 - 0.11
I c,/r, 1.00 - 1.55 1.30
A,/A, Not Available 0.04
AMA,r 0.10 - 0.25 0.10
IW/I” 0.0015 - 0.014 0.05
Iglln Not Available 1.0
Kg/Iw Not Available 1.0
a 0.0 - 10.0 0 - 10
Adflg Not Available 0.183 - 0.91

 

 

 

 

0 - 0.509 0.319 I

 

 

39

Table 2 Distribution of Stresses vs. Loading in Curved Beams
(L = 600’, a = 1.0).

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Stress SAM 1SYM SXM I
4.72 1.0 0.0 0.0
1.83 1.0 0.0 0.0
8.78 0.65 0.0 0.35
25.25 0.03 0.97 0.0
3.01 0.57 0.43 0.0
11.77 0.23 0.66 0.11
x-only 36.16 0.02 0.98 0.0
3 (11) y-only 3.85 0.42 0.58 0.0
I z-only 3.99 0.03 0.70 0.27
x-only 31.36 0.03 0.97 0.0 l
4 (12) y-only 4.36 0.36 0.64 0.0 l
5.79 0.34 0.51 0.15 =
8.75 0.98 0.01 0.0
4.55 0.34 0.66
12.6 0.15 0.70
25.59 0.03 0.97
4.41 0.36 0.63
5.09 0.02 0.74
42.06 0.03 0.97
3.94 0.43 0.57
5.83 0.45 0.35
40.19 0.04 0.96
3.13 0.58 0.42
15.19 0.37 0.52 0.11

 

 

40

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 3 Natural Frequencies (cps).
or Modes. L = 200’ L = 600’ L = 1000'
1 in 0.634 out 0.250 out 0.126
‘2 out 0.811 out 0.274 out 0.133
' 3 out' 0.845 in 0.367 in 0.284
0,0 4 in‘ 1.559 ’out. 0.767 out 0.315
5 in 2.103 in 0.902 in 0.699
Tin (s) 1.59 2.70 3.52
T“ (s) 1.23 4.00 7.94
1 in 0.634 out 0.292 out 0.163
2 out 0.901 in 0.367 out 0.194
3 out 1.057 out 0.362 in 0.284
' 4 in 1.559 out 0.802 out 0.339
0'1 5 out 2.107 in 0.902 in 0.699
__Eé(s) 1.58 2.70 3.52
T,é (s) 1.11 3.42 6.13
1 in 0.635 in 0.367 out 0.209
2 out 1.169 out 0.375 in 0.284
3 in 1.559 out 0.585 out 0.328
4 out 1.631 in 0.902 out 0.446
0'5 5 out 2.127 out 0.947 in 0.699
1__Eé.(s) 1.57 2.70 3.52
:59.(s) 0.86 2.63 4.78
1 in 0.635 in 0.367 out 0.233
2 in 1.559 out 0.470 in 0.284
3 out 1f911 in 0.902 out 0.604
4 out 2.675 out 1.287 in 0.699
9'0 5-. in 2.903 in 1.680 out 0.921
__35,(s) 1.57 2.70 3.52
To, (5) 0.52 2.13 4.29

 

 

 

 

 

 

 

41

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4 Comparison of Wind and Seismic Stresses.

stress a=0 . 1 (i=0 . 5 0r=1‘. 0 a=2 . 0 o=3 . 0 a=8 . 0

L=1000’ wind max 9.49 7.59 7.15 6.89 6.80 6.68
seis max 1.76 1.73 1.47 1.46 1.68 1.53

wind 1/4 5.38 4.26 4.00 3.85 3.80 3.73

seis 1/4 2.42 2.42 2.28 2.24 2.18 2.26

L=600’ wind max 10.24 7.25 6.33 5.74 5.52 5.23
seis max 7.55 9.51 6.88 6.75 9.22 9.25

wind 1/4 4.20 2.85 2.43 2.16 2.06 1.93

"seis 1/4 1.87 2.77 2.19 2.44 2.44 2.26

L=200’ wind max 6.57 3.96 2.82 1.97 1.62 1.13
seis max 17.42 20.22 21.98 28.74 28.50 17.56

wind 1/4 3.09 1.79 1.21 0.79 0.62 0.37

seis 1/4 7.52 9.48 11.86 13.84 14.25 9.17

 

 

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10.

LIST OF REFERENCES

Dusseau, R.A., ”Seismic Analysis of Two Steel Deck Arch
Bridges," M.S. Thesis, Department of Civil Engineering,
Michigan State University, 1981.

Dusseau, R.A., and Wen, R.K., "Responses of a Long
Span Arch Bridge to Differential Abutement Motions”,
Proceedings of the Third U.S. National Conference on
Earthquake Engineering, Charleston, South Carolina,
August 1986.

Dusseau, R.A., and Wen, R.K., "Seismic Responses of
Deck-Type Arch Bridges,” Earthquake Engineering and
Structural Dynamics, Vol. 18, 1989, pp. 701-715.

Kuranishi, S., and Nakajima, A., ”Strength Character-
istics of Steel Arch Bridges Subjected to Longitudinal
Acceleration”, Structural Engineering Earthquake
Engineering, JSCE, Vol. 3, 1986, pp. 2875-2955.

Lee, C.H., "Nonlinear Seismic Analysis of Steel Arch
Bridges," Ph.D. Dissertation, Michigan State University,
E. Lansing, MI, 1990.

Hedallah, K., and Wen, R.K., ”Elastic Stability of
Deck-Type Arch Bridges,” Journal of Structural
Engineering, ASCE, Vol. 113, No. 4, April 1987, pp.
757-768.

Merrit, F., "Structural Steel Designers’ Handbook",
McGraw-Hill-Book Co.,New York, 1972, pp. 13-1-13-41.

Raithel, A., and Franciosi, C., "Dynamic Response of
Arches Using Lagrangian Approach", Journal of
Structural Engineering, April 1984, pp. 847-858.

Standard Specifications For Highway Bridges 1983.
American Association of State‘Highway and.Transportation
Officials. Section 3.15. Washington, D.C.

Thakkar, S.K., and Arya, A., "Dynamic Response of
Arches Under Seismic Forces”, Proceedings of the Fifth
World Conference on Earthquake Engineering, Rome,
Italy, 1973.

73

74

LIST OF REFERENCES (Continued)

11.

12.

13.

14.

Wen, R.K., Notes on research. 1/5/91.

Wen, R.K., ”Seismic Behavior and Design of Arch
Bridges,” Proceedings of the 4th U. 8. National
Conference On Earthquake Engineering, Palm Springs,
California,May 1990, Vol. 1, pp. 1027-1036.

Wen, R.K., and Suhendro, B., "Nonlinear Curved-Beam
Element for Arch Structures", Journal of Structural

Engineering, Vol. 117, No. 11, Nov. 1991, pp. 3496-
3515.

U.S. Department of Transporation, "Arch Bridges,”Series
No. 2, Washington D.C., 1977.

mm
W

APPENDIX

GLOSSARY OF SYMBOLS

A = cross-sectional area
Ac = ground acceleration coefficient
A, = area of cross beam

AP = amplification factor

A, = cross-sectional area of the rib

A, = cross-sectional area of the deck stringers

Ax = cross-sectional area of the rib cross-bracing
c; = seismic response coefficient

:3 = one half of the width of the rib cross-section

one half of the depth of the rib cross-section

«0
ll

I%.= the maximum tower z-displacement to the maximum arch
crown z-displacement

D1 = z-displacement

E = Young's modulus

F = force

acceleration of gravity
H = rise (height)
I” = moment of inertia about cross beam local y-axis

I¢ = moment of inertia about cross beam local x-axis

75

 

 

76

GLOSSARY OF SYMBOLS (Continued)

moment of inertia about rib local x-axis

In

I moment of inertia about rib local y-axis

n
k = stiffness

K, = torsional constant of the cross beam
L = span length

M = total mass per foot of span length
Mr = mass of the rib total mass

lg = the bending moment about y-axis

IQ = the bending moment about x-axis

= number of panels

P

N
compressive force

R = dynamic response

Rfl.= static response

I; = radius of gyration about rib local x-axis
13 = radius of gyration about rib local y-axis
S = section modulus

S

dimensionless soil coefficient
SAM = ratio of axial stress to total stress

SF,”5 = scaling factor

SR = maximum stress ratio (dynamic stress to static stress)

SWR - ratio of the seismic stress to the wind stress

SYM = ratio of stress due to bending moment about the y-axis

to total stress

SXH

ratio of stress due to bending moment about the x-axis

77

GLOSSARY OF SYMBOLS (Continued)

to total stress

T = period of the bridge

I; = fundamental in-plane period

EL, = fundamental out-of-plane period

W = width of the bridge

x, Y, z = global coordinate axes in the x, y, and z direction

X-bracing = cross-bracing

x-beams = cross-beams

:9 8 ground acceleration in the X-direction

y, = ground acceleration in the Y—direction

z = displacement in Z-direction

z-force = force in the z-direction

z, a ground acceleration in the Z-direction

a = ratio of end tower lateral stiffness to ribtsystem.1ateral'

stiffness

cy1= ratio of the z-force through a single tower to the 2-
force through the rib system I

E = critical damping coefficient

2:“. = total stress