f 'vv‘r

 

 

ave-5mm RESF'GNS‘E OF CANNLEVER BRIDGES
TO MGWNG FORCES

‘Ehesis for the Dear-u o! M. S.
MECHlGAN STATE UNIVERSITY
Taoktisfos Toridis
1961

This is to certify that the

thesis entitled

DYNAMIC RESPONSE OF CANTILEVER BRIDGES
TO MOVING FORCES

presented by

Teoktistos Toridis

has been accepted towards fulfillment
of the requirements for

M. S. degree in Civil Engineering

 

Major professor

 

Date June 30. 1961.

0-169

LIBRARY

Michigan State
University

 

ABSTRACT

DYNAMIC RESPONSE OF CANTILEVER

BRIDGES TO MOVING FORCES
by Teoktistos Toridis

An analytical study is made of the dynamic behavior of three-span

(symmetrical) cantilever bridges under moving forces. In the study, the

bridge is represented by a beam with uniform flexural rigidity, and the
mass of the beam is lumped at five locations.

The variables considered are the speed of the moving force, the
stiffness and weight of the bridge, and the relative lengths of the various
spans of the structure.

A mathematical analysis based on the method of "modal analysis"

is programmed for numerical solutions of problems on the MISTIC, the

digital computer of Michigan State University. The numerical data

obtained for the study cover nearly all bridges having practical propor-
tions. AmOng the results presented is a set of graphs by use of which
the first five natural periods of vibration of this type of bridges can be
easily obtained.

It is found that the dynamic effects can be as much as 56% greater

than the maximum static effects. In general, the dynamic response tends

to increase with an increase in the speed of the moving force. For a given

Speed, the amount of excitation to which the bridge is subjected increases

with the duration of the travel of the moving load on the bridge.

 

Teoktistos Toridis
Results also indicate that the response of the structure is composed of
mainly the first, second, and to a lesser degree, the third normal modes

of vibration.

DYNAMIC RESPONSE OF CANTILEVER

BRIDGES TO MOVING FORCES

B y a“
f , L

TE OK TISTOS TORIDIS

A THESIS

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

NIASTER OF SCIENCE

Department of Civil Engineering

1961

ii

ACKNOW LEDGMENTS

The work reported in this thesis was carried out as part of a
research project sponsored by the Division of Engineering Research
to study the dynamic behavior of cantilever bridges.

The investigation was under the immediate supervision of Dr.
R. K. Wen, the author's advisor, to whom gratitude is extended for

his valuable guidance during all phases of the work.

AC KNOW LEDGMEN TS .....................................

LIST OF TABLE S

LIST OF FIGURES

I.

II.

III.

IV.

TABLE OF CONTENTS

1. 1. General
1. 2. Object and Scope
1. 3. Past Work

1. 4. Notation

MATHEMATICAL ANALY SIS ..........................

2. 1. System Considered and Assumptions
2. 2. Equations of Motion
2. 3. Free Vibrations

2. 4. Forced Vibrations

2. 5. Dimensionless Form of Differential Equations
2. 6. Parameters of Problem
COMPUTER PROGRAM ...............................

3. 1. Outline of Program

3. 2. Time Increment and Convergence Considerations
3. 3. Detailed Operations in Computer Program
RESULTS OF INVESTIGATION ........................

4. 1. Natural Periods and Normal Modes of Vibration

4. 2. History Curves

Page

ii

11
13
13
14
16
21
21

22

iv

Page
4. 3. Spectrum Curves with Respect to a 24
4. 4. Maximum Response 28
V. SUMMARY AND CONCLUDING REMARKS .............. 33
LIST OF REFERENCES .................................... 35

FIGURES (l — 23) .......................................... 36

LIST OF TABLES

 

Number Page
1 Effect of magnitude of increments .................... 15
2a Maximum amplification factors with respect to T and a

for i = o. 10 ..................................... 29
12
2b Maximum amplification factors with respect to T and a
for = 0. 20 ................................... 3O
2
2c Maximum amplification factors with respect to T and a
for i = 0. 4o ..................................... 31
12

3 Absolute maximum A. F. ........................... 32

Number

10

11

12

13

14

15

16

17

18

System Considered .................................

Relations

Relations

Relations

Relations

Relations

Normal modes of vibration ..........................
History curve for m

History curve for m ..............................

2
History curve for m3 ..............................
History curve for m4 ..............................
History curve for m5 ..............................
1l a
Spectrum curves for m and m (— = O. 50, — = 0.10) .
l 5 l 1
2 2
Spectrum curves for m and m (— = 0.50, —- = O. 40). .
1 5 1 l
2 2
1 a
Spectrum curves for m and m (— = O. 90, — = 0.10),
2 4 l l
2 Z
1l a
Spectrum curves for m and m (— = O 50, — = 0 10)
2 4 l 1
2 2
11 a
Spectrum curve for m (— = 0. 50. ‘_ = 0. 10) .......
3 l 1
2 2
11 a
Spectrum curve for m (— = 1- 30. —‘ = 0- 20) .......
3 l2 12

LIST OF FIGURES

between T and T ........................

between T and T ........................

between T and T ........................

between T and TS ........................

1

vi

Page
36
37
38
39
4O
41
42
43
44
45
46
47

48

49

5O

51'

52

53

Number

19

20

21

22

23

Maximum

Maximum

Maximum

Maxi mum

Maximum

amplification factor
amplification factor
amplification factor
amplification factor

amplification factor

f
or m1

for m

OOOOOOOOOOOOOOOO

56

57

58

I. INTRODUCTION

1. 1 I General

The behavior of highway bridges under moving traffic is a compli-
cated dynamic phenomenon. It depends on a large number of variables
such as the type and weight of the structure and the vehicle, the speed
of the vehicle, the unevenness of the approach roadway and of the bridge
deck itself, and the damping characteristics of the bridge and vehicle.
Because of a lack of sufficient knowledge on the relative importance of
these variables, provisions for the dynamic effects in the design of
highway bridges have been based on entirely empirical rules. For
example, the "Impact Formula" given in the ”Standard Specifications
for Highway Bridges" by the American Association of State Highway
Officials implies that the dynamic effects depend on only one variable,
namely, the length of the bridge.

There have been a ,great deal of research activities in recent years
in the study of the dynamic behavior of highway bridges. The goals
appear to be to assess the relative importance of the different variables
that enter into the problem and, ultimately, to develop a more rational
procedure for making provisions concerning the dynamic effects in bridges.

In the past, researches on the subject have been largely focused on
simple span and continuous bridges. But it was noted in a recent series
of field tests (3) that it was the cantilever bridges that were most sus-

ceptible to vibrations. Since 1959, an investigation into the dynamic

behavior of this type of bridges has been conducted at Michigan State

University. This thesis represents a phase of the investigation.

1. 2 Object and Scope

The purpose of this investigation has been to study analytically the
dynamic response of three-span symmetrical cantilever bridges to
moving loads.

The variables considered in the study are the speed of the moving
load, stiffness and weight of the bridge, and the relative span lengths
of the various spans in a cantilever bridge.

In the analysis, the bridge is represented by a beam. The mass of
the beam is lumped at a finite number of points. However, the structure
is considered to possess uniform flexibility along its length. The moving
load is idealized as a moving constant force.

The mathematical analysis used has been based on a method
described in Reference (1). The method deals with the dynamic analysis
of elastic beams and frames by the "modal analysis" approach. It can be
applied to cases in which the point of action of the exciting force does not
necessarily coincide with one of the lumped mass points.

The present investigation includes the following three phases:

1. The application of the mathematical theory in Ref. (1) to develop a
computer program for use on the MISTIC (the digital computer of Michi-
gan State University).

2. The use of the program developed to obtain numerical results

covering certain appropriate ranges of the variables of the problem.

 

3. The study of the numerical results obtained with the purpose of
understanding better the dynamic behavior of this type of bridges and

determining the effects and importance of the variables of the problem.

1. 3 Past Work

 

A mathematical analysis of the dynamic behavior of cantilever
bridges was reported in Ref. (6), but no numerical results were given
for the moving load problem. Some numerical results concerning the
dynamic response of cantilever bridges to moving loads were reported
in Ref. (4). However, they were limited to one fixed set of relative span
lengths of the bridge. In the present study, as already mentioned, these

span lengths have been treated as variables.

1. 4 Notation

The symbols used in this report are defined in the text where they

first appear.

 

 

II. MATHEMATICAL ANALYSIS

The mathematical analysis of the problem presented in this chapter
is essentially the same as that given in Ref. (1). It is included here for

the sake of completeness.

2. l. Sistem Considered and Assumptions

The physical system is shown in Fig. 1a. It consists of an elastic
system of beams with two side or anchor spans, two cantilever arms, and
a suspended span. The suspended span is connected to the two cantilever
arms by means of frictionless hinges. The mass of the beam is concen-
trated at five points in the manner shown in Fig. 1b. However, the
flexibility of the beam is considered to be uniformly distributed.

In the analysis, the usual beam theory is assumed: that is, the
bending moment is proportional to the second derivative of the deflection
curve, and deformations due to shearing stresses are negligible.

As mentioned previously, the moving load is represented by a

constant force, traveling from the left to the right. No damping has been

considered in the system.

2. 2 Equations of Motion

Consider the beam and mass system shown in Fig. 1b. Applying
d'Alembert's principle, the deflection of some point mass mi may be
expressed as:

n
= - >3 " d , + ,
vi [1'21 mJ-YJ. iJ] pglz (1)

 

in which

iz

deflection of m. measured from the static equilibrium posi-
1
tion of the beam under the action of its own weight
second derivative of the deflection of m. with respect to time
J

influence coefficient for deflection at m. due to a unit load
1

at m.
J

weight of moving load

influence coefficient for deflection at mi due to a unit load at
a distance 2 from the left—most support, specifying the
instantaneous position of the moving load

number of point masses (five in the present case)

i

The first part on the right hand side of Eq. (1) shows the effect of

the inertia forces on the deflection of mi, while the second one represents

the effect of the moving load.

An equation of the form of Eq. (1) can be written for each concen—

trated mass, m. .
1

The number of equations, 11, obtained in this manner is

equal to the number of concentrated masses, or degrees of freedom of the

system.

where

These equations may be expressed in the following matrix form:

Y=-DMY+pG (2)

 

 

 

 

 

 

 

 

 

 

1
Y2
=def1ection matrix

yn
- d . . . T

dll 12 d1n

d21 d22 d2n

= flexibility matrix
d
_ n1 n2 nn_
0 . __

I- ml 0

0 m 0

= mass matrix

0 0 m
— L1...
III”

3’2

= acceleration matrix

_. Yn _.

glz

gZZ

= flexibility matrix for the moving load
g

 

2. 3. Free Vibrations

 

The expression for free vibrations of the structure can be obtained
by considering the forcing function in Eq. (2) to be equal to zero. Eq. (2)

then reduces to the following form:
Y = - DMY _ (3)

The solution of the above second order differential equation is

accomplished by letting:

 

 

Y = W sinx/T' t (4)
in which

le "

“’2
W = = a particular deflection configuration of the structure

(= normal mode)

.Wn 4
X = a constant to be determined (= natural angular frequency

squared)
t = time (independent variable)

Substituting Eq. (4) into Eq. (3), premultiplying both sides by M, and
simplifying, one obtains the equation:

MW = MW (5)

in which K = MDM. It may be noted that both M and K are symmetric

matrices. A solution of Eq. (5) yields the natural frequencies and normal

modes of vibration of the system.

 

2. 4. Forced Vibrations

 

Premultiplying both sides of Eq. (2) by M one obtains:
MY 2 - KY + pMG (6)

To solve this equation the "modal analysis" method is used by

 

 

letting:
Y = Ué (7)
where
U : [wlw2 . wn]
_ 17
¢
2
<1>
Q :
n
_4> J

The elements of the §> matrix are time-dependent, while those of

U are not consequently:
it = Us: (8)
Substituting Eq. (7) and (8) into Eq. (6), one obtains the expression:

MU§> = -KU§> + pMG (9)

Using the orthogonality relationships of the normal modes and
Performing some manipulations which can be followed in Ref. 1, Eq. (9) is

separated into a set of second order differential equations of the type:

: ()N‘j)l MG (10)

4,1 Hip ' . p
(WJ) KwJ

 

9

j ' 1 .
where (W ) = transpose of W . In this equation both the numerator and

the denominator on the right hand side are scalars. It is therefore a
scalar equation involving only one dependent variable, (if). The right hand

side of Eq. (10) becomes a constant for any given location of the moving

load, and thus its solution is possible.

2. 5. Dimensionless Form of Differential Equations
Eq. (5) may be put into the form:

(M—XK)W=O

The dimensionless version of this equation is:

 

 

(M -)\K)W=0 (11)
s s s
inwhich
l
Ms'ioom1
K :48EIK
s 2 3
m L
l
3
mlL
k

:—————— X
s 48OOEI

The subscript "s" in this case stands for "scaled" quantities, and

m1 is defined in Fig. lb.
Since Eq. (10) has the dimension of acceleration it is rendered
dimensionless by dividing through by the gravitational acceleration, g.

The notation and the scaling factors used throughout the operations below

are the same as in Ref. 1.

 

First, the independent variable t in Eq. (10) is transformed into

a dimensionless form by letting:

T :Z t
—_‘C_
dT—ZLdt

Substituting into Eq. (10):

 

2. .,
(v )Zd «:9 Hjp‘ng) MG
2L 2 j' j

'r (W)KW

Next, the dependent variable 4) is rendered dimensionless, letting:

3

L L

(P ~A¢ ’ o 48EI
O

 

where m is the mass per unit length of the beam.

With these substitutions and the division by g, one obtains:

-. A . . .l

Ao v 2 dzctJ o J-J: (WJ) MG 2
—(--—) —-——— +— X <1) —————

g 2L d'r2 g I

. . g
(WJ) KWJ

Introducing a new parameter ”a, " the above equatlon becomes:

.—

 

 

2—' . . j'
2d 4.3 1-3 = (W) MG 2 (12)
a 2+). <1) ' . g
where
A
V O _
a =—-— —- -
2L g
. A .
x1 : _° )9
g

10

 

11

Finally, the right hand side of Eq. (12) is made dimensionless,

 

 

 

using:
M Z—J—EM
b
- 48E
G = 31G
L
12 =MDM
- 4
D : 831..
L
I3 ‘ 1 p
’ L
mb g

from which the following equation results:
ZJ’ J" - -

d -' -' W G -

a ct + )9 $1 _ ( ) M p

2 " .: .
d7 (WJ) KwJ

(13)

2. 6. Parameters of Problem

 

It may be noted that in order to solve Eq. (13) the following quantities

have to be given or determined: (2, I6], WJ, M, G, E. However, once the
1

parameters 1-1 , 13- and a are specified all the remaining coefficients and
2 2

constants in the above equation can be computed by using the values of these

parameters. The range of the parameters and a brief description of their

significance are given below:

2. 6.1. Span Lengths
To determine the range of the span lengths, a study of seven three-

Span cantilever bridges given in Ref. (3) was made. In order to include

 

12
nearly all bridges of practical proportions, the following range was

finally selected:

11
'1— = 0.50m 2.00
2
3— = 0.10 to 0.40
12

2. 6. 2. Speed Parameter a

Although this parameter incorporates in it the effect of the stiffness
of the bridge, for a given bridge with a uniform E1 the only variable in the
expression for a is the speed of the moving force.

For a given speed, the value of a does not remain the same within
the range of the span lengths considered. To establish the limiting values
of this parameter, the characteristics of the previously mentioned seven
cantilever bridge have been used. It has been found that the speed range
of 15 to 75 mph in an actual bridge-vehicle system corresponds approxi-
mately to the range of values of a from O. 0225 to 0.1125.

In passing, it may be pointed out that the magnitude of the moving

force is not a parameter. Variations in the magnitude of the force produce

linearly proportional effects in the response of the, beam. Therefore, in
obtaining all numerical results a value ofp = O. 3150 has been used. This

value of 13 corresponds to approximately 20 tons in a representative actual

bridge—vehicle system.

 

13

III. COMPUTER PROGRAM

3.1. Outline of Program

 

The analysis of the problem is programmed for numerical solutions
on the MISTIC. The sequence of the basic operations in the program is
outlined as follows.

1. The computer is fed the data for the problem, specifying the

S.___.

characteristics of the bridge and load, and the time interval for use in

the numerical integration process.

 

2. The matrices Ms, D, and KS (see Eq. 11) are formed.

3. The natural frequencies and normal modes of vibration of the bridge
are calculated and printed out as part of the output of the computer.

4. The constant coefficients in Eq. (l3)--the governing differential
equation-~are formed.

5. For the particular time (or location of the load on the bridge under
consideration) the coefficient involving G is formed. The elements of matrix
G represent the static deflections at the various mass points for that par-

ticular location of the load.

6. The differential equations are integrated numerically for a pre-set

time increment.
7. The dynamic deflections are obtained by use of Eq. (7).

8. Steps (5) through (7) are repeated for the complete passage of the

load over the bridge.

 

 

14
9a. As output of the program, after each increment of time in the
numerical integration process, the computer prints out the dynamic
deflection and the corresponding static deflection for each of the five
mass-points.
9b. A modified version of the program will print out only the following
data for every mass-point: The maximum dynamic deflection and the
location of the load (1') producing this maximum; the maximum static
deflection and the location of the load producing this maximum; the
"maximum amplification factor with respect to T, " defined to be the
ratio of the maximum dynamic deflection to the maximum static deflection.

It takes the computer approximately one hour to compute and print
out the dynamic and static deflections for every step of integration (the
total number of steps covering the crossing of the bridge is equal to 1000).
If only the maximum responses are printed out, the corresponding com-
puter time is reduced to about 8 minutes.

Some detailed information related to the programming of the problem

is further presented in Art. 3. 3.

3. 2. Time Increment and Convergence Considerations

For the numerical integration the "Linear Acceleration" method has
been used. In this method the time increment should be small enough to
satisfy the criteria of stability and convergence. A smaller increment
Will in general give a more accurate answer, but the number of steps of

integration, and hence the computation time will be increased.

 

 

 

15
In Ref. (5), the convergence criterion for this method is given as:
At = 0. 389 T

where At is the maximum time increment and T is the shortest natural
period of vibration of the system. It is also stated that if the convergence
criterion is satisfied the requirement for stability is automatically met.

In this investigation the interval of integration was chosen to meet
the above requirements for nearly all bridges within the range of the
parameters considered. As shown in Art. 2. 5. , the independent variable
was put into a dimensionless form by introducing the variable T. It was
found that an increment (AT) of 0. 0005 gives satisfactory results in all
cases where convergence can be obtained. The number of integrations
corresponding to this increment is 1000. A comparison of the results

obtained by using different increments is shown below:

TABLE 1. Effect of magnitude of increments

 

Number of Max1mum dynamic

A1- : increment . . deflections at m
integrations , ,
(dimen51onless)
0. 0005 1000 0. 011968143
0. 00061 746 0. 011967531
0. 0010 500 0. 011968675

It is seen that the deflections corresponding to the three different
increments do not vary appreciably. However, in certain cases, for low
7
a values, using the increments of O. 0006‘ and 0. 0010 has caused very

Slow rates of convergence, or sometimes no convergence at all. Hence

 

 

16

AT was chosen as 0. 0005. Considering an average bridge, this increment
corresponds to a time equal to .110 of the fifth natural period of vibration.
Although the increment of 0. 0005 has been satisfactory in obtaining

numerical results within practically the whole range of the parameters, it

1
did not meet the convergence criterion in the following cases: l—- = 1. 30,
1 2
i = 0.10, a = 0.0225 — 0.1125;—.1 2 1.50, —a— = 0.10, a = 0.0225 — 0.1125;
1 1 l
2 2 2
1
1 a 11 a .
--- =1. 70; --- = 0.10, a = 0. 0225; ~—--- = 1.90, —- O. 10, a = 0.1125- USlng
12 l2 12 12

a smaller increment of 0. 0001 has been satisfactory in obtaining convergence
and hence numerical results in the latter two cases. Also, it has been found
possible to obtain convergence in the first two cases for high values of (1,
using AT = 0. 0001 or 0. 00007. But for smaller values of a it seems that
even a smaller AT would be necessary. Hence, no numerical solutions

have been obtained for these two cases, because they would require a great

amount of computer time.

3. 3. Detailed Operations in Computer Program

 

The basic steps of the computer program have been outlined in Art.
3. 1. In this article is presented some detailed information related to the

Programming of the problem.

3. 3.1. Natural Frequencies and Normal Modes

These quantities are obtained by solving Eq. (11). For this purpose,
matrices MS and KS have to be formed. KS results from the multiplication
0f three matrices (NISDMS) and a numerical constant used for scaling pur—

Poses (see Art. 2. 5.).

 

 

17

The elements of matrix D are computed by using a "deflection"
subroutine. This subroutine has been prepared for calculating the
deflections at the five mass points shown in Fig. lb due to a vertical
unit load applied at any point on the beam.

Having formed MS and K8, the natural frequencies and modes of
vibration of the structure are obtained by use of a library routine
(M5 - 139). This routine has been modified so that it would use the
matrices already stored in the computer as its data for computation.
In this manner, a data tape that would normally be required was
eliminated.

The characteristic values (1.2) given by the computer represent

the natural angular frequencies of vibration ()xJ) scaled by the factor

J and i] [XJ appears in

shown in Art. 2. 5. The relationship between 1.
Eq. (13)] has been given in Art. 2. 5. For coding purposes, the actual
value of the natural frequency (Xj) has not been needed. Since kjs is
stored in the memory of the computer, ii can be formed using the

following equation:

11 s

 

where

L 2 overall length of beam

11 = length of side span

The normal modes as printed out by the computer are scaled in the

manner indicated below:

 

18

 

w. = (14)

WJ = normal modes printed out by the computer

normal modes of vibration of the beam

sh.

Substituting the above expression for WJ into Eq. (13) and cancelling

out the scaling factors one obtains:

Z-j . . -j' - -

2 d - - _

a .3372 + 04.3 :ME— (15)
dT (wJ)KwJ

Eq. (15) represents the final form of the differential equations

solved in the computer program.

3.3.2. Coefficients of Eq. (15)

The value of the speed parameter a is a constant (for a given
speed) as is Xj, since ij is a characteristic property of the bridge under
consideration. Also, the right hand side of Eq. (15) is a constant for
a specified location of the load. This makes the solution of the equation
Possible by numerical integration. To form the denominator on the right
hand side of Eq. (15) matrix R has to be calculated. This results from
the matrix multiplication MDM. The denominator is a scalar, and its
value is fixed for a given beam.

The numerator on the right hand side of Eq. (15) is also a scalar.
It has to be computed for each step of integration, because the column

matrix G is a function of the position of the load. G is calculated by the

 

19

same "deflection" subroutine that forms D.

3.3.3. Integration of Eq. (15)

The integration of Eq. (15) has been carried out by use of Library

routine F-3. The time required for the moving load to cross the beam has
been divided into a number of intervals. The instant at which the load is
exactly over the left-most support of the beam is referred to as "zero

time. " At that instant the initial conditions (displacement, velocity, and

acceleration) of all the mass-points have to be specified. The computer

prOgram can handle initial conditions other than zero.

3.3.4. Dynamic Deflections and A. F.

Considering a single mass, Eq. (7) may be written as:

n J J
(y.) = 23 W 4) (16)
1 (1 J21 l
where the subscript ”d“ stands for “dynamic. "
Upon using the dimensionless variable ((33 (see Art. 2. 5.), the
following expression is obtained:
4
n _ , mbgL n . ~.
:A Z) J‘J:______ZW.JJ 17
(yi)d o jzl Wi (P 48EI jzl 1 <13 ( )

The corresponding static deflections are readily obtained using the

elements of matrix G. This is shown below (the subscript “st” stands for

 

“static");
3 gL4
m
_ " _ b " "
(y')st ‘ p 48131 (giz) ’ 48EI p(gm) ”81

1

20

The mathematical expression of the amplification factor obtained

from Eqs. (17) and (18), after cancelling out the common factors is:

j 'J'
Wi 4;
(l9)

 

3.3.5. Check by Desk Calculator

The computer program has been checked by comparing numerical

results obtained (for a test problem) by using the program and a desk

calculator.

 

21

IV. RESULTS OF INVESTIGATION

The numerical results presented in this chapter comprise the
free vibration behavior and the dynamic response of these bridges to
a single force moving across the spans with a uniform speed. The
behavior of the bridge has been investigated for the time interval
Starting from the entry of the load on the left-most span to its departure
from the right-most span.
A description of the computer program used in obtaining the

numerical data has been given in Chapter III.

4.1. Natural Periods and Normal Modes of Vibration

 

The relations between the fundamental period and the next four

higher periods of vibration are shown in Figs. 2 through 5. These
1

figures cover the following range of parameters: l—l— = 0. 50 to 2. 00,
2
a
“-1- = 0.10 to 0. 40. An increment of 0.10 was used for both parameters.

2
It may be observed that the graphs in Figs. 2-5 exhibit regular patterns.

Hence, they may be used, through interpolation if necessary, to obtain
the ratios of the first five natural periods for bridges that fall within
the above—mentioned range of parameters.

In order to obtain the first five natural periods of a bridge by use
of Figs. 2-5, Fig. 6 has been prepared. This figure relates T1, the
fundamental period to a reference quantity T5. The latter quantity is

numerically equal to the fundamental period of vibration of the suspended

Span treated as a simply supported beam. Therefore, with the aid of

22
the curves in Figs. 2-6 one needs only to compute the quantity Ts' in
order to obtain the values of the first five periods of a three-span
cantilever bridge.

The first five normal modes of vibration of a typical bridge are

shown in Fig. 7.

4. 2. History Curves

 

Presented in Figs. 8-12 are the "history curves" for the five point
masses that represent the lumped masses of the bridge. The term history
curve as used here designates a plot of the variation of the deflection of
a mass as a function of time. The bridge considered is characterized by

. 1l a
the followmg values of parameters: -—— = 0. 90, --— = 0. 20. The speed

12 1‘2

parameter a is equal to 0.1035, which corresponds to about 69 mph in an
actual bridge-vehicle system. The above numerical values of the
parameters were chosen for presentation herein because of the distinctive-
ness of the dynamic effects that they produced.

Plotted in each figure are both the static and the dynamic deflections

as a function of T = EYE . The variable T may be regarded as a dimension—
less measure of time, or a dimensionless measure of the location of the
moving force (the force is assumed to move from the left to the right).
During the movement of the load across the bridge the dynamic
deflection as well as the static deflection of a given mass changes. The

dynamic effect shows itself in the form of oscillations superposed on the

static deflection curve.

23

It may be noted that in all the history curves the dynamic effects
are relatively small when the load is on the first portion of the bridge,
that is, on the left side-span and left cantilever arm. As soon as the
load crosses the left or first hinge the vibrations start to grow larger in
amplitude.

In general, the largest amplitude of oscillation occurs when the
load is on the right side-span. This behavior had been noted in experi-
ments (1) and it was explained by the fact that before the load reached the
right side-span, this span was already excited to motion by the load
moving on the other spans. The general features of the history curves
mentioned in the preceding are somewhat less marked for m3 (the mass
at the center of the suspended span) than for the other masses.

The dynamic oscillations are not entirely periodic but in nearly
all the history curves one can identify sets of waves that possess rather
distinct periods. The fundamental and third natural period of vibration

can be noticed in the history curve for m An examination of the normal

3.
modes as given in Fig. 7 shows that the second and fourth modes should

be absent in the response curve of m since it happens to be on a node-

3,

point of these modes. In all the history curves, except that for m there

3}

can be found distinct trains of waves having periods that lie between the
fundamental and second natural period. The average period of such waves

, 2L . .
is approximately equal to 0. 0450 7, which is close to the second period

2
T2 (0. 0352 -2-—I'—J-) than is to the fundamental period T1 (0. 0589 1711’).
v

24

The fourth or the fifth mode appears in a few locations, particularly
near the middle of Figs. 8 and 9. It is impracticable to try to distinguish
between these two modes, because in this particular case they are very
close to each other. The contribution of these two modes is. however,
somewhat insignificant. The response of the structure is essentially
dominated by the first. second and, to a lesser degree, the third mode.
It should be born in mind. however. that this judgment is based on one
example. More data are needed to allow one to draw general conclusions
concerning the relative contributions of the various modes.

The oscillations of m1 (the mass at the center of the left side-span)
and m (the mass at the left hinge) in many instances have a phase dif-

2

ference of 180 degrees. The same phenomenon exists in the case of m4
(the mass at the right hinge) and m5 (the mass at the center of the right

side-span). It may be observed from Fig. 7 that such phase differences

exist only in the first three modes of vibration.

4. 3. Sgctrum Curves with Respect to a

 

4.3.1. General

While the complete history curve for a particular mass gives the
detailed picture of the behavior of that mass at a specific value of the load
speed (or a), the most significant point on a history curve is obviously its
maximum ordinate, or the maximum dynamic deflection. The ratio of
the maximum dynamic deflection to the maximum static deflection on a

history curve is called the "maximum amplification factor with respect

 

 

i I l.
,fllflluhls. .ltlll. III

 

 

25
to T, " and will be denoted by the symbol (MAF)T. There is a (MAF)T
corresponding to a given value of a. A plot of (MAF)T against a is
referred to as a "spectrum curve with respect to a, " or simply, "spectrum
curve."

For this study spectrum curves with respect to a have been obtained
for the following range of parameters: a from 0. 0225 to 0.1125 at

intervals of 0. 0045 (corresponding to approximately 15 mph to 75 mph

11
at intervals of 3 mph); -1-— from 0. 50 to 1. 90 at intervals of O. 20; and for
2
three values of-l-i, 0.10, 0. 20 and 0. 40.
2

Representative spectrum curves are presented in Figs. 13 through
18. A common characteristic to all these curves is their undulating
patterns. This pattern has been noted in the case of simple span bridges

, and m bear a

(2). In fact the general shapes of the curves for m1, m2 3

distinct resemblance to those for simple spans, the corresponding
characteristics being the increaseing "amplitudes" and ”periods" of the
undulations. The general appearance of the curves for m4 and m5
resembles less those for simple spans. However, all curves do show

the general trend of increasing (MAF)T as a is increased. More detailed

interpretation of the data will be presented in the following.

4.3.2. Comparison of Spectrum Curves for m1 and ms

It is of interest to compare the spectrum curves for m1 and m

because they are symmetrically located with respect to the center of the

bridge. For the majority of cases examined, the spectrum curve for m5

26

generally lies above that for m1, signifying a larger dynamic effect for
m than for m . This situation is illustrated in Fig. 13. In these cases

5 l

the maximum dynamic deflection of m generally occurs when the moving

1
load is near the middle of the left side-span. The time interval between
the initiation of the motion of this mass and the occurrence of the maximum
dynamic deflection is relatively small. On the other hand, in the case of
m5 the maximum dynamic deflection does not occur until the moving load
has traveled a great length on the bridge. Thus on the basis of the dura—
tion of excitation, it is reasonable to expect that the dynamic effect for

m5 would be larger than that for m1.

There are, however, some exceptions to the feature mentioned in

1
the preceding. For-i—l- = 0. 50 to 1.10 and large values of the fa— ratio the
2 2
spectrum curves for m1 and m5 are located quite close to each other,

intersecting at several points. This is illustrated in Fig. 14. An
explanation is given as follows: In the above-mentioned range of parameters
the cantilever arm is relatively long. This causes the maximum dynamic
deflection of m1 to take place after the moving load has crossed the left-
side—span and is near the left hinge. Thus the maximum dynamic deflection

of m1 is associated with a longer duration of excitation and consequently a

larger magnitude .

4.3.3. Comparison of Spectrum Curves for m2 and m4

1 l
1 . 1
Presented in Fig. 15 (T- = 0. 90. '13— : 0.10) and Fig. 16 (T = 0. 50,
a 2 2 2
T = 0.10) are typical spectrum curves for m2 and m4.

2

27

Comparisons of spectrum curves for m2 and m4 indicate similar

2

relationships between m and m4 to those between ml and m5. Whenever
the maximum dynamic deflection of mZ takes place as the load is traversing
the left side-span, the spectrum curve of m4 lies above that of m2 (see

Fig. 15). If the maximum dynamic deflection occurs when the load is

near the left hinge, the two spectrum curves are close to each other (see
Fig. 16). This feature can be explained, as for the case of m1 and m , in

5

terms of the dependence of the dynamic effects on the duration of excitation.

4.3.4. Spectrum Curves for m3

1
are given in Fig. 17 (—1-i = 0. 50,

2

Typical spectrum curves for m3

a

. 11 a
1 = O. 10) and F1g. 18(T :1. 30. '1‘-

2 2 2
The resemblance of the curve in Fig. 17 to a typical spectrum curve

2 0. 20).

for the mid-span deflection of a simply supported beam (2) is quite
striking. The reason is thought to be that for the structure considered

in this figure the suspended span is long relative to both the side span and
the cantilever arm. It follows that the masses on the latter spans are

small as compared to m and hence they have a relatively small influence

3
on its motion. On the other hand, the stiffness of the side-span and the

cantilever arm is relatively large because of their short lengths. Under
these conditions the center span could be expected to behave essentially

as a simply supported beam.

In Fig. 18 it may be noted that the spectrum curve for the structure

involved is substantially different from that for a simply supported span.

28

This may be explained by the fact that the structure concerned has a

long side-span. Hence the conditions related in the preceding paragraph
are essentially reversed. In this case the side—spans could have a fairly
large influence on the behavior of the suspended span from the standpoint
of both the inertia forces and stiffness. Consequently, the suspended span
no longer behaves like a simply supported beam.

In passing, it may be noted that Figs. l3, l6, and 17 constitute a

1
complete set of spectrum curves for the bridge with -i— = 0. 50 and
2
3— - 0 10
1 _ o I
2

4.3.5. Negative Amplification Factors

Whenever the maximum dynamic and static deflections do not
occur when the load is on the same span, negative amplification factors
result (see Tables 2b and 2c). This follows from the fact that the sign
of the maximum dynamic deflection is opposite to that of the static. For

example, in the case of m , while the maximum static deflection occurs

1
when the load is on the left side-span, the maximum dynamic deflection

may take place when the load is on the left cantilever arm, or vice versa.

4. 4. Maximum Response

 

The maximum ordinate of each of the spectrum curves obtained
for this study is listed in Tables 2a, 2b and 2c. The values of T and a
at which this maximum occurs are also given. For convenience the data

in these tables are also presented graphically in Figs. 19 through 23.

Table 2a.

1/.

Maximum amplification factors with respect to ’7' and a,

for—‘1- = o. 10

Point

1.14
.0560
.1125
1.16
.0675
.1125
1.13
.0795
.1125
1.15
.0865
.1035
1.17
.1000
.0945
1.17
.1005
.0900

Maximum A. F.

Point

m2

1.28
.1410
.1125
1.17
.0695
.1125
1.13
.0825
.1125
1.19
.0940
.1125
1.20
.1130
.1080
1”20
.1160
.1035

Point

In

3

1.14
.2255
.1125
1.13
.2545
.0855
1.10
.2540
.1125
1.22
.2470
.1125
1.13
.2550
.0990
1.13
.2545
.0630

Point

in
4

1.20

.3660
.0900.

.4335
.0945

.4080

.1080

1.31
.4065
.0945
1.19
.3930
.0810
1.20
.4005
.0900

Point

Ink

Table 2b. Maximum amplification factors with l‘£;r:l.‘.é.C‘1‘ to T and 0.,

(1
for [—-: 0.20
2

Maximum A. F.

fly Point 'Point Point Point Point
I 2 ml mZ m 3 m4 m ,.
at 1.25 1.19 1.08 1.23 1.28
0.50 7 . 1570 .1560 .2495 .3560 3630
a .1080 .1125 .0675 .0990 1035
at -1.12* 1.20 1.06 1.20 1.32
0.70 7 .1710 .1690 .2540 .3430 4470
a .0945 .1125 .0720 .0945 0945
at 1.06 1.27 1.10 1.13 1.35
0.90 7 .0720 .1790 .2420 .3325 4490
a .0450 .0990 .1035 .0810 1035
at 1.12 1.19 1.24 -1.20* 1.27
1.10 7 .0800 .1870 .2490 .4065 4180
a .0945 .1125 .1125 .1125 0810
at 1.15 1.20 1.19 1.56 1.43
1.30 7 .0875 .0995 .2530 .4145 .4125
a .0990 .1125 .1125 .1125 .1125
at 1.17 1.20 1.19 1.41 1.30
1.50 7 .0930 .1035 .2525 .3925 .4130
a .0990 .1080 .0855 .0900 .1080
at 1.17 1.21 1.15 1.28 1.25
1.70 7 .0945 .1115 .2495 .4070 .4150
a .0945 .1080 .0810 .1125 .0720
at 1.17 1.21 1.16 1.34 1.34
1-90 7 .0995 .1125 .2605 .4165 .4155
a .0945 .1080 .0810 .1125 .1125

*See explanation in the text.

 

 

Table 2c.

1

0.50

0.70

0.90

1.30

1.50

1.70

1.90

*See explanation in the text.

y .
12

Maximum amplification factors with respect to ’T and a,

O.
for ——-=

f

0.40

Point

m1

1.24
.1750
.1125
1.24
.1855
.1080
1.28
.1950
.1035
1.27
.2025
.1080

-l.19*

.2045
.0900
1.10

.0815
.0810
1.13

.0860
.0855
1.15

.0910
.0900

Maximum A. F.

Point

rn2

1.22
.1780
.1125
1.19
.1865
.1125
1.20
.1940
.1080
1.20
.1995
.1125
1.17
.2035
.0945
1.20
.2085
.1125
1.25
.2105
.1125
1.28
.2135
.1125

Point

m

3

1.13
.2500
.0720
1.13
.2625
.0900
1.17
.2570
.0765
1.16
.2595
.0855
1.20
.2710
.0990
1.18
.2520
.0765
1.20
.2600
.0810
1.20
.2260
.1125

Point

m4

1.20
.3220
.1125
1.25
.3370
.0855
1.22
.3230
.0810
1.27
.3200
.0855
1.23
.2985
.1125
1.27
.3000
.1125
1.27
.3005
.1125

-1.36*

.4060
.1125

Point

In

5

1.24
.3465
.0810
1.31
.3385
.0855
1.30
.3320
.0900
1.39
.3215
.0855
1.37
.4025
.0900
1.43
.4180
.0765
1.49
.4025
.1125
1.54
.4055
.1125

32

Most curves in Figs. 19-23 show no distinct trends. This may be

due to insufficient data. It may be noted, however, that curves for m3,

a
m and m and — = O. 40, generally indicate an increase in dynamic
1
effects with an increase in the values of-l— . Another feature that may
2

be noted from these curves is that the ranges of variation of the maximum
amplification factors are larger for ml, m4 and m5 than for m2 and m3.

The absolute maximum amplification factor for each mass (maximum

1
A. F. with respect to T, a, 1—1 and 130 is listed below:
2 2

TABLE 3. Absolute maximum A. F.

Mass oi t Absolute : .3.
p n maximum A.F. 1 l
2 2
l 1. 28 0. 90 0. 40
2 1. 28 1. 90 0. 40
3 1. 24 1.10 0. 20
4 1. 56 1. 30 0. 20
5 1. 54 1. 90 0. 40

From the above table it may be noted that the absolute maximum

A. F. does not vary appreciably among m

much greater for m and m .

of 1. 56 for m4 and 1. 54 for m

the given loading condition, i. e. , a constant moving force.

4 5

5

,m andm.

2

However, it is
The value of the absolute maximum A. F.

must be considered to be striking under

 

 

33

V. SUMMARY AND CONCLUDING REMARKS

An analytical study of the dynamic response of three-span
cantilever bridges has been presented. The major findings of the
study are summarized and discussed as follows:

1. The dynamic effects in this type of bridges as caused by moving
loads can be quite large. The greatest dynamic effect found within the
data obtained is 56 per cent larger than the corresponding maximum
static effect. This is quite striking, particularly if one considers the
fact that the investigation has not included the influence of such factors
as the vertical springing motion of the vehicle and the unevenness of
the bridge deck surface. The latter factors all tend to increase the
dynamic effects of the moving load.

2. The dynamic effects generally increase as the speed of the moving
force increases.

3. From a study of a limited number of history curves at different
sections of the bridge, it has been found that the major contributions to
the dynamic deflection of the bridge come from the participation of the
first, second and to a lesser degree the third normal mode of vibration.
Response in the 4th and 5th modes is scarce and of relatively minor
importance.

4. In general, the dynamic effects are larger for the latter portion

Of the bridge than for the first portion. This seems to indicate that for
a given speed the amount of excitation to which the bridge is subjected

increases with the duration of the travel of the moving load on the bridge.

 

 

34

The constant force representation of the moving load may be
considered as a preliminary step for an investigation of the dynamic
behavior of cantilever bridges. In the author's opinion, future studies
should commence with an extension of the present analysis to consider
more realistic representations of highway vehicles. For example a
vehicle may be idealized as a mass supported by two springs, repre-
senting the two axles of a vehicle. In this manner, the vertical bouncing
motion and angular pitching motion may be taken into account in the
analysis. In addition, for a more comprehensive study of the problem
the effect of the unevenness of the bridge deck needs to be accounted for.
Last but not the least, for applications to the design of such structures,
the dynamic bending moments in the bridge. in addition to the dynamic

deflections, should be included in future studies.

 

35

LIST OF REFERENCES

Wen. R. K. L. , "Dynamic Behavior of Cantilever Bridges Under
Moving Loads, " Progress Report No. 1, Division of Engineering
Research, College of Engineering, Michigan State University,
February, 1960.

Wen, R. K. L. , ”Dynamic Behavior of Simple Span Highway Bridges
Traversed by Two-Axle Vehicles, ” Civil Engineering Studies,
Structural Research Series No. 142, University of Illinois, 1957.

Oehler, LeRoy, T. , "Vibration Susceptibilities of Various Highway
Bridge Types, " Journal of the Structural Division, Proceedings of
ASCE, July 1957.

Huang, T. , and Veletsos, A. S. , "A Study of Dynamic Response of
Cantilever Highway Bridges, " Civil Engineering Studies, Structural
Research Series No. 206, University of Illinois, 1960.

Newmark, N. M. , "A Method of Computation of Structural Dynamics, "
Journal of Engineering Mechanics Division, Proceedings, ASCE,

July 1959, pp. 67-94.

Konishi, I. and Komatsu, S. , ”Vibration Behavior of Gerber Beam, "
Technical Report No. 29, Vol. 6, No. 3, The Engineering Research
Institute, Kyoto University, Kyoto, Japan, 1956.

36

3 ti
direction of ymme r 031 about

movement of force '
1Hinges

  

 

 

 

2.2.: f

 

 

 

 

 

s ide -s pan 8 pended s pan 3 ide-s pan
can ilevJE cantiIever
arm arm
L

 

 

 

(a) Cantilever bridge type

 

 

 

 

 

 

 

 

mass per unit length of beam

1
1 111.5 a tab (21)

B
I

B
I

1
?m2 - mb-'¥mb (E;_+.%)
m3 = n1.b (23 )

(b) Concentrated point masses

FIG. 1 - SYSTEM CONSIDERED

37

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.110
3.00
2.60 ,
. 1
__ a 0010 — 1
l
2.20 2
1.80
% B 0020‘
2
2. =3 0030—
1.80 12
2'. 9 0011.0-
12
L-o—ttfv'“ ’
1.00
O 60
. 001.10 0.80 1020 1.60 2000

' r T and T
Fig. 2 - Relations between 1 2

 

38

 

3.90

 

 

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/ \ / ,

 

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°\ "4%“ ,xwor’p’

 

 

 

 

 

 

 

 

 

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1.50
1 10 ,1 9 W“
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g . .' T “1.11551 T2
Fig. 3 — Relations netreen 1 ‘ J

a.

1b.oo

12.00

10.00

6.00

b.00

39

 

 

 

0.h0

 

 

 

 

 

 

 

 

 

 

 

Fig. h.-’Re1ations between T1

T
andh

2.00

40

 

 

 

 

 

 

 

 

 

 

 

 

 

111.00
12.00
10.00 I
a a 0.13
B 00140 .i.
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11
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Fig. 5 -Relations between T1 and T5

 

8.50

h.00

3.50

3.00

2.50

2.00

1.50

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 6 - Relations between.T1

 

and Ts

41

 

 

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42

 

 

 

 

 

"’5

a Relative Periods

\/ T1
T2 = 0075 T1
\,_/ 3 1
\ rx\ T14 a 0,32 111

/‘\ r - 0.31 r

/ \ 5 1

 

 

 

 

 

 

 

 

 

l1
—- - 0-90. g .- 0.20
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Fig. 7 - Norml modes of vibration

 

 

43

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H- . -
omoo I MU m8 and HE .3.“ 02150 flagpoomm I 4n .3?”

 

 

 

 

 

0mH.0 0HH.0 000.0 040.0 000.0 0m0.0 0H0.0
00.H
, w Ly,
\m L/ \r \w 41KA»
\, (x ’0
- ,- 4 4% .741 .
m \w11-1L \ \ 0H H
aIZ \
//<
4 \L \
4L L \ om.”
\ r/KL
HEIIVL
0m.H
ma 0:.H
04.0 n .m
m
3.01%
H 0m.H

 

 

 

 

 

 

 

 

z 04 4°0dso: Q%IN-°J°V’W“WEX‘N

50

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. m m 4 ma a .
H q . IH. a .0 .3 00:50 0.398% ma mam
AOHOO I .dI 00 O I HHV ”an M m
8 hopuadhdm voomm
00H.0 0HH.0 000.0 000.0 000.0 . 000.0 0H0.0
00.H
c .
.\\ 0H.H
Na |JQ_ \
H
k 00.H
C on;
H. ..

0:.H

NH

0H.0 u .m

m

H

00.0 a m:.

H
00.H

‘2 on noodsoa qa;n.°J°V’mnmtxvn

51

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

- N ‘ .
A36 u NW .om.o u .mMJ 4.: can we .8.“ 03.30 05.50on ... 0H ..me
8 nova—aha dim
00H.0 0HH.0 000.0 000.0 000.0 000.0 0H0.0
. 8.0
ohod .035 253.0on a
w o O 02 .5. 8m 0
H , ..H 4 3H H/
H x _ 00.H
H H,
.H _ ,
AH ,. H» \ HH 0H.H
H / \
b H
x / H
r H H
/ H
< ON.H
NE
NH on;
2.0 .. a
.NH.
00.0 n .H|..
H
OJoa

 

a 04 twain-I mu °a°v mummy!

52

1 N . -
NH H u .0 _
ASN.0 u M. a0.00.0 I a ma .3.“ 3.30 08.50on 5 Ha“

8 .Hopoadhdm 000mm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. , . . . 000.0 0H0.0
00H.0 0HH 0 000 0 000 0 000 0 00.H
, 0
0a 0H.H m.
\ \a m
.\ w.
”a
u.
ON.H 1.
U.
r.
O
S
.m
.0
“4'
00.H m.
a
- . 0:.H
m
H
3.0 u .m
NH
00.0 .. .H.....
H 00.H

53

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a N N n. u
8N6 u m... .omJ I My ma you Ea... 00$»...on I ma .mah
.8 .3983»de 000mm
00H.0 0HH.0 000.0 000.0 000.0 000.0 0H0.0
_ 00.H
0 \ >d> oH.H
0.. I/
e\
om; _
00.H
NH. 04 H
8.0 I d
00.H .. mH
HH
00.H

 

 

.2 09. swede“ um u'v mmrm

Maximum A.F. for ml with respect to z' and at

1.60

1.50

1.h0

1.30

1.20

1.10

1.00

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 positive max. A.F.
a negative max. A.F.
//p~“‘e\
/ x a .- 0.1.0
30-“ {‘72
‘ \
\ \
\‘ k
\ g 8 0.10 \\ O’CW
12 >/,o/\/‘ I
”V /' \\ ,0”,
‘k f/ \I §’/
“\ I a - 0.h0
‘prg I 0020 1
O 30 0.70 1.10 1.50 1.90
11
T;
Fig 19 - Maximum amplification factor for 1:11

54

.3155

Ww‘gg... a. , _ J,” “I. .1.
it..- '— ‘ .1

 

Maximum A.F. for 32 with respect to r: and 0c

,1.20

1.60

 

 

1.50

 

1.140

1.30

 

 

 

 

 

1.10

 

 

 

 

 

 

1.00
0.30

0.70 1.10 1.50 1 .90

11

‘3

Fig. 20 - Maximum amplification factor for me

55

Maximu- A.F. for In} with respect to z- and at

1.60

1.50

1.h0

1.30

1.20

1.10

1.00

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.\‘
\9’"
0030 0.70 1.10 1.50 1.90
1
l
12

Fig. 21 - Maximum amplification factor for 1:13

56

Nazism: A.F. for mu with respect to c' and as.

 

 

 

 

 

 

 

 

 

 

 

 

 

1.60
1.50
s 0.20
1.1.0
)
I
1.30 V/
>—--—c(
LE, - 00140
. 1.20 0
Lg . 0.10
1.10
a positive max. A.F.
a negative max. A.F.
1.00 .
0.30 0.70 1.10 1.50 1.90
3.1..
12

Fig. 22 - Iaximm amplification factor far um

57

Maximal! A.F. for ms with respect to r and at.

1.60

1.50

1.140

1.30

1.20

1.10

1.00

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[g . 0.10
12
0.30 0.70 1.10 1.50 1.90
3.1.
1
2

Fig. 23 - Iaximun amplification factor for m)-

58

.. nil“?
..L .. . .
. ..l"!.Iu—' i. «C:

.l ..I. Irlmil.“

1.1.1 oEJI-lll. .
. . . Duh-ham?!» Q's-IIIIIF‘ '41 7
. n . y.

HHIEIP a.-- ...

. ....ixllnbil.|ll.lelfill.. . 'n‘lI-i‘ulu. . .

.....0‘..1.l.|. .Illlr'i.
. . I ., I) .. .v

. v 4 Z . x
.unu b IILI1‘cl.zc.I.lI.l 4a.. -‘Hfi...\.fl .4...

 

-.I., r «III‘VI

......-.l.. . fi.‘ .

,_ 4. (Han; P—J-x’ *'

 

 

 

 

 

 

 

 

TA

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