THE EFFECT OF DIAGONAL STAYS
ON THE
NATURAL FREQUENCIES OF SUSPENSION BRIDGES
by
Charles Howard Kahn
B.C.E.,
North Carolina State College
1948
and
Jack C. McCormac
B.S.,
The Citadel
1948
Submitted in Partial Fulfillment of the
Requirements for the Degree of
MASTER OF SCIENCE
From the
Massachusetts Institute of Technology
1949
Signatures of Authors.....
Department of Civil and Sanitary(
Engineering, May 20, 1949.....
Signature of Professor
in Charge of Research..
Signature of Chairman of Department
Committee on Graduate Stud7 ts....
0
*@
Contents
Acknowledgment................................ page 1
2
..................................
I. Summary
Foreword.................................*..
3
Purpose....................................
3
Model
Used.. ..... ..............
..........
Scope......................................
Results
...
5
...............................
Recommendations
4
4
........................
Method..........
3
5
...........................
II. Historical Background...................4..
6
III. Theoretical Considerations.................
13
Nomenclature...........................
14
Energy
16
Method.........
Exact Method...
...........
....
. .*...................
Effect of Stays......00.00...
IV. Procedure....
.......... .......
........
..........
16
17
20
Dimensions of the Model................
21
Description of App.ratus...............
23
V. Results and Recommendations................
37
Repetition of the Work With a
Single Truss.............
38
Work Culminating in the Failure
of the Eridge. 0. 0. .
0 0 ...
0
39
30 454
I
u----*
-,
-
Final Completion of Data.................page 43
Suggestions for Further Work.............
Bibliography
Appendix
. . ..
:....
Appendix B...
...
...
. ..
........
..........
. ..
..
..
. ..
55
...........
...
...
.....
47
56
59
___-~
Figures
1
General View of the Model....................page 25
*
..
& 0
0*
0
0
#
26
0
0
28
3
Tachometer Circuit.............
4
Vibrating Mechanism............
0*
5
Driving Mechanism..............
00
6
Flexible Coupling, Type I......
0*
7
Flexible Coupling, Type II.....
8
Diagonal Stay, Type I..........
9
Diagonal Stay, Type II.........
29
0
29
0*
30
.
30
..
32
00
33
*0
..
..
10
Bridge After Failure...........
11
Bridge After Failure...........
12
Bridge After Failure...........
13
First Mode, Single Truss.......
00
14
First Mode Shape, Double Truss.
@0
15
First Torsional Mode Shape.....
16
Second Mode Shape..............
17
Second TorsionalMode........................
0 0
0*
42
*0
42
..
42
0 0
.
50
#
51
52
0
53
0000*00a
54
Tables
I
Comparison of Theoretical and Experimental
Frequencies...**..........
*
39
II
Effect of Stays on Frequencies...............
45
III
Comparison of Experimental and Theoretical
Frequencies with End Stays...........
46
If
Acknowledgment
The authors wish to express their sincere
appreciation to Professor C. H. Norris for his
suggestion for the topic for the dissertation
and for his continuous help throughout the
investigation.
They also wish to express the
indebtedness to Mr. Donald Gunn without whose
suggestions and technical assistance the model
could not have been successfully assembled.
LLMR
I. SUMMARY
0
Summary
Forevard
For quite a few years a controversy has raged in
engineering circles concerning the theory behind the
design of suspension bridges.
There were proponents
of both the "Flexible" and the "Rigid" schools of design
thought.
Until the failure of the Tacoma Narrows Bridge,
however, both theories were considered applicable to safe
design.
With the failure of the Tacoma Bridge, suspension
design was jolted into the classification of a dynamic
structure, and a flood of research was precipitated whici'
had as its purpose the investigation of the behavior of
suspension bridges under dynamic loads.
Purpose
The purpose of this investigation was to check the
accuracy of the theoretical expressions for tie mode shapes
and natural frequencies of suspension bridges developed
by Edward C.
Holt,
Jr.,
in his Master's thesis at M.I.T.
1
The effect of two types of diagonal stays on both the
frequency and the amplitude of vibration, including mode
shape determinations, was also investigated.
Modei Used
The model previously used in the investigation by Holt
1 "The
Effect of Diagonal stays on the Natural Frequencies
of Suspension Bridges" by Holt, 1947
4
and originally constructed by James W. Greely and W. Carter
McClure2 was used as the basis of an expanded model which
consisted of a two-cable bridge instead of the original
one-cable model.
The same physical constants were maintained
for the new model with the exception of several small changes
in cable and suspender dimensions.
These dimensions were
changed in order to improve the dynamic characteristics of
the model.
Scope
An attempt was made to determine the frequencies and mode
shapes for the same modes as had been investigated by Holt.
however, because of the action of the bridge, it was found
possible to get only the first two natural and torsional
modes.
These shapes were measured under three conditions:
vithout stays;
with diagonal stays running from the tops of
the towers to the trusses; and With diagonal stays running
from the towers at the elevation of the roadway to the cables.
Methocd
The vibrations were impressed upon the bridge by means
of a rotating eccentric veight driven by a DC motor.
The
weights were attached directly to the truss of the bridge.
To measure deflections, vibrating reeds were mounted on
2 "The
Determination of Suspension Bridge Stresses by Model
Study" by Greely-McClure, 1938
Ames Dials with micrometer controls at the bottoms.
The
frequency of the vibration was controlled by the use of
a rheostat in the motor circuit.
Results
Because of the paucity of data,
it
is impossible
to draw any all-inclusive conclusions concerning the
validity of the formulas derived by Holt.
That data
which was taken checks the theoretical values obtained by
the exact method fairly well.
Because of the time consumed
in the construction of the model and because of the behavior
of the model in the range where further data was wanted,
it was not possible for the authors to procure the volume
of data which they had hoped for.
Recommendations
Additional verification of the theoretical equations
With slight additions to and modifications
is imperative.
of the model,
these data could be achieved.
The derivation
of a theoretical equation for the torsional frequencies is
also warranted.
II.
HISTORICAL BACKGROUNID
-
- -- -
- -----
<----~---~--~
-~
In the past several decades, during the development
and perfection of suspension bridge design, two divergent
channels of thought have grown up concerning the basic
premise of design.
One S&hool of thought holds that sus-
pension bridges should be made as flexible as possible,
the other that they should be made, to a certain extent,
rigid structures.
Although there were sufficient examples of bridge action
and even bridge failure to warrant a closer scrutiny of the
problem,
the suspension bridge was not jolted into its true
classification as a dynamic structure until November 7, 1940,
when the Tacoma Narrows Bridge failed.
The Tacoma Narrows Bridge was built across Puget Sound
near the city of Tacoma, Waashington, to connect the Olympic
Peninsula with local highways.
The Sound is 4,600 feet vide
at its narrovest point, and the waterway is deep, with swift
tidal currents.
1940.
The bridge was opened to traffic on July 1,
It had a cehter span of 2,800 feet and two side spans
of 1,100 feet each.
There were also plate-girder approach
spans which,
with the anchorages,
5,939 feet.
The roadway was twenty-six feet vide with side-
made the over-all length
valks 4 feet 9 inches vide on each side.
This gave a total
vidth, center-to-center of stiffening girders, of 39 feet.
The main towers were 420 feet high, and were composed of
two shafts, each uniform in section, connected by horizontal
struts and fixed at the bases.
The shafts were spaced thirty-
nine foot centers at the top and fifty foot at the bottom.
The depth of stiffening girders was 8 feet, giving a ratio of
span length to depth of stiffening girders of 350 to 1, an
extreme ratio for suspension bridges, and one which made the
bridge by far the most flexible of the modern suspension
bridges.
The deck weighed 110 pounds per square foot.
The
suspender cables had an average test strength of 165,000 pounds
per rope.
They had a factor of safety of approximately 6.5
under dead load.
Laterals were arranged in K-system and
were designed to resist, in conjunction with the main cables,
a wind pressure of 620 pounds per lineal foot of bridge.
For several days before the failure of the bridge, motions
of the structure were noticed and were under investigation.
Motions in the side spans were damped by means of 1 9-16 inch
cables connected to a point on each side span and to the ground.
The cables extended from the deck of the side spans to anchors
consisting of 50 cubic-yard concrete blocks placed 30 feet out
from the anchorages.
Methods of damping the center span
were under consideration.
On November 7, 1940, at 10 A.M.,
however, before a decision could be reached, the center
span of the bridge developed an extreme torsional movement,
and, soon after, the span failed.
Prior to November 7,
the girder motion was vertical
and undulating, both sides of the girder moving up and
down in unison.
The span oscillated in two loops with a
node at midspan, the portions on either side moving in opposite phase.
seconds.
The period of the motion was approximately 12
On the day of the collapse, the mode of motion
started off as usual, but rather suddenly the twisting mode
developed,
the sides of the girder moving up and down in
opposite phase,
as well as the two halves of the main span,
with a faster period of about 4 seconds.
The two cables
were executing longitudinal "rolling" or pendular oscillations which,
just before the collapse, were also in op-
posite phase.
This gave rise to the angular twisting motion
of the deck which increased until the transverse section of
the roadway inclined at an angle of 45 degrees to the horizontal.
The wind on the day of failure was 42 miles per hour.
This was not unusual, as the span had survived higher winds.
This wind, however, caused a vertical wave motion that developed
the lag or phase difference between the opposite sides of the
bridge.
The immediate cause of the failure of the bridge was the
slipping of the center cable band on the north side.
In the
view of the FWA board of consultants, the critic~l slippage
was probably caused by the short diagonal ties which connected
the center band to the stiffening girders.
This slippage,
it is believed, initiated a torsi6nal vibration which, once
established,
tended to increase.
This torsional movement
caused bending stresses in the concrete floor, stressed
thbe structural members beyond their elastic limits, and
created impact loads on the suspender cables under which
one of them snapped.
A progressive snapping of the cables
then followed.
The suspender cable ends jerked high in the air above
the main cables, while sections of the floor system several
hundred feet in length fell out successively, breaking up
the roadway toward the towers until only stubs remained.
The tie-down cables,
placed on the side spans to damp out
the vertical vaveimotions previously observed on the bridge,
are credited with having prevented the violent waves developed
in the center span from being transmitted to the side spans,
After the failure of the center section, however, the check
reins became inoperative.
This permitted a movement which
buckled the stiffening girders and deformed the steel in
the floor system.
The side spans sagged 30 feet after the
center section of the deck fell,
but despite this sag,
their
5 1/4 inch concrete roadway slabs remained intact except
where broken by impact at the transverse joints.
Because
of the unbalanced pull of the side-span cables, the towers
were pulled back 12 feet toward the shores.
An 18-inch
transverse kink was found in one stiffening girder of
the Tacoma side span after the failure.
The failure of tile bridge exposed the inadequacy
of the conventional method of designing suspension bridges
against wind action.
It inaugurated a new era in which
dynamics of wind action was forced on the attention of the
Civil Engineer, and the rigid school of design thought
was thrust to the forefront.
Strangely enough, the Tacoma failure was not the first
of the kind to be recorded.
On May 17, 1854, Charles Elletts
suspension bridge across the Ohio River at Wheeling was
destroyed by wind forces.
Eyewitness accounts follow remarkably
well the same procedure as in the Tacoma failure.
In addition
to this, quite a few other bridges, not of the suspension type,
had failed in storms, obviously because of wind loads, and
reports of some of these also follow the pattern of the Tacoma
failure.
In addition to this accumulation of evidence, Ethe BronxWhitestone Bridge in New York had evidenced undulatory motions
in winds which led to corrective measures being taken that
resulted in damping out the vibrations on that bridge.
It
seems almost inconceivable that these portents of trouble
should have been ignored.
The failure of the bridge inaugurated a feverish period
of activity in experimental analysis of suspension bridges.
Research was carried on at The University of Washington,
12
Northwestern University and Virginia Polytechnic Institute,
and by several private consulting engineers.
These inves-
tigations have led to improvements in design and have
broadened the thinking in the suspension bridge field
considerably.
The investigations continue, and it is to
this 'groving library of experimental data that te authors
humbly add the fruits of their labor.
III,
theoretical Considerations
14-
Nomenclature
All the quantities listed in the table below are in
the fundamental units of feet, pounds, slugs, and seconds.
a1 ,a2
Absolute amplitude of the two ends of a stay
am
Maximum amplitude of Vibration (energy method)
Ac
Cross-sectional area of the cable
As
Cross-sectional area of a stay
A,B,C,D
Constants of integration
ClC2,C3,C4
Constants of integration
E
Modulus of elasticity of the stiffening truss
Ec
Modulus of elasticity of the cable
Es
Modulus of elasticity of a stay
h
Cable sag
A h
H
Change in cable sag due to AH or mode shape
Horizontal component of cable stress due to
dead load only
Increment of H due to inertia loads
I
Moment of inertia of the stiffening truss
k
Frequency function defined in equation (4e)
K.E.
A K.E.
Kinetic energy in the mean position
Increment in K.E.
due to a stay
1
length of center span
1
length of side span
length of a stay
AI-
Change of length of the cable in the- extreme position
A Ls
Change in length of a stay in extreme position
LT
Constant of the structure defined in equation (5a)
in
Mass per unit length of the main span
n
Integer used to denote, the number of vaves
in mode shape
N
Frequency in cycles per second
A(N2)
Change in N2 due to a stay
P.E.
I P.E.
Potential energy in extreme position
Change in P.E.
due to a stay
=
r
Differential operator,
v
Maximum velocity of vibration at any second
V
Weight per foot of main span
A V
Inertia load per foot of main span
x
Distance along the main span
Measured from one end for the energy method
Measured from the center for the exact method
y
Dead load deflection of cable from cable
chord at any section
Constant of the structure defined in eq. (4c)
P
Frequency function defined in equation (4d)
I
Frequency function defined in equation (4a)
{
Slope of the backstays
&
Frequency function defined in equation (4b)
G
Maximum displacement at any section
Slope of a stay
W
Frequency in radians per second
A&A) Change in V2 due to a stay
-I
The equations listed in this section and the assumptions
used in the derivations of these equations can be found
in the thesis by Holt.
3
For convenience of reference,
however, the equations so derived will be repeated here,
and, vith the table of nomenclature on the preceding tvo
pages, should enable the reader to follovw vork performed
in this thesis.
Method
Energ
Assuming that the shape of vibration of a suspension
bridge is a sine curve and that the span vibrates in n
then
equal segments,
This equation has been derived under the assumption
that AH is equal to zero.
Since this is true only for the
antisymmetric modes where n is even, it becomes necessary
to derive another equation for the symmetric modes vhere
A H is not zero.
For this condition, the following equation
applies:
t jl~
~4
n4
rIh(~ZnII
El
Exact Method
In the exact method,
Y(=A
3 Holt,
op. cit.
in Sx()
for the antisymmetric modes,
In this equation, "A"
is an arbitrary constant and
This equation defines
is defined in equation (4b) below.
mode shapes identical with those assumed in the derivation
of equation (1), and, therefore, equation (1) exactly expresses the frequencies of the antisymmetric modes.
For the smetric modes, equation (3) becomes:
-
-
aa-
8
(4)
fx
SA52
05k
Cc0; z
c .
+S'
COA z
co~(4a)
.(4b)
ZFc=
2(4c)
4 +4mEILJL
(A-)
21-I
8kA 4
(4a
Note that this equak is valid only when A H has a
value.
by trial,
The equation for frequency, which must be solved
A plot of the right and left
is listed below.
members against
for the model under consideration is
given in Appendix (B)
of Holt's thesis.
k2
Err
~-4C'64
LrLA~sCa
r)v-i
(5)
r
Gh
-
(Q
Ef fect of Stays
By the use of the energy method,
an expression for the
cbange in frequency due to the use of one diagonal stay was
derived.
It is:
F.2
4Holt, op. cit.
AsI ,-coj
si
(6)
r-19
The assumptions made in deriving were admittedly
seriously in error, and a more accurate set of equations
was developed by Holt in work subsequent to his thesis.
For the antisymmetric modes:
b5 in n'
in !EY/
L
+SI
_ smSx
+
T sinh r
5Wnh Irj
inkFo i 0!5 x < x.
-F-, 4- 5n SxX)-(Sl
_
-xnh
* in
p_?
rx-x
y~b -x
.
Foe X"x.x
z
The corresponding frequency equation is:
5i n
smri6&
-x,)
2-()smh Yx~sInh (i-K.) g
_
i
s \ n/ sih
_)__
sin
4
For the symmetric modes:
_
i
sin
__sin
Sx
{
Cos x si
C05 I
yi
0c
(
-
A
S(12
-cos
Jx
A-~
Cos
Cos
Y-x*
cosh -rx
Fow- 0n
La05~
cosh
x.)
I
Cash r
+:)_
sin Y
N(Y +Y)
4: inh
( Y)cosh
)
LY+.) c05
(9)
s"nh
'r
Szcosh rx
cor) rL
cosh
x !S x.
X)
P
l
The corresponding frequency equation is:
=
-
~k
(i 0)
(1)
~
X
(l=+x-~,
_L5
co
2.
:
-
cobh
rx.
(10 Q)
'u/I cosh 4
(SOC+6) GI
A5 E5
cos 4
SCosh
cL+h
"(x- sinh r(i -x
C051
+ tan st -4
rn14 LT (Wn)
7
)
(to 6)
('Oc)
19
Additional nomenclature introduced by these exprestions are:
x,0
Distance from the center of the span to
the point of attachment of the end stay
to the stiffening truss
V
Frequency function defined in equation (10c)
X
Frequency function defined in equation (10a)
Frequency function defined in equatio.n (10b)
Fs
Axial stress in a stay for maximum deflection
20
IV. PhOCEDTRE
r
21
Dimensions of the 16del
Length, center span.....
.D
.....
.
..
*.........12t-
.
Length, side span.......
.
Height of towers........
Elevation of stiffening trus
Cable
.
0..
.31-0"
....
0..
ounda t ons
from ff
s
sag............
000
Cable diameter..........
00000
...
0
...
0
...
0
...
4
**e*OtO7l
***s*
.0
Hangers, 2 wires@.......
82-5
......
S .
00000
o
.....
0o000
... 000...
O.1.
0..
.0
Width of roadway........
0.0.
***.........3''76
.0
Moment of inertia of the trus So0
0.0..
0 0 0 0
0.00l9o8"4
.....
O. 0
.0
Modulus of elasticity off the cable.
.0..00
...
0
.29,500,000 psi.
Modulus of elasticity off the truss.
000.0
...
0
.29,500,000 psi.
...
0
0.0.0.000
Total weight of the main span .0 0 0 00
Stays:
.....
(Tower to truss)
Length
...
Diamet er, 1 wire
Slope
00.0
.
0
0 0
0 0
00
0
0
00
.0..0000.000000000000
00
0
0
000
0 000.00-0-0
0
Points of attachment to truss from
00000
end of bridge ......
Stays:
93.4
. .
. . ..
--
- - --
0
0
.
.
.
...26 1/8"
.... 0
.0*014
0.**33-40
.24" from
tower
000-----.-
(Tower to cable)
- - . . . . 00000.25"
Length
Diameter, 1 wire@................. - - - - - Slope ............
-
-
-
-- --. -
0
000...00-*
.23*
Points of attachment to truss from
- ----.---. At tower
.......
end of bridge
Points of attAchment to cable (cable
height)..... .0.00000000.10" 2
202
In addition to the preceding dimensions, the following
constants of the model are given
AC
= 0.003959 sq. in.
H = 116.8 pounds
V
= 7.57 pounds per foot
'; = 38050?
m
= 0,235 slugs per foot
As=
0.0001539 sq. in.
99'
Procedure
Description of Apparatus
The model of the suspension bridge used in this investigation already existed as a single-cable model which
had been constructed by James W. Greely and W. Carter
McClure.
A discussion of the design and construction
of the bridge can be found in their thesis.
5
The same
moedl had been used by Edward C. Holt, Jr., for the first
portion of the investigation which the authors have continued in this thesis.
in his thesis. 6
Holt
t
s work and results can be found
Holt also used a single-cable affair but
suggested that for future studies, especially for studies
of torsional characteristics, it would be desirable to construct the complete model of a two-cable bridge.
This was
done, holding the dimensions of the second cable system
equivalent to those of the existing system.
The towers
were constructed of the same aluminum alloy used in the previous work,
and were welded together in two places above
the elevation of the roadway,
the roadway.
8 inches and 16 inches above
An attempt was made todevise a road system
to connect the two sides of the roadway with the hope that
the hinges at the bottoms of the towers could be eliminated
and replaced by rollers at the tops of the towers.
In the
opinion of the authors, this condition would more nearly
approximate the action of a true suspension bridge.
5 Greely
6Holt,
and McClure,
op.cit.
op.cit.
It was
found, however,
that the torsional resistance of the stif-
fening trusses themselves was so great that if anything
were added between the two sides, the possibility of developing torsional vibrations was very small.
It was finally
decided to omit any floor system, leave the hinges at the
bottoms of the towers, and thereby transmit the vibrations
from one truss to the other mainly through the towers.
The
size of the main cables was reduced slightly because of the
unavailability of the size used in the previous model.
In
the case of the hangers, .a different problem was encountered,
The original hangers used by Greely and McClure were high.carbon, cold-drawn wire 0',009 in diameter.
by Holt,
Therefore,
As previously noted
these snapped quite"Peadily under sustained vibrations.
he replaced them with a low-carbon, hot-drawn wire
O'.'028 in diameter.
This was found to be unsatisfactory by
the authors because the wires were given to easy crimping.
This changed their lengths and, consequently, the stress they
carried.
It also had a slight damping effect on the vibrations.
In the present model,
piano wire 0','014 in diameter was used,
suspended from the cable in the same way as were the previous
wires.
These wires proved very satisfactory, being very strong
and not easily crimped.
In the previous model, the dead load
weights bad been hung from the suspender wires below the connection of the roadway.
two-cable model.
This was very unsatisfactory for a
During large vibrations, the weights would
swing together and disrupt the mode of vibration.
Also, at
25
higher modes,
where the bridge vibrations were in excess of
the acceleration of gravity, the suspenders would go slack,
thereby taking their load off the cables.
This condition
was corrected by connecting the weights directly to the floor
system, and, at their lover extremities, transversely to
eadh other by means of wire rods.
The dimensions of the model
used are given in tabular form on. pbp 19 and an over-all
view of the model is shon in figure (1).
Filure
1
An eccentric weight attached directly to the stiffening
truss and driven, through flexible couplings, by a variablespeed motor through a gear box was used to vibrate the model.
Holt, in his investigation, used a Wisconsin, series-wound AC
motor, type A.
This was found to be entirely unsatisfactory.
Over the entire range of operational voltages, speed control
wA5 VMMY ODPr-ICVIPr TO MAjWTAIN,
resulted in erratic data.
&wDO
%S LACK or- COWTV.O..
In the lov range of voltages,
the power output was very poor, and it was found to be impossible
to drive all three gears in the gear box simultaneously.
This is very understandable, since the motor was rated at
110 volts and was receiving, at most, 16 volts.
It was
apparent that a DC motor was the easiest and most satisfactory method of correcting both of these difficulties.
The entire AC circuit was, therefore, replaced by a much
simpler DC circuit vhich proved very satisfactory.
series-vound,
was used.
A
GE DC motor producing one-fortieth HP at 1800 RPM
A schematic diagram of this circuit is shown in
Figure (2).
_
0
DC
Molor
'
Filure 2:tMOTOK CONTROL
ClkCULT
At first it was thought necessary to connect a gear box
to the motor drive in addition to the existing box in order
to develop full power without the accompanying disadvantage of
high-speed operation.
Upon investigation, however, it was
found that the rheostat control was sufficient, when used with
the present drive gear box, to provide sufficient power over
the entire range of operating speeds.
set-up was satisfactory.
Speed. control on this
An improvement that could have been
made, however, would have been to have placed a smaller
rheostat in the circuit to act as a micrometer adjustment for
pover settings.
It
was very difficult to achieve small adjust-
ments with the large rheostat alone.
The same method used by Holt to measure frequencies vas
employed by the authors.
The system consisted of a magneto-
voltmeter tachometer constructed to run off the rear end of
the motor shaft.
This tachometer consisted of a Delco 27-volt
DC shunt motor and a Triplett voltmeter, model 321.
A constant
voltage was maintained across the field, and the voltage generated
by the armature was measured.
proportional to the speed.
For a DC motor this voltage is
The tachometer Was calibrated
against a Weston tachometer, model 44.
is given in Appendix (A).
This calibration data
A schematic drawing of the tachometer
circuit is shown in Figure (3).
As in the previous work on the model, the eccentric veights
were connected directly to the truss of the model.
consisted of 1/2" square steel blocks.
The weights
These blocks were mounted
at varying eccentricities and in various combinations to provide
readable deflections.
-
---
-
,------
~
-----------------
-
-~
2i~
G
1oDC
200mCL
volt0
ef
ma~e~oH
M
L
Fijure3 : TACHOMETER CIRCUIT
in the natural and torsional modes.
Unfortunately, this
procedure precluded the possibility of comparing relative
magnitudes of vibration under the same exciting force.
Comparative magnitudes are, therefore, limited here to
those that could be made by eye.
The eccentric was driven
through a flexible coupling and a gear box which provided
gear rations in addition to direct drive of 1:2 and 1:9.6.
A photograph of the driving mechanism is shown in Figure (5)
and a diagram of the gear box is shown in Figure (4).
Two
different types of flexible couplings were used between
the gear box and the eccentric.
At low frequencies, it
was found that any type of connection that permitted torsional
movements in the coupling built up pulsating-type forces which
destroyed mode shapes and made readings impossible.
In this
speed range, therefore, a rigid connection such as that
illustrated in Figure (6) was used.
At higher frequencies,
29
FI ll l l
3'
r?
_
_
I
I il l
f1x5e
ecceniric
C
w"i Il
6 1
bearin U0A
fallacged
lo
Iruss)
Fure4: VIBRATING MECHANISM
it was possible to use stiff wire shafts sucb as the one
shown in Figure (7).
Between the two trusses, whenever they
Filure 5
30
vere connected, a coupling of the type shown in Figure (6)
wvasused.
To achieve torsional vibrations, the eccentric
veights on each side of the bridge were mounted 180 degrees
out of phase with each other.
Fi9 ure 6
Amplitude readings vere made by using Ames dials with
vibrating reeds mounted on their tops and with screw attachments
at their bases.
Dials with 0"001 readings were used.
Filure
7
After
repeating some of the experiments conducted by Holt in the
previous work on the bridge in order to get the "feel" of
taking data and also to check the respective accuracy of
the operators, it was felt that much was lacking in the procedure for measuring deflections, both as to the ease of measurement and as to the accuracy of measurement.
It was thought
by the authors that an electric contact system would prove
the best solution to the problem.
An attempt was made to
use a Magic-Eye tube to indicate contact between the reed and
the bridge truss.
To do this it was necessary to impress a
voltage across the bridge.
This failed because the bridge
had just been painted with aluminum paint which served effectively as an insulator for the truss.
The authors con-
sidered that enough time was not at their disposal to remove
the coat of paint from the members of the bridge.
In the future,
however, this method would undoubtedly prove much more satisfactory than the vibrating reed procedure.
A great disadvantage
of the reed technique was that if the reed were allowed to strike
the truss with any more than a very light blow, a vibration of
the first mode would be superimposed upon the mode in which
the bridge was vibrating.
When this happened, readings had
to be suspended until the extraneous mode damped out.
Two types of stays were used on the bridge.
The first
consisted of diagonal stays connected from points near the tops
of the towers to points on the truss.
The point of attachment
32
to the truss was made as near as possible to the position
of maximum displacement so that the effect of the stays on
the mode shapes would be as great as possible.
This vas,
of course, limited by the fact that attachment vas dictated
by the location of suspender connections in order that the
attachment could be made without changing the structural
properties of the truss.
The second set of stays consisted
of diagonals running from the towers at the level of the
roadway to the cable above the points of maximum deflection.
In the opinion of Dr. D.
B. Steinman 7 , these would have the
greatest effect on damping out vibrations.
A viev of these
tvo connections can be seen in Figures (8) and (9).
Fiure 8
7 Dr.
D.B. Steinman, "The Romance of Bridges," MIT lecture,
January, 1949
33
An initial tension vas jacked into the stays in
an attempt to prevent them from going slack during any
period of the vibrations.
unsuccessful.
This, to a great extent, proved
The amount of initial tension placed in. the
stays was limited by the fact that high tensions tended
to lift
the truss from its horizontal position vith the
result that the suspenders in the vicinity of the point
of attachment of the stay vould go slack.
A happy balance
between stay tension and suspender tension had to be found,
Filure
9
and this balance did permit the stays to go slack at times
during the more violent vibrations.
The same wire was used
for the stays as was used for the suspenders.
The stays were
connected to the truss and to the towers by pinned toggles,
and to the cable over the same connection which was used for
the suspenders.
It was found that the tension in the stays
which ran from the towers to the cable tended to pull the
suspender connection over which the stay was hung up the cable,
thereby increasing the tension in the suspender at that point
while reducing the tension in the stay.
This was prevented
by placing holding clamps above each point of connection
on the cable.
All of these details can be seen in Figures
(8) and (9).
The method of securing Ames dial readings, which proved
at times very difficult, warrants a more detailed explanation
than has been given previously.
In the work done on the model,
with one cable, frequency determinations were effected by
taking one of the Ames dials along the bridge and determining
simultaneous deflection and frequency readings.
This was
done by setting a frequency by the use of the rheostat control
and screwing up the Ames dial until the vibrating reed first
ticked the bridge.
In the determination of mode shapes, however, it was found
that,
because of the variation in the amplitude of the bridge
during the mode under investigation, it was necessary to use
one Ames dial as a base reading,
taking a reading on it
siiul-
taneously with a reading on each of the other Ames dials.
In this way, relative amplitudes were found and a true picture
of the mode shape could be determined.
As has been stated
previously, because of the lov structural damping in the first
mode, care had to be taken not to strike the bridge too hard
a blov vith the vibrating reed in order that the mode being
measured would not be corrupted by extraneous modes.
The determination of frequencies for a single-cable
model was a very simple matter.
For a forced vibration, as
the frequency approaches one of the natural frequencies of
the bridge, a sharp increase in the amplitude of vibration
is encountered.
Hence, all that was necessary in this case
to determine the frequency was to take a series of readings on
one of the Ames dials over a range of frequencies.
By plotting
this data and picking off the peak point, the frequency could
be determined.
Unfortunately, this was not the case with the
two-cable model.
It was discovered early in the series of
test runs made on the latter model that the maximum amplitude
of vibration rarely fell on one of the natural frequencies
of the bridge.
The authors were unable to find a way to determine
the natural frequencies other than direct obversation of the
behavior of the bridge and the relationship between the speed
of the eccentric veight and the vibration of the bridge.
This,
at first glance, seems to be a very unscientific method of
determining the frequencies, but in the tests performed, it
proved surprisingly satisfactory.
This was attested to by
the maintainance of a constant phase relationship between the
two sides of the bridge and constant frequency readings.
In
future tests conducted on the bridge, however, it would be
desirable to find a more exact method of frequency determination.
Q6
As has been mentiohed, mode shapes were deterined
by making simultaneous readings on a base Ames dial and
on each other dial in the system.
This was of the utmost
importance with the AC-drive system on the eccentric, since
constant frequencies were very hard to maintain.
It was
of a lesser importance with the DC-drive system, and it
is the opinion of the authors that with the addition
to the driving circuit of a micrometer rheostat as has
been suggested previously, this procedure could be eliminated,
greatly simplifying shape readings.
In order to plot the
mode shapes, each amplitude reading was divided by its corresponding control point amplitude reading.
This gave a
table of relative amplitudes of vibration for various points
along the bridge with the base reading as 1.000 which, upon
plotting, yidded the required mode shapes.
37
V.
RFSULTS AND 1ECONMENDATIONS
The experimental work on the model was actually divided.
into four separate phases.
These were:
1) Repetition of the
work with the single truss; 2) Construction of the present
model; 3) Work culminating in the failure of the bridge; and
4) Reconstruction of the model and final completion of the
testing program.
With the exception of the second step which
has already been discussed in detail in Section IV, these
steps will be discussed separately.
Repetition of thd Work vith aISingle'Tiuss
In running the experiment conducted by Holt, all that
was expected to be achieved by the repxtition of the data
was the achievement of some sort of facility in the taking of
data.
Because of the state of the equipment, which had not
been used in several years, and because the proposed construction
program was quite extensive, it was decided that readings for
the first mode only vould be taken.
The resulting frequency
for the first mode was 2.06 cycles per second.
This does not
compare too favorably vith Holt's experimental value of 2.48
c.p.s. nor with his theoretical values of 2.46 c.p.s.
In the
opinion of the authors this discrepancy is due entirely to the
poor state of the model during the original readings, and
this conclusion is borne out by the results of the experiments
conducted with the new tvo-cable model.
The mode shape curve
for the first mode may be seen in Figure (13).
This shape curve
iblt, but this can be
differs from the shape curve given by
attributed to the same reason as Was the discrepancy in
the frequency.
Work Culminating in tIie Feilure of the Bridge
From the very beginning of the work with the two-cable
model, troubles were encountered which tended to slow up
and, at some times, to completely halt the testing procedure.
The first
such obstacle was the problem of the driving mechanism.
It was impossible to drive all three gears in the gear box at
Consequently, the unused gear had to be detached
the same time.
The determination of the first natural
during each vibration.
and torsional modes proceeded uneventfully.
A comparison of
the experimental and theoretical values for the first natural
There was no'existing theoretical
mode may be seen in Table I.
equation for torsional modes and this will have to be developed
at some future time.
Tao/e I
Ccrmpari on o F Theo rek&/ca/
'e Xpe
e
'r7en-la/
rlece
a nd
.p
T/?2eor-e Ica
xac t-
,nF~
Me*bod
2.1/
.2
2.'YG
.
Pe rcep?
EIs,
#-o
2.o0?
29./_I-
y
MeHod
Er
2.'6
perce nt
-
As can be seen, the results were fairly good.
When the
frequency of vibration was increased to the point where
the first torsional mode was expected to be, no mode
was encountered.
By varying the frequency over quite a
range, the first torsional mode was found to exist at a
frequency below that of the first natural mode.
This
completely amazed the authors, as the result was definitely
not what was to be expected.
In the light of what happened
later, it is not very unusual that this should have happened.
With the completion of the first torsional mode, the frequency
was increased until the bridge vibrated in the second mode.
Very shortly after commencing this vibration, it was noticed
that the violence of the vibration had shaken the Southern end
foundation loose from the floor of the laboratory.
This, in
effect, made the tests experiments on the vibratory characteristics of a simple beam with some wires strung on it.
No
permanent method was discovered to anchor this foundation
to the floor.
The temporary method used was to load the
beam above the pier with weights until it remained stationary
during the vibrations.
The next thing noticed was a peculiarity
in the shape of the vibration of the second mode.
The West
truss of the bridge vibrated very well in a mode shape resembling that of the expected shape of the mode.
however,
behaved in a most peculiar manner.
The East truss,
While the two
ends of the East truss vibrated in phase with the respective
ends of the West truss, the center of the East truss remained
practically stationary.
After trying unsuccessfully to correct
this condition, Prof. J. P. Den Hartog of the Mechanical Engineering Department of M.I.T. was approachled in the 1nope
that he would be able to determine the difficulty.
Prof.
Den Hartog kindly consented to inspect the bridge.
It was
his opinion that, while the two trusses appeared not be
vibrating in the same phase, they actually were.
Vhat made
them appear to be otit of phase was the fact that the amplitude of one truss was large with respect to the distance
between the trusses and also with respect to the amplitude
of the other.
As Prof. Den Hartog explained this, it was
due to the fact that it is practically impossible to construct
two vibrating systems which are exactly equivalent.
This small
differential in the physical properties of the two trusses
could make the vibration characteristics seem to be quite
different.
It was immediately after this, while Prof. Den Hartog
was still inspecting the vibration, that the bridge failed.
The South end of the West cable snapped at the tower connection with the resulting destruction of the West span.
Pictures
of the model after failure are shown in Figures (10), (11) and
(12).
Just prior to failure, it had been noticed that the
pin connection to the East truss at the North end, which was
supposed to act as a slotted connection, was not actually
acting as such.
The pin was resting, with a good deal of force,
against the truss connection, thereby placing an axial compression
42A
Fiu.re
F1u re
10
(Il(
Figu-e
LZ
in the East truss.
This was in fact a dissimilarity of
the kind that Prof. Den Hartog had been describing, and
it was considered that this was undoubtedly the cause of
the peculiar action of the center of that span.
After re-
constructing the bridge, this condition was eliminated.
Only one reason for the failure of the cable could be de-"
termined other than a suspicion of fatigue failure.
The
connection at the top of the tower at the point of failure
was slightly out of line with the fore-aft axis of the
bridge.
This condition was also eliminated in the recon-
struction of the bridge.
Final Conpletion of Da t a
In his thesis,
Holt came to the conclusion that it
would
be the best practice to assume the experimental frequencies
for the condition without stays to be the correct values,
and to calculate percent errors for the theoretical frequencies.
At first glance, this does not seem to be a justifiable conclusion.
As Holt points out, however, the mode shapes computed
with observed frequencies check the observed mode shapes much
more accurately than do the shapes computed using the theoretical frequencies.
It is also to be noted that the observed
frequency readings are never very far off.
The tachometer
dial has as its smallest division a two percent interval.
Estimating the nearest percent, this admits the possibility
of an error of one-half of a percent in the dial reading.
Carrying the computation through for the increment of frequency
caused by this error, using the middle-range tachometer dial
and the 2/1 drive ratio, it
will be seen that the resulting
error in frequency will be 0.00948 cycles per second.
This
error will be slightly greater on the high-range dial, but it
can be easily seen that neither will be of appreciable importance.
The same practice, therefore, has been continued
in this work.
It is puzzling to the authors how Holt got frequency
readings for so many modes without a resulting destruction of
the model.
For work with the two-cable model, it was discovered,
as described below,
that beginning with the second mode,
the
vibrations were so violent that the safety of the model was
It may be that the added mass of
continually in question.
the extra cable and truss system made the vibrations excessive where they would not have been excessive in a onecable system.
A description of the action of the bridge during the
second mode is probably warranted to justify not continuing
with the data past the first
mode.
Without stays,
the vibra-
tion in the second mode was as violent as that that had resulted
in the destruction of the bridge.
the towers to the truss,
For the stays that ran from
the following condition was encountered.
Since the second mode is a symmetrical mode, during the half
of the vibration when the ends of the bridge went up, the stays
vould go slack and on the downsving would impart impact
forces to the trusses which completely disrupted the vibrations.
On the upsving of the symmetrical modes for the case with
the stays that ran from the roadway to the cable, the suspenders in the vicinity of the point of attachment of the
stay to the cable would go slack.
On the downsving part
of the cycle these suspenders were subjected to large impact
loads which, in the opinion of the authors., they could not
have long survived.
The effect of the stays on the frequencies is shown in
Table II.
The number 2 type stays have the greater effect on
Tale _E
EF f-ec4i
S+ay s
of
t/ End
Sl-ays on
Perce n,"
2. 1- 1.
#
Fr/Eyercie5
A 2 End
"''
./
I
Towet
2
To we-
*o Roadwoy
to Cob/v
the frequency substantiating the statement by Dr. Steinman.8
It would be desirable to re-run the frequency determination
of the first torsional mode without stays since it seems entirely
probable that this reading is seriously in error,
8 Dr.
D. B. Steinman, op.cit.
Table III compares the experimental frequencies with
stays with those computed by the use of Equation 8.
As can
be seen, the theoretical value checks the frequency for
the first torsional mode better than it does the first
natural mode.
The authors at this time are unable to
say whether this is due to the fact that te equation applies to the torsional mode or whether it
is completely due
to chance, to be clarified later by a derivation of an equation
Both types of stays
for the frequency of the torsional modes.
COm.aIi
Theoa-e -Icui
Tah/e JZ[
of EAper
o,
wiiH
Freya.encies
Iea)
And
Theo,-e
CEQc-
First nde
Fl-J
5,76
Mebb-oci
't .2 /
5+ays
End
CI
Percen /
Eer-o-
5. 91
Mode'..
F 0 ,rs-.
damped the amplitudes of vibration quite well, the number 2
type stays having the greater effect.
For a beneficial effect
during all modes of vibration, however, it might prove expedient to place both types of stays on the model at once.
Due to a lack of time, the authors were unable to investigate
this condition.
The mode shapes are shown in Figures (14), (15), (16)
and (17).
The natural modes check symmetry fairly well, and
upon comparison with Holt's theoretical curves are seen to
j ~:
agree very well.
The lack of symmetry for the shapes when
stays were used is due to the fact that it
to put the same initial
was not possible
tension in both stays.
This would
move the node point off center tovard the side which had the
stay with the greater initial tension in it.
It is to be noted that the mode shapes for the first
natural mode should look approximately like those for the
first torsional mode in the regions near the end of the span.
The drawing-dovn effect, however, was not justified in the
plotting of the curves,
hence the dashed lines to show the
approximate actual "Shape.
Before any definite conclusions can be drawn as to
the applicability of Holt's formulas, more experimental data
is necessary and a formula for the torsional modes is necessary
to eliminate the confusion that nov exists.
Sample data for
the calculations and sample calculations are given in Appendix
B.
Suggestions tor Further Work
In so far as the model is concerned,
there is still
to be desired in so far as future work is concerned.
much
The changes
that are suggested by the authors, for brevity, &re tabulated
below:
1.
An attempt should be made to simulate a roadway
between the two trusses.
If this can be done, the hinges
at the bases of the towersshould be replaced with rockers
at the tops of the towers and the bases should be fixed.
2.
The foundations should be fixed, with an attempt
to attach them permanently to the floor of the laboratory
instead of having them held down by their dead weight
alone as is now the case.
3.
A method of applying a sufficient constant force
to the bridge that would give measurable amplitudes under
all conditions of testing should be devised.
If this is
.not possible, a relation should be developed between the
force applied and the amplitude achieved.
4.
The electrical contact system explained previously
should be installed.
5.
A new type of flexible coupling other than the one
used between the trusses in this work should be found.
A
more flexible type is needed.
6.
It
is also suggested that roller bearings should be
placed on the gear box to eliminate the friction there that
was a constant source of trouble.
7.
The rbeostat addition to the tachometer circuit
for micrometer adjustments of frequencies should be installed.
This is the type of work that probably could be done
for the fulfillment of the Bachelor thesis requirements.
Further experimentation should include a continuation of
the work done in this th esis,
higher frequency readings.
with an attempt made to achieve
Another possibility is the inves-
tigation of deflections under static loads.
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Bibliography
"The Failure of the Tecoma Narrows Bridge"
A report to the Hon. John M. Carmody, Administrator of
the Federal Works Agency, Washington, D. C., March 28,
1941, by a board of engineers consisting of 0. H. Ammann,
Theodor von Karman and Glenn B. Woodruff
"Failure of the Tacoma Narrows Bridge; Report of the Special
Committee of the Board of Direction," Proceedings,
A.S.C.E.,
December,
1943
"Mechanical Vibrations"
J. P. Den Hartog, McGraw-Hill, New York, 1940
"The Determination of Suspension Bridge Stresses by Model Study"
J. W. Greeley and W. C. McClure, Thesis submitted to
the Massachusetts Institute of Technology, 1938
"The Effect of Diagonal Stays on the Natural Frequencies of
Suspension Bridges"
E. C. Holt, Jr., Thesis submitted to the Massachusetts
Institute of Technology, 1947
"Rigidity and Aerodynamic Stability of Suspension Bridges"
D. B. Steinman, Trans., A.S.C.E., Vol. 110 (1945), p. 439 ff.
"Oscillations of Suspension Bridges"
Hans Reissner, Journal of Applied Mathematics, A.S.M.E.,
March, 1943
56
Appendix A
57
Tochometer Co Ii brat ion Data
Middle Rang e
V
RPM
Low Rang e
V
60
RPM
32
H i gh Range
V
RPM
.3/0
30
3/F
50
5/6
50
6-10
60
6/
10
/000
'320
80
/390
70
90
/00
5/7
50
70
80
735
1022-
/00
h/ P M' read
al)
/-500
90
Tchomx-l-ervo/f 'eAer Jef/ec
Wes tolfn
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V-d4e
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ion.
6062
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I
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A=
--1
}
11
7I
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59
Appendix B
-
60
Sample Frequency Determination
Repro Sa-ion
o
ComoI-ka/,on of
Feequeocy
( A/-o e: The Ame s D/
Reada'ys
f:
LatoioAr /0
give,? hep-e
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61
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