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Roebling Suspension Bridge. I: Finite-Element Model and
Free Vibration Response
Wei-Xin Ren1; George E. Blandford, M.ASCE2; and Issam E. Harik, M.ASCE3
Abstract: This first part of a two-part paper on the John A. Roebling suspension bridge 共1867兲across the Ohio River is an analytical
investigation, whereas Part II focuses on the experimental investigation of the bridge. The primary objectives of the investigation are to
assess the bridge’s load-carrying capacity and compare this capacity with current standards of safety. Dynamics-based evaluation is used,
which requires combining finite-element bridge analysis and field testing. A 3D finite-element model is developed to represent the bridge
and to establish its deformed equilibrium configuration due to dead loading. Starting from the deformed configuration, a modal analysis
is performed to provide the frequencies and mode shapes. Transverse vibration modes dominate the low-frequency response. It is
demonstrated that cable stress stiffening plays an important role in both the static and dynamic responses of the bridge. Inclusion of large
deflection behavior is shown to have a limited effect on the member forces and bridge deflections. Parametric studies are performed using
the developed finite-element model. The outcome of the investigation is to provide structural information that will assist in the preser-
vation of the historic John A. Roebling suspension bridge, though the developed methodology could be applied to a wide range of
cable-supported bridges.
DOI: 10.1061/共ASCE兲1084-0702共2004兲9:2共110兲
CE Database subject headings: Bridges, suspension; Finite element method; Three-dimensional models; Vibration; Natural
frequency; Dead load; Equilibrium; Model analysis.
Introduction
Many of the suspension bridges built in the United States in the
19th Century are still in use today but were obviously designed
for live loads quite different from the vehicular traffic they are
subjected to today. A good example is the John A. Roebling sus-
pension bridge, completed in 1867, over the Ohio River between
Covington, Kentucky, and Cincinnati, Ohio. To continue using
these historic bridges, it is necessary to evaluate their load-
bearing capacity so that traffic loads are managed to ensure their
continued safe operation 共Spyrakos et al. 1999兲. Preservation of
these historic bridges is important since they are regarded as na-
tional treasures.
The unique structural style of suspension bridges permits
longer span lengths, which are aesthetically pleasing but also add
to the difficulties in performing accurate structural analysis. De-
sign of the suspension bridges built in the early 19th Century was
based on a geometrically linear theory with linear-elastic stress-
strain behavior. Such a theory is sufficiently accurate for shorter
spans or for designing relatively deep, rigid stiffening systems
that limit the deflections to a small fraction of the span length.
However, a geometrically linear theory is not well adapted to the
design of suspension bridges with long spans, shallow trusses, or
a large dead load. A more exact theory is required that takes into
account the deformed configuration of the structure.
In modern practice, finite-element 共FE兲analysis is effective in
performing the geometric nonlinear analysis of suspension
bridges. Geometric nonlinear theory can include the nonlinear
effects inherent in suspension bridges: cable sags, large deflec-
tions, and axial force and bending moment interaction with the
bridge stiffness. Two- or three-dimensional finite-element 共FE兲
models with beam and truss elements are often used for both the
superstructure and the substructure of cable-supported bridges
共Nazmy and Abdel-Ghaffar 1990;Wilson and Gravelle 1991; Lall
1992; Ren 1999; Spyrakos et al. 1999兲.
Another area where FE analysis has had a major impact re-
garding suspension bridge analysis is in predicting the vibration
response of such bridges under wind, traffic, and earthquake load-
ings 共Abdel-Ghaffar and Rubin 1982; Abdel-Ghaffar and Nazmy
1991; Boonyapinyo et al. 1999; Ren and Obata 1999兲. In addi-
tion, major efforts have been expended to predict the lateral
共Abdel-Ghaffar 1978兲, torsional 共Abdel-Ghaffar 1979兲, and verti-
cal 共Abdel-Ghaffar 1980兲vibrations of suspension bridges to pre-
dict their dynamic behavior. FE parametric studies 共West et al.
1984兲have demonstrated the variation in the modal frequencies
and shapes of stiffened suspension bridges.
Structural evaluations using dynamics-based methods have be-
come an increasingly utilized procedure for nondestructive testing
共Friswell and Mottershead 1995; Brownjohn and Xia 2000兲.A
difficulty with dynamics-based methods is establishing an accu-
rate FE model for the aging structure. FE models typically pro-
1Professor, Dept. of Civil Engineering, Fuzhou Univ., Fuzhou, Fujian
Province, People’s Republic of China; and Professor, Dept. of Civil En-
gineering, Central South Univ., Changsha, Hunan Province, People’s Re-
public of China. E-mail: renwx@yahoo.com
2Professor, Dept. of Civil Engineering, Univ. of Kentucky, Lexington,
KY 40506-0281.
3Professor, Dept. of Civil Engineering, Univ. of Kentucky, Lexington,
KY 40506-0281 共corresponding author兲. E-mail: iharik@engr.uky.edu
Note. Discussion open until August 1, 2004. Separate discussions
must be submitted for individual papers. To extend the closing date by
one month, a written request must be filed with the ASCE Managing
Editor. The manuscript for this paper was submitted for review and pos-
sible publication on November 28, 2001; approved on November 19,
2002. This paper is part of the Journal of Bridge Engineering, Vol. 9,
No. 2, March 1, 2004. ©ASCE, ISSN 1084-0702/2004/2-
110–118/$18.00.
110 / JOURNAL OF BRIDGE ENGINEERING © ASCE / MARCH/APRIL 2004
vide dynamic performance predictions that exhibit relatively large
frequency differences when compared with the experimental fre-
quencies and, to a lesser extent, the models also predict differ-
ences in the modes of response. These differences come not only
from the modeling errors resulting from simplified assumptions
made in modeling the complicated structures, but also from pa-
rameter errors due to structural damage and uncertainties in ma-
terial and geometric properties. Dynamics-based evaluation is
therefore based on a comparison of the experimental modal analy-
sis data obtained from in situ field tests with the FE predictions.
To improve the FE predictions, the FE model must be realistically
updated 共calibrated兲to produce the experimental observed dy-
namic measurements 共Friswell and Mottershead 1995兲. Thus the
scope of this study on the dynamics-based evaluation of the Roe-
bling suspension bridge is composed of several tasks: FE model-
ing, modal analysis, in situ ambient vibration testing, FE model
updating, and bridge capacity evaluation under live loading.
This paper presents the results of the first two tasks in the
dynamics-based evaluation scheme of the Roebling suspension
bridge. A 3D FE model is developed for the ANSYS 共1999兲com-
mercial FE computer program. All the geometric nonlinear
sources discussed previously are included in the model. A static
dead-load analysis is carried out to achieve the deformed equilib-
rium configuration. Starting from this deformed configuration, a
modal analysis is performed to provide the frequencies and mode
shapes that strongly affect the free vibration response of the
bridge. Parametric studies are performed to determine the signifi-
cant material and structural parameters. Results of the FE modal
analysis are compared with ambient vibration measurements in
the accompanying paper 共Ren et al. 2004兲. This FE model, after
being updated 共calibrated兲based on the experimental measure-
ments, will serve as the baseline for the structural evaluation of
the bridge. The baseline structural evaluations provide important
design information that will assist in the preservation of the Roe-
bling suspension bridge, and furthermore, the methodology devel-
oped in these two papers can be applied to a wide range of his-
toric cable-supported bridges.
Bridge Description and Historic Background
The John A. Roebling suspension bridge, shown in Fig. 1, carries
KY 17 over the Ohio River between Covington, Kentucky, and
Cincinnati, Ohio. The 321.9 m 共1,056 ft兲main span of the sus-
pension bridge carries a two-lane, 8.53 m 共28 ft兲wide steel grid
deck roadway with 2.59 m 共8ft6in.兲wide sidewalks cantilevered
from the trusses. The towers are 73.15 m 共240 ft兲tall and
25⫻15.85 m 共82⫻52 ft兲at their base and encompass 11,320 m3
共400,000 cu ft兲of masonry. Towers bear on timber mat founda-
tions that are 33.53⫻22.86 m 共110⫻75 ft兲and 3.66 m 共12 ft兲
thick. The suspension bridge system is composed of two sets of
suspension cables restrained by massive masonry anchorages.
Stay cables radiate diagonally from the towers to the upper chords
of the stiffening trusses. Deck loads are transferred from the
stringers and floor beams to the suspenders, trusses, and stays and
then to the suspension cables, which then transfer the loads to the
anchorages and towers. The approach span roadway varies from 6
to 7.25 m 共20 to 24 ft兲wide and is composed of a concrete deck
supported by riveted steel plate girders. The plan and elevation
views of the Roebling suspension bridge are shown in Fig. 2.
The John A. Roebling suspension bridge 共completed in 1867兲
was the first permanent bridge to span the Ohio River between
Fig. 1. John A. Roebling suspension bridge
Fig. 2. Layout of Roebling suspension bridge
JOURNAL OF BRIDGE ENGINEERING © ASCE / MARCH/APRIL 2004 / 111
Covington and Cincinnati. At the time of its opening, the soaring
masonry towers represented a new construction method that sup-
ported a state-of-the-art iron-wire cable technology. This monu-
ment to civil engineering of the 1800s represented the longest
span in the world at the time of its opening. Today, this nationally
historic bridge 共designated as such in 1975兲remains the second
longest span in Kentucky, and after 133 years of service, the
bridge still carries average daily traffic of 21,843 vehicles 共Par-
sons et al. 1988兲. The bridge is currently posted at 133.4 kN 共15
t兲for two-axle trucks and 97.9 kN 共22 t兲for three-, four-, and
five-axle trucks.
Finite-Element Modeling
For the purposes of this study, a complete 3D FE model has been
developed as shown in Fig. 3共a兲. This model is used for both
static and dynamic analyses. The FE model of the Roebling sus-
pension bridge is briefly described herein, with greater details
provided in Ren et al. 共2003兲.
Element Types
A suspension bridge is a complex structural system in which each
member plays a different role. In the FE model, four types of FEs
used to model the different structural members: main cables and
suspenders, stiffening trusses, floor beams and stringers, and tow-
ers.All cable members of the Roebling suspension bridge 共primary
cables, secondary cables, suspenders, stay cables, and stabilizer
cables兲are designed to sustain tension force only and are modeled
using a single 3D tension-only truss element between joints 关Fig.
3共c兲兴, which allows for the simulation of slack compression
cables. Both stress stiffening and large displacement modeling are
available. Stress stiffening modeling is needed for the cables since
cable stiffness is dependent on the tension force magnitude. Cable
sag can also be modeled within the stress stiffening modeling. An
important input parameter is the initial element strain, which is
used in calculating the initial stress stiffness matrix.
The stiffening truss is modeled as a 3D truss composed of a
single beam or truss element between joints 关Fig. 3共b兲兴. Top and
bottom chords of the truss are modeled as 3D elastic beam ele-
ments since they are continuous across many panels. Vertical truss
members are also modeled as 3D elastic beam elements, whereas
the diagonals are modeled as 3D truss elements since they are
pinned connected and do not provide much bending stiffness. Tie
rods that connect the primary and secondary cables are also mod-
eled as 3D truss elements.
Tower columns are modeled as 3D elastic beam elements,
whereas the web walls of towers above and below the deck are
modeled as three-node quadrilateral membrane shell elements, as
shown in Fig. 3共c兲, because the bending of these walls is of sec-
ondary importance.
The deck is simplified as stringers and floor beams in the
analytical model; that is, the principal load-bearing structural el-
ements of the deck are the stringers and floor beams. These can be
subjected to tension, compression, bending, and torsion, and con-
sequently each one is modeled using a single 3D elastic beam
element between joints. Three-dimensional FE discretization of
the Roebling suspension bridge consists of 1,756 nodes and 3,482
elements, resulting in 7,515 active degrees of freedom.
Material and Cross-Section Properties
Basic materials used in the Roebling suspension bridge are struc-
tural steel, masonry towers, and iron cables. The material con-
stants used are summarized in Table 1. Note that the stringer and
floor beam mass densities include the contribution from the
bridge deck weight and sidewalks, as well as the lateral bracing
system contribution.
In addition to the material properties of Table 1, cross-
sectional properties and initial strains are required. Cross-
sectional constants are used to model the structural member fea-
tures described below:
• Stiffening truss: Top chord is a built-up member with a solid
cover plate, and bottom chord is a built-up member with top
and bottom lacing bars. The top and bottom chords have riv-
eted joints but employ pin connections at each panel point for
the verticals, which are latticed columns. Diagonals are steel
eye bars.
Fig. 3. 3D finite-element model of Roebling bridge: 共a兲full 3D
elevation; 共b兲part 3D elevation 共span center and stiffness truss兲;共c兲
part 3D elevation 共tower and cables兲
112 / JOURNAL OF BRIDGE ENGINEERING © ASCE / MARCH/APRIL 2004
• Primary cables: Composed of seven strands, each containing
740 No. 9 gauge cold-blast charcoal iron wires, for a total of
5,180 wires. These wires are parallel to each other and form a
cable that is 313.27 mm 共12 1/3 in.兲in diameter with an ef-
fective area of 55,920 mm2共86.67 in.2兲. A total of 4,671.2 kN
共1,050.2 kips兲of cable wire was used, which also includes the
wrapping wire. Design ultimate strength of one wire is 7,206
N共1,620 lb兲; therefore the design ultimate strength per pri-
mary cable is 37,326 kN 共8,391.6 kips兲.
• Secondary cables: Composed of 21 strands, including 7 that
contain 134 wires each and 14 that contain 92 wires each,
resulting in 2,226 wires. These No. 6 gauge ungalvanized steel
wires are parallel to each other and form a cable that is 266.7
mm 共10.5 in.兲in diameter, resulting in an effective area of
43,086 mm2共66.78 in.2兲. Design ultimate strength of one wire
is approximately 24,000 N 共5,400 lb兲. Therefore, the design
ultimate strength per cable is 53,467 kN 共12,020 kips兲.
• Suspenders: Composed of three helical wire ropes in which the
outer pair of wrought iron wire ropes is 38.1 mm 共1.5 in.兲in
diameter and is part of the original construction. These pairs of
ropes at 1.52 m 共5ft兲spacing supported the original truss and
floor system. In 1897, a third rope 57.1 mm 共2 1/4 in.兲in
diameter was added, and these additional ropes are spaced at
4.57 m 共15 ft兲intervals. The combined ultimate strength is
2,517.6 kN 共566 kips兲.
• Stay wires: There are 72 stay cables. These stays are 57.1 mm
共2 1/4 in.兲diameter, helical iron wire ropes with an ultimate
strength of 8,000 kN 共1,800 kips兲each.
• Floor beams and stringers: In the suspension span, each 127
mm 共5 in.兲open steel grid deck is supported by C10⫻20
crossbeams spaced at 1.143 m 共3ft9in.兲, resting on six string-
ers spaced at approximately 1.6 m 共5ft3in.兲. The four outer-
most stringers are 381 mm 共15 in.兲I-sections that weigh 729.5
N/m 共50 lb/ft兲, and the two center stringers are 508 mm 共20
in.兲I-sections that weigh 948.4 N/m 共65 lb/ft兲. Floor beams are
riveted, built-up steel sections with a web plate 914.4 mm 共36
in.兲deep with four flange angles riveted to it.
Boundary Conditions
The towers of the Roebling suspension bridge are modeled as
fixed at the base, whereas the cable 共both primary and secondary兲
ends are modeled as fixed at the anchorages. The stiffener truss
and stringer beams are assumed to have a hinge support at the left
and right masonry supports but they are continuous at the towers
to simulate the actual structure. In addition, the stiffener truss for
the Roebling suspension bridge is an uncommon one-hinge de-
sign placed in the center of the span to provide for temperature
expansion. The hinge was modeled by defining separately coinci-
dent nodes in the top as well as the bottom chords at the midspan.
Coupling the vertical and transverse displacements of the coinci-
dent nodes while permitting them to move independently in the
horizontal direction simulates the expansion hinge effect. In ad-
dition, the twisting and in-plane rotations are constrained to dis-
place equally, but the rotation about the lateral 共z-axis兲of the
bridge is discontinuous.
Static Analysis—Dead Load
In the design of suspension bridges, the dead load often contrib-
utes most of the loading. It was realized as early as the 1850s that
the dead load has a significant influence on the stiffness of a
suspension bridge. In the FE analysis, this influence can be in-
cluded through static analysis under dead loading before the live
load or dynamic analysis is carried out. The objective of the static
analysis process is to achieve the deformed equilibrium configu-
ration of the bridge due to dead loads in which the structural
members are prestressed.
For the static analysis of the Roebling suspension bridge under
dead loading, the value of the deck dead load is taken to be 36.49
kN/m 共2,500 lb/ft兲, which is taken from the report by Hazelet and
Erdal 共1953兲. In the FE model, the dead load is applied directly
on each node of both inner stringers. The distributed load is
equivalent to a 166.81 kN 共37.5 kips兲point load applied on each
node of the inner stringers.
Table 1. Material Properties
Group
number Young’s modulus
关MPa 共lb/ft2兲兴
Poisson’s
ratio
Mass
density
关kg/m3
共lb/ft3兲兴
Structural
member
1 2.1⫻105(4.386⫻109) 0.30 7,849 共490兲Stiffening
trusses
2 2.0⫻105(4.177⫻109) 0.30 7,849 共490兲Cables
3 2.0⫻105(4.177⫻109) 0.30 7,849 共490兲Suspenders
4 2.0⫻105(4.177⫻109) 0.30 7,849 共490兲Stay wires
and tie rods
5 2.0⫻104(4.177⫻108) 0.15 2,500 共156兲Tower
6 2.1⫻105(4.386⫻109) 0.30 19,575 共1,222兲Floor
beams and
stringers
Table 2. Influence of Cable Prestrain on Maximum Axial Forces and Main-Span Deflections
Prestrain
Bottom chord 共kN兲Top chord 共kN兲Cable members 共kN兲
Deflection
共m兲Panel 30 Panel 55 Panel 40 Panel 55 Primary
cable Secondary
cable Suspender
0.0 ⫺1,771.0 720.4 2,830.8 ⫺37.5 6,992.7 5,004.9 101.6 0.967
0.1⫻10⫺5⫺1,769.0 719.8 2,827.6 ⫺37.4 6,996.7 5,006.6 101.6 0.966
0.1⫻10⫺4⫺1,718.0 713.6 2,799.0 ⫺37.1 7,028.7 5,033.4 102.1 0.956
0.1⫻10⫺3⫺1,574.9 651.3 2,514.1 ⫺33.4 7,351.2 5,292.7 107.2 0.859
0.5⫻10⫺3⫺794.7 367.2 1,259.5 0.0 8,773.7 6,452.7 129.7 0.431
0.6⫻10⫺3⫺600.6 295.8 951.2 ⫺12.6 9,130.9 6,745.4 135.4 0.325
0.7⫻10⫺3⫺406.8 224.6 645.8 ⫺8.2 9,489.8 7,044.3 141.1 0.221
0.8⫻10⫺3⫺213.8 153.6 343.7 ⫺3.7 9,850.5 7,336.1 146.8 0.118
Note: One panel⫽4.572 m 共15 ft兲.
JOURNAL OF BRIDGE ENGINEERING © ASCE / MARCH/APRIL 2004 / 113
Initial Tension in Main Cables
A cable-supported bridge directly derives its stiffness from cable
tension. For a completed suspension bridge, the initial position of
the cable and bridge is not known; only the final geometry of the
bridge due to the dead loading is known. The initial bridge ge-
ometry has been modeled based on the dead-load deflected shape
of the bridge. When the bridge is erected, the truss is initially
unstressed. The dead load is borne completely by the cables,
which is a key assumption. It turns out that the ideal FE model of
a suspension bridge should be such that on application of the dead
load, the geometry of the bridge does not change, since this is
indeed the geometry of the bridge. Furthermore, no forces should
be induced in the stiffening structure. This can be approximately
realized by manipulating the initial tension force in the main
cable that is specified as an input prestrain in the cable elements.
The initial tension in the cables can be achieved by trial and error
until a value is found that leads to minimum deck deflection and
minimum stresses in the stiffening trusses.
The maximum axial force and main span deflection variations
versus cable prestrain are summarized in Table 2. Panel locations
for panels 30, 40, and 55 are shown in Fig. 2 as P30, P40, and
P55, respectively. Deck deflection profiles for varying prestrains
in the cables are plotted in Fig. 4. It is clear that the deck deflec-
tions and the forces in the stiffening truss decrease with increas-
ing cable prestrain, whereas forces in the cables and suspenders
increase with increasing cable prestrain. It is observed that
smaller cable prestrain 共below 0.1⫻10⫺3) has almost no effect on
the deflections and forces of the bridge.
It is evident that for a prestrain of 0.8⫻10⫺3in both primary
and secondary cables the deflections of the deck are nominal. In
the computer model, the deck deflections cannot be reduced fur-
ther by increasing the prestrain without causing an upward deflec-
tion of the deck at some points. Although the maximum deflection
at the deck center with a prestrain of 0.8⫻10⫺3in both primary
and secondary cables is about 118.9 mm 共4.68 in.兲, it is consid-
ered an adequate simulation of the dead-load deflected shape of
the bridge. Even though this leads to initial stresses in the stiff-
ening truss, the magnitude of the stresses is reduced to a mini-
mum since the cables carry most of the dead load, as is evident
from the forces in the suspenders. The presence of initial stresses
in the truss model is conservative as far as estimating the capacity
of the truss.
With cable prestrain of 0.8⫻10⫺3, the force in the suspenders
of the main span due to dead load alone is typically 146.8 kN 共33
kips兲. This means that of the 166.8 kN 共37.5 kips兲force applied at
each panel point along the bridge deck, 146.8 kN 共33 kips兲are
transferred to the main cable. Thus the use of a prestrain of 0.8
⫻10⫺3in the primary and secondary cables is about 90% effi-
cient in keeping the truss stress free under gravity loading. In
addition, the total 共primary plus secondary兲cable tension of
17,187 kN 共3,864 kips兲determined by the computer analysis is
close to the 15,568 kN 共3,500 kips兲reported by Hazelet and Erdal
共1953兲. Therefore, a model with an initial prestrain of
0.8⫻10⫺3in the cable elements is considered the correct analyti-
cal model.
Another interesting feature of the Roebling suspension bridge
is the inclined stays. In the original design, Roebling felt that the
use of stays was the most economical and efficient means of
providing stiffness to long-span bridges. These stays also carry
approximately 10% of the total bridge load 共Hazelet and Erdal
1953兲. Deck deflection comparisons for the model with and with-
out inclined stays are given in Table 3 and Fig. 5. These numeri-
cal results demonstrate that the stay wires reduce the central deck
Fig. 4. Main-span deck deflections for various cable prestrain levels Fig. 5. Main-span deck deflections with and without stay wires
Table 3. Influence of Stays on Maximum Axial Forces and Main-Span Deflections
Stays
Bottom chord 共kN兲Top chord 共kN兲Cable members 共kN兲Deflection 共m兲
Panel 30 Panel 55 Panel 40 Panel 55 Primary
cable Secondary
cable Suspender Side
span Main
span
Without
stays
⫺78.09 37.06 19.22 2.95 10,121 7,585.6 151.53 0.0253 0.1298
With
stays
⫺213.78 153.62 343.72 ⫺3.69 9,851 7,336.1 146.80 0.0116 0.1180
Note: One panel⫽4.572 m 共15 ft兲.
114 / JOURNAL OF BRIDGE ENGINEERING © ASCE / MARCH/APRIL 2004
deflection by about 55% in the side spans, but only a 10% reduc-
tion is observed in the main span. The results also show that
initial strain in the stay wires only contributes slightly to deck
deflection reduction. Thus, prestrain in the elements representing
the inclined stays is neglected in the analytical model.
An inspection of the axial forces induced in the bottom and top
chords of the stiffening truss shows that the force pattern changes
along the bridge deck under dead loading. As shown in Table 3,
axial force in the bottom chords of the main span goes from
compression at the towers to tension at the span center, while the
top chords follow the opposite pattern, that is, from tension at the
towers to compression at the span center. This force pattern dis-
tribution is consistent with the bending-moment distribution of a
continuous beam subjected to a gravity load with an internal
hinge at the center.
Geometric Nonlinearity
It is well known that a long-span cable-supported bridge exhibits
geometric nonlinearity that is reflected in the nonlinear load-
deflection behavior. Geometric nonlinear sources include 共1兲large
deflection effect due to changes in geometry; 共2兲combined axial
force and bending moment interaction; and 共3兲sag effect due to
changes in cable tension.
Large deflections can be accounted for by recalculating the
stiffness matrices in terms of the updated structural geometry.
Large deflection of a structure is characterized by large displace-
ments and rotations but small strains. Interaction between axial
force and bending moment can be included through the inclusion
of a structure geometric stiffness. Sagging cables require the in-
clusion of an explicit stress-stiffness matrix in the mathematical
formulation in order to provide the numerically stabilizing initial
stiffness. Introducing preaxial strains in the cables and then run-
ning a static stress-stiffening analysis to determine an equilibrium
configuration of the prestressed cables can include cable sag.
Stress stiffening is an effect that causes a stiffness change in the
element due to the loading or stress within the element. The
stress-stiffening capability is needed for analysis of structures for
which the stiffness is a function of the tension force magnitude, as
is the case with cables.
The FE model described previously is used to determine the
large deflection effect on the structural behavior of the Roebling
suspension bridge due to dead loading. Table 4 compares the
maximum axial forces in typical members and the maximum deck
deflection at the span center for small and large deflection analy-
ses. The stress-stiffening capability is included in both analyses to
ensure a convergent solution. It is clearly shown that large deflec-
tions have almost no effect on the member forces and deck de-
flection due to dead load alone. This is consistent with the obser-
vation that the maximum deck deflection of the bridge is very
limited 共about 118.9 mm兲due to introducing prestrain of
0.8⫻10⫺3in the cables, which results in a relatively stiff bridge.
Further comparison between small and large deflection analyses
without cable prestrain, as shown in Table 5, demonstrates that
large deflections do not change the member forces and deck de-
flection significantly, even though the maximum deck deflection
of the bridge reaches about 0.945 m. It can be concluded that the
large deflection analysis is not necessary in determining the initial
equilibrium configuration of the bridge due to dead load and a
small deflection analysis is sufficient, but stress stiffening must be
included. However, convergent 3D nonlinear simulation of the
Roebling suspension bridge with both primary and secondary
cables required a large deflection solution along with the stress-
stiffening behavior with convergence determined using displace-
ments.
In the FE modeling of a suspension bridge, the cable between
two suspenders is discretized as a single tension-only truss ele-
ment 共cable element兲. Truss elements are also used to model sus-
penders and tie rods connecting the primary and secondary
cables, which do not provide sufficient restraints at each cable
element node in the transverse 共lateral or z-axis兲direction since
they are in the x–yplane. This limitation is solved by constrain-
ing the transverse displacement of each cable node to equal the
transverse displacement of the corresponding node at the bottom
chord of the stiffening truss, which should be fairly close to the
physical response of the suspension bridge.
Table 4. Influence of Deformation Analysis on Axial Forces and Main-Span Deflections Including Cable Prestrain
Analysis type
Bottom chord 共kN兲Top chord 共kN兲Cable members 共kN兲
Deflection 共m兲Panel 30 Panel 55 Panel 40 Panel 55 Primary
cable Secondary
cable Suspender
Small
deformation
⫺213.8 153.6 343.7 ⫺3.69 9,851 7,336 146.8 0.1180
Large
deformation
⫺212.3 159.7 344.1 ⫺3.75 9,864 7,337 146.8 0.1177
Note: One panel⫽4.572 m 共15 ft兲.
Table 5. Influence of Deformation Analysis on Axial Forces and Main-Span Deflections Excluding Cable Prestrain
Analysis type
Bottom chord 共kN兲Top chord 共kN兲Cable members 共kN兲
Deflection
共m兲Panel 30 Panel 55 Panel 40 Panel 55 Primary
cable Secondary
cable Suspender
Small
deformation
⫺1,771 720.4 2,831 ⫺37.47 6,993 5,005 101.6 0.9668
Large
deformation
⫺1679, 795.2 2,856 ⫺35.73 7,083 5,050 102.5 0.9449
Note: One panel⫽4.572 m 共15 ft兲.
JOURNAL OF BRIDGE ENGINEERING © ASCE / MARCH/APRIL 2004 / 115
Modal Analysis—Free Vibration
Modal analysis is needed to determine the natural frequencies and
mode shapes of free vibration. A shifted Block-Lanczos method
共Grimes et al. 1994兲in ANSYS is chosen to extract the eigenvalue/
eigenvector pairs. As mentioned previously, the modal analysis of
a suspension bridge should include two steps: static analysis
under dead loading, followed by a prestressed modal analysis. To
investigate the effect of the static analysis and cable prestrain on
the dynamic properties of the Roebling suspension bridge, the
following three cases are considered:
• Case 1: Modal analysis without dead load effect based on the
undeformed configuration;
• Case 2: Prestressed modal analysis that follows a dead-load
linear static analysis, but without the prestrain in the cables;
and
• Case 3: Prestressed modal analysis that includes the dead-load
linear-static analysis results with a cable prestrain of
0.8⫻10⫺3.
A frequency comparison for these three cases is summarized in
Table 6, where self-weight is clearly shown to improve stiffness.
The frequencies in Table 6 show that the inclusion of self-weight
共Case 2兲resulted in an increased transverse 共lateral兲natural fre-
quency of nearly 20%, whereas the increase in the Case 2 versus
Case 1 vertical natural frequency was only about 5%. This obser-
vation shows that the transverse 共lateral兲stiffness of the bridge is
more significantly impacted than is the vertical stiffness of the
bridge. Therefore, modal analyses without a dead-load static
analysis will result in the underestimation of the cable-supported
bridge capacity.
Comparing Cases 2 and 3 shows that the prestrain in the cables
only slightly increased the natural frequencies of the suspension
bridge. Thus, it is prestress induced by dead loading, which con-
tributes significant stiffness improvement rather than the initial
equilibrium configuration. However, the initial equilibrium con-
figuration is essential in determining the dynamic response under
wind or seismic loadings 关for example, Abdel-Ghaffar and Nazmy
共1991兲; Ren and Obata 共1999兲兴.
Case 3 共prestressed modal analysis starting from the dead-load
equilibrium configuration with a prestrain of 0.8⫻10⫺3in the
cables兲is closer to the actual situation and has been implemented
here to evaluate the modal properties of the Roebling suspension
bridge. Since the bridge is modeled as a complete 3D structure,
all possible modes could be obtained. Typical transverse, vertical,
and torsional mode shapes are shown in Fig. 6, and a coupled
transverse-torsion mode shape is shown in Fig. 7. Table 6 shows
that the dominant free vibration modes in the low-frequency 共0–
1.0 Hz兲range are in the transverse direction. This may be ex-
plained by the fact that the lateral load-resisting system of the
Roebling bridge is a single truss in the plane of the bottom stiff-
ener truss chords 共Fig. 2兲, unlike the lateral systems of modern
bridges, which have major lateral load-resisting systems consist-
ing of two lateral trusses. Furthermore, guy wires in the horizon-
tal plane of the lower chords, which were meant to add lateral
stability, are slack and thus ineffective.
Parametric Studies
As mentioned previously, a major advantage of FE modeling and
analysis is in performing parametric studies. Structural and mate-
rial parameters that may significantly impact the modal properties
can be identified through parametric studies. Structural and mate-
rial parameters of the Roebling suspension bridge include deck
Fig. 6. Some typical vibration modes: 共a兲plan view of third trans-
verse mode (f⫽0.614 Hz) ; 共b兲elevation view of third vertical mode
(f⫽1.574 Hz); 共c兲elevation view of first torsional mode (f
⫽1.513 Hz); 共d兲elevation view of second torsional mode (f
⫽2.008 Hz)
Fig. 7. Transverse/torsional coupled mode (f⫽1.546 Hz):
共a兲elevation view; 共b兲plan view
Table 6. Frequency Results
Mode
number Case 1
共Hz兲
Case 2
共Hz兲
Case 3
共Hz兲
Dominant
mode
1 0.152 0.191 0.196 1st transverse
2 0.334 0.412 0.420 2nd transverse
3 0.493 0.599 0.614 3rd transverse
4 0.647 0.684 0.686 1st vertical
5 0.714 0.841 0.869 4th transverse
6 0.879 1.032 1.069 5th transverse
7 1.116 1.243 1.246 2nd vertical
8 1.121 1.294 1.336 6th transverse
9 1.294 1.500 1.513 1st torsional
10 1.488 1.515 1.546 Coupled mode
11 1.518 1.571 1.574 3rd vertical
12 1.561 1.782 1.839 7th transverse
13 1.744 1.989 2.008 2nd torsional
14 1.872 2.004 2.051 8th transverse
15 2.031 2.300 2.314 3rd torsional
16 2.232 2.310 2.364 9th transverse
116 / JOURNAL OF BRIDGE ENGINEERING © ASCE / MARCH/APRIL 2004
self-weight, cable tension stiffness, suspender tension stiffness,
the stiffness of the stiffening trusses, and vertical and transverse
bending stiffness of the deck.
For cables, the cable tension stiffness depends on both the
cross-sectional area Aand elastic modulus E. Incrementing the
cable cross-sectional area implies a larger tension stiffness, which
is supposed to increase the frequencies. However, the cable
weight increases proportionately, which results in reducing the
frequencies. These two effects tend to cancel each other, resulting
in frequencies that remain essentially unchanged.
Both transverse and vertical frequencies increase smoothly
with increasing cable elastic modulus, as shown in Fig. 8, in
which the relative cable elastic modulus E
¯
is defined by E
¯
⫽E/E0where E0is the basic elastic modulus of the deck used in
the initial model. The exception is in the range of E
¯
⫽1.0– 1.5 for
the second vertical 共the first asymmetric兲mode. This observation
is also true for the third vertical 共the second symmetric兲mode. In
addition, variation in the elastic modulus of cables as well as
cable cross-section area resulted in a reordering of the dominant
mode shapes as they relate to the sequential order of natural fre-
quencies.
It has been observed that the vertical frequencies increase
smoothly when the suspender stiffness increases though almost no
variation in the transverse frequencies was found. These results
are consistent with the observation that suspenders of a suspen-
sion bridge provide stiffness in its geometric plane, which is ver-
tical for the Roebling bridge.
For the stiffener trusses, both transverse and vertical frequen-
cies increase with increasing stiffness, as shown in Fig. 9, espe-
cially for the higher modes. A reduction in truss stiffness leads to
modal reordering. Mode numbers of the torsion and higher num-
bered vertical modes increase for the large truss stiffness models.
Results reported in Ren et al. 共2003兲demonstrate that the ver-
tical bending stiffness 共moment of inertia兲of the deck does not
contribute to either transverse or vertical frequencies, even though
the deck vertical bending stiffness is increased fivefold. This re-
sult is consistent with the fact that the deck design does not pro-
vide vertical bending stiffness to the whole bridge. However, in-
creasing the lateral bending stiffness moment of inertia of the
deck does increase the transverse frequencies, as shown in Fig.
10, but does not contribute to vertical frequencies as anticipated.
A variation in the lateral bending stiffness of the deck also leads
to a reordering of the dominant mode shapes in the sequential
order of natural frequencies given in Table 6.
Conclusions
A complete 3D FE model has been developed for the J. A. Roe-
bling suspension bridge in order to start the evaluation of this
historic bridge. From the dead-load static analysis, the prestressed
modal analysis, and parametric studies, the following conclusions
and comments are offered:
1. The static analysis of a suspension bridge is geometrically
nonlinear due to the cable sagging effect. Stress stiffening of
cable elements plays an important role in both the static and
dynamic analysis of a suspension bridge. Nonlinear static
analysis without stress stiffening leads to an aborted com-
puter analysis due to divergent oscillations in the solution.
Large deflection analyses have demonstrated that this effect
on the member forces and deck deflection under dead loads
is minimal. Upon introducing proper initial strains in the
cables, the static analysis of the Roebling suspension bridge
can be based on elastic, small-deflection theory.
2. It has been demonstrated that a suspension bridge is a highly
prestressed structure. Furthermore, all dynamic analyses
must start from the deformed equilibrium configuration due
to dead loading. It has been clearly shown that self-weight
can improve the stiffness of a suspension bridge. In the case
of the Roebling suspension bridge, the transverse 共lateral兲
stiffness increases are much more significant than are in-
Fig. 8. First two vertical frequencies versus cable elastic modulus Fig. 9. First two vertical frequencies versus truss stiffness
Fig. 10. First two transverse frequencies versus deck lateral bending
stiffness
JOURNAL OF BRIDGE ENGINEERING © ASCE / MARCH/APRIL 2004 / 117
creases in the vertical stiffness. Inclusion of dead-load effect
resulted in transverse natural frequency increases of nearly
20%, but the vertical natural frequencies only increased by
approximately 5%.
3. Dominant modes for the Roebling suspension bridge in the
low-frequency 共0–1.0 Hz兲range have been shown to be in
the transverse direction; the lowest transverse frequency is
about 0.19 Hz. This illustrates that the lateral stiffness is
relatively weak: only a single truss is used in the Roebling
bridge.
4. Throughout the parametric studies, the key parameters af-
fecting the vertical modal properties of the Roebling suspen-
sion bridge are mass, cable-elastic modulus, and stiffening
truss stiffness. Key parameters affecting the transverse
modal properties are mass, cable-elastic modulus, stiffening
truss stiffness, and the deck system transverse-bending stiff-
ness. Stiffness parameter variations have been shown to
cause some reordering in the sequencing of the natural
modes of vibration. FE model updating is carried out in the
companion paper 共Ren et al. 2004兲by adjusting these design
parameters so that the live-loaded analytical frequencies and
mode shapes match the ambient field test frequencies and
mode shapes.
5. It is observed that the effect of decreasing the truss or cable
stiffness by 50% does not lead to a significant decrease in the
bridge natural frequencies. This fact points to the importance
of the cables in governing the stiffness of a suspension
bridge.
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