Rainbow Bridge Analysis

Khang Tran, University of Portland, December 16, 2015

Abstract: This report discuss the analysis method and results of the loading scenario that the Rainbow

Bridge in New York has to go under on a daily basis using the analysis tools provided in Autodesk

Simulation Mechanical. Displacement, shear forces, and moments were calculated for the midpoint

node of the bridge, as well as the maximum values for said categories to determine whether the bridge

can withstand the loading of current traffic, comparing to when it was estimated in 1940, the year it was

built.

I. Introduction

The collapse of Hyatt Regency hotel

walkway at the Hyatt Regency Kansas City in

Kansas City, Missouri on July 17th, 1981 has

proven that tiny compromises in the integrity of

structural designs can become fatal destruction

[1]. In addition to that, there are many other

factors going into a design of a structure that

engineers need to consider before construction

can begin. However, due to resources

limitation, project scale, and other

circumstances, many of these factors such as

abrupt changes in weather, supporting types,

and loading scenarios were underestimated or

neglected [2]. This was the main motivation for

the Autodesk Simulation Mechanical (ASM)

software analysis of the Rainbow Bridge (Figure

1), which was built in 1940 when analysis tools

of such type weren’t available.

Figure 1: Rainbow Bridge, Niagara Falls

ASM analysis tool allows users to model

engineering structures in the software, apply

loading scenarios that they think will happen to

the structures, and analyze under those

conditions to determine the stress

concentration, displacement, and failures of

several type at the desired location in order to

adjust accordingly in the actual design. More

importantly, for large scale structures such as

steel bridges, it is crucial to consider all the

aspects that could affect the structural integrity

and loading supports, and ASM provides the

tools necessary to bridge the unknown gaps [3].

The goal of this project is to model the

design of the bridge as well as its corresponding

loading scenario in ASM to provide a better

understanding of the displacement and

stresses, where they occur, and compare the

results to a set of hand-calculation answers.

These details will be mentioned more

specifically in the sections below.

II. Methods

In order to model the design of the

Rainbow Bridge in ASM, actual lengths of the

bridge span and the supporting ends were

determined to be at 950 feet and 202 feet,

respectively [4]. Since the structure is an arch

bridge that is symmetrical about the central

axis, individual lengths of the supporting beams

on one side of the bridge were determined

using conversion factor based on a large scaled

image and the dimensions found above.

At the mid-section of the bridge on the

image, there are 4 beams of the same length,

including beam with index number 12 on table

1, and hence the lengths were assumed to be

4.04 feet. The lengths of the supporting beams

on the other half of the structure, based on the

symmetry of it, were determined accordingly.

Based on the dimensions calculated above,

a model of the bridge was then drawn in ASM

using line option. The design was split up into 3

parts, which are the arcs at the bottom of the

bridge, the supporting beams in the middle, and

the deck of the bridge. The material and

element type were chosen as Steel ASTM – A36

and Beam, accordingly, for all three parts of the

model. For the cross sectional areas of the

parts, the arcs and the supporting beams were

assumed to be made out of hollow rectangular

beams.

The deck of the bridge was assumed to be

supported by W44x335 I-Beam. In addition to

that, the top and bottom nodes of the far most

supporting ends on both sides of the bridge

were assumed to be fixed to represent the

support that the bridge has in reality from the

concrete structures at the river banks that can

be seen in Figure 1.

For loading scenario, the worst case was

assumed to determine whether the structure

would be able to withstand criteria that might

not be considered when it was built in 1940.

The deck of the bridge was designed to have 4

lanes for vehicular purposes [4]. The side length

of a typical semi-truck, along with its weight,

were determined to be at about 75 feet long,

and 80,000 lbs, accordingly [5]. Using the length

of the bridge span found above, a distributed

load of 365 psi, equivalent to 52 semi-trucks all

parking on the deck of the bridge, was used in

the analysis of the structure. Figure 2 provides a

closer look to the ASM model of the bridge

before the analysis began. The actual weight of

the bridge, however, was not considered in the

setting of the distributed load, due to limited

resource on that particular area.

Figure 2: ASM Model with Loading Scenario

After the analysis was completed in ASM,

shear and moment diagrams for the structure

were determined and can be found in the

results section.

The displacement, shear force, moment,

and axial stress were also determined for the

values at the fixed nodes of the far end

supporting beams, and the maximum and

minimum values were also found for said

categories. These outcomes, along with the

comparison with the hand-calculation data, can

be found in the results section below.

III. Results

Table 1 and 2 provide the results from

height and length calculation of the supporting

beams, as well as other sections of the bridge in

which the methods of calculation were

discussed above.

Index Numbers (Starting from left

side of the bridge)

Lengths measured

from image (cm)

Approximated actual lengths

(feet)

1 5 202

2 3.5 141.4

3 2.8 113.12

4 2.3 92.92

5 1.8 72.72

6 1.4 56.56

7 1.1 44.44

8 0.8 32.32

9 0.5 20.2

10 0.3 12.12

11 0.2 8.08

12 0.1 4.04

Table 1: Lengths of Supporting Beams

Height 1 (in)

Base 1 (in)

Height 2 (in)

Base 2

(in)

Supporting Beams

48.48 48.48 30 30

Arcs 193.92 193.92 120 120

Table 2: Dimensions of Parts

The hand calculations for the displacement,

shear forces, and moment were done using the

following equations [6]:

{𝐹} = [𝐾]{𝑑} − {𝐹˳}

Where:

{F} : the concentrated nodal forces

vector

{Fo} : the equivalent nodal forces vector

[K] : the global stiffness matrix

{d} : vector of unknown degrees of

freedom

For a uniformly distributed load acting over

the beam element, {Fo} was chosen as the

following vector [6]:

{𝐹˳} =

{

−𝑤𝐿

2−𝑤𝐿2

12−𝑤𝐿

2𝑤𝐿2

12 }

Where:

w : the uniformly distributed load

L : the length of the beam member

The stiffness matrix was determined using

the following matrix form [6]:

[𝐾] =𝐸𝐼

𝐿3[

12 6𝐿 −12 6𝐿6𝐿 4𝐿2 −6𝐿 2𝐿2

−12 −6𝐿 12 −6𝐿6𝐿 2𝐿2 −6𝐿 4𝐿2

]

Where:

E : Young’s modulus of elasticity

I : Moment of Inertia

L : Length of the beam member

The vector {d} was also calculated under the

form [6]:

{𝑑} = {

𝑣₁∅₁𝑣₂∅₂

}

Where:

v1 : displacement at the node

∅1 : rotation at the node

Figure 3 and 4 display the results for the

shear and moment diagram of the bridge

structure analyzed in ASM, respectively.

Figure 3: Shear Diagram

Figure 4: Moment Diagram

From the equations listed above, the set of

hand calculation results were determined, and

the comparison between those and the ASM

results can be found in table 3.

Categories Center Node

Hand Calculation

Displacement 0.0003185 in

0.00345 in

Shear Forces 66266 lb

128943 lb

Moments 8461 ksi 14500 ksi

Table 3: Data comparison table

Figure 5 displays the stress concentration of

the bridge.

Figure 5: Stress Concentration

Between the results determined from the

hand calculation and those from the ASM

software, it was determined that there was a

huge difference between the two in every

category. The displacement calculated from the

hand calculation, although quite small, was

about ten times as large as the one calculated in

ASM. In addition to that, ASM provided an

insight look to the location of the stress

concentration on the bridge, whereas the hand

calculation did not have this tool.

IV. Discussion

This project has proven to be an interesting,

yet very challenging task to do. The design of

the bridge in ASM has been reduced to a much

simpler form to reduce run time simulation,

which caused the results to distance themselves

from the actual scenarios that the structure

goes under. A lot of assumptions and design

decision were made during the analysis of the

structure, including the material type for the

beam elements, the size of the beams, the

loading scenario, and the simplification of the

process. This was due to the fact that no data

was present during the research process of this

project to determine the correct information

for said categories.

The hand calculation could have been more

extensive. The structure was simplified into one

beam with support reaction at the ends and the

same loading scenario on top, which was

completely different from the modeling in ASM,

and from the actual scenario of the bridge,

which was the main cause to why the answers

from the two methods were so far apart. In

addition to that, the bridge was built in 1940s

and so there weren’t a lot of construction

notes, and data from analysis before

construction broke ground to compare the

answers to.

V. Conclusions

This project has proven that analyzing

structures of huge scale such as steel bridges is

a very daunting task to accomplish. The actual

design of the bridge is very complicated and the

capacity of the software poses a limitation on

how much analysis an engineer can do.

Following the analysis and hand calculation,

the analysis could be broken down into

individual components for separate analysis to

determine the effect of the macro loading on

the structure to the micro components of the

bridge before the analysis of the bridge as a

whole can be executed. After all, designing and

building large scale engineering structures such

as this would require the work of a lot more

people, going into much deeper research and

effort.

VI. Works Cited

1. Troitsky MS. Planning and Designing of

Bridges. Montreal, Canada: John Wiley

& Sons, Inc; 1994.

2. Chen WF, Duan L. Bridge Engineering

Substructure Design. Boca Raton, Fl :

CRC Press LLC; 2003.

3. Xanthakos PP. Theory and Design of

Bridges. New York, NY: John Wiley &

Sons, Inc; 1994.

4. Rainbow Bridge (Niagara Falls).

Wikipedia Web site.

https://en.wikipedia.org/wiki/Rainbow_

Bridge_%28Niagara_Falls%29. Modified

2015. Accessed December 2, 2015.

5. Barker GM, Staebler J. Service Ability

Limits and Economical Steel Bridge

Design. FHWA Office of Bridge

Technology. 2011; 1-

20.https://www.fhwa.dot.gov/bridge/st

eel/pubs/hif11044/hif11044.pdf

6. Logan DL. Development of Beam

Equations. In: A First Course in the Finite

Element Method. 3rd ed. Stamford, CT:

Stamford, CT : Cengage Learning;

2012:166-223.