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ABSTRACT

INELASTIC BEHAVIOR OF TWO-SPAN CONTINUOUS BEAMS
SUBJECTED TO MOVING LOADS

by Shien Tsun Wang

The purpose of the investigation is to study
experimentally the elasto-inelastic behavior of two—span
symmetrical continuous beams subjected to a single
unsprung load or sprung load. All beams tested were
made of mild steel (average yield stress 27,500 psi),

6 ft. long (3 ft. each span), and have a rectangular cross-
section of l" x 1/4".

In the test program the weight of the load was
varied from 0.56 to 1.8 times the minimum static load
that would first cause yielding in the beam; the speed
was varied from 0.2 fps to 15.3 fps.

Results show that a heavy load moving at a
relatively slow speed can cross the beam smoothly. The
resulting permanent displacement is larger at the first
mid-span than at the second mid-span. In tests that
involved either a heavy load or a high speed, the load
"jumps" as it approaches the middle support, i.e., it
loses contact with the surface of the beam, and then

"landed" back on the beam at a short distance from the

 

 

middle
this ca
span.
at the
tests i

set is

Shien Tsun Wang

middle support. The resulting permanent displacement in
this case is larger in the second span than in the first
span. The maximum value of the permanent displacement

at the first and the second mid-spans observed in all the

tests is 0.6426". In most cases, however, the permanent

set is less than 0.14":

INELASTIC BEHAVIOR OF TWO-SPAN
CONTINUOUS BEAMS SUBJECTED

TO MOVING LOADS

BY

SHIEN TSUN WANG

A THESIS

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

MASTER OF SCIENCE

Department of Civil Engineering

1964

 

investig
to fiO‘lf
Engineer

been SEC

Grant N:

Research

experime
R- Hills
Used if}
Steel Cc
of that

bars USE

which ha
we” to w
instruct
also to

their Co

ACKNOWLEDGMENTS

The work reported herein was done as part of an
investigation of the inelastic behavior of beams subjected
to moving loads conducted in the Department of Civil
Engineering of Michigan State University. The project has
been sponsored by the National Science Foundation under
Grant No. 612143 administered by the Division of Engineering
Research.

This investigation represents a continuation of the
experimental study on simply supported beams made by Mr.

R. Hills who had designed most of the experimental set—up
used in this study. Thanks are expressed to Bethlehem
Steel Company and Dr. I. M. Viest, Structural Engineer

of that organization, for arranging to supply the steel
bars used in this study.

This report also constitutes the author's thesis
which has been written under the direction of Dr. R. K.

Wen to whom gratitude is extended for his guidance and
instruction. The author wishes to express his appreciation
also to the Machine Shop of the College of Engineering for

their cooperation.

ii

 

II.

('1'

TABLE OF CONTENTS

Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . l
1.1. Object and Scope 1
1.2. Notation 3
II. EXPERIMENTAL WORK . . . . . . . . . . . 4
2.1. Experiment Set-up 4
2.1.1. General 4
2.1.2. Test Beams 4
2.1.3. Load Carriage 5
2.1.4. Driving System 6
2.1.5 Load Arrester 6
2.1.6. Abutments, Accelerating,
Decelerating Tracks, Monorails
and Guide Plates 6
2.2. Measurements and Instrumentation 7
2.2.1. Beam Deflections 7
2.2.2. Carriage Reaction 7
2.2.3. Load Position 8
2.3. Stress-Strain and Moment—Curvature
Relations 8
2 3 1 Stress—Strain Relation 8
2.3.2. Moment-Curvature Relation 9
2.4. Static Behavior of Continuous Beam 10
2.4.1. Theoretical Analysis 10
2.4.2. Experiment Behavior 12
2.5. Test Program 12
2.5L1. Parameters ofStudy 12
2.5.2. Unsprung Load Test 13
2 5 3. Sprung Load Test 14

iii

Chapter Page

III. RESULTS OF INVESTIGATION . . . . . . . . . . 15
3.1. Unsprung Loads 15
3.1.1. Typical Patterns of Behavior 15
3.1.1.1. Behavior with "Medium"Va1ue of
a and "Medium" Value of B 15
3.1.1.2. Behavior with "Small" Value of
a and "Large" Value of B 16

3.1.1.3. Behavior with "Medium" Value

of a and "Large" Value of

B 18
3.1.1.4. Behavior with "Large" Value

of a and "Medium" Value of

B 20

3.1.2. "Jump" Phenomenon 20
3.1.3. Maximum Mid-Span Dynamic

Deflections 22
3.1.4. Mid-Span Permanent Deflections 23
3.1.5. Maximum Acceleration of Load 24
3.1.6. Deformation Under Repeated

Loading 25
3.1.7. Permanent Curvature and “Hinge

Length" 26
3.2. Sprung Loads 29
3.2.1. Typical Patterns of Behavior 29

3.2.1.1. Behavior with "Medium" Value

of a and "Medium" Value

of B 29
3.2.1.2. Behavior with "Small" Value

of a and "Large" Value

of B 31
3.2.1.3. Behavior with "Medium" Value

of a and "Large" Value of

B 31
3.2.1.4. Behavior with "Large" Value

of a and "Medium" Value

of B 33

3.2.2. "Jump Line" 33
3.2.3. Maximum Mid—Span Dynamic

Deflections 33

3.2.4. Maximum Spring Force 34

iv

{WA

TH E‘

M‘ ”IV-17M If

 

 

 

Chapte:

IV.

LIST OF

FIGURES

Chapter Page
IV. SUMMARY AND CONCLUSION . . . . . . . . . . . 35

4.1. Summary 35
4.2. Discussion and Conclusion 37

LIST OF REFERENCES . . . . . . . . . . . . . . . . . 39

FIGURES . . . . . . . . . . . . . . . . . . . . . . . 40

10.

ll.

12.

13.

14.

15.

l6.

17.

18.

19.

20.

LIST OF FIGURES

Overall View of test set-up . . . . . . . . .

Load carriage . . . . . . . . . . . . . . .
Test beam and differential transformers
Driving system . . . . . . . . . . . . . . .
Load arrestor . . . . . . . . . . . .

Typical stress-strain curve

Moment-curvature diagram . . . . . . .
Curves of yield load and collapse load
Deflection at point of loading . . . . . . .
Static behavior of two-span continuous beam .

Dynamic test record for unsprung case:
a = 0.05623. B = 1.0 . . . . . . . . .

Dynamic test record for unsprung case:
a = 0.04468. B = 1.6 . . . . .

Distribution of residual moment . . . . . . .
Final deformed shapes . . . . . . . . . . . .

Dynamic test record for unsprung case:
a = 0.06108. B = 1.6 . . . . . . . . . .

"Jump lines" . . . . . . . . . . . . . .
Location of starting position of jump . . . .

Maximum deflection of first mid-span
(unsprung case) . . . . . . . .

Maximum deflection of second mid-span
(unsprung case) . . . . . . . . . . .

Location of load for maximum deflection of
first mid-span (unsprung case) . . . . .

vi

Page
41
41
42
42
43
44
44
45
46

47

48

49
50

50

51
52

53

54

55

56

Figure

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

Location of load for maximum deflection of
second mid—span (unsprung case) . . . . .

Permanent deflection of first and second
mid-spans O o o o o o o o o o o o o o o 0

Maximum vertical acceleration of load on
first Span I O O C O C O O O O O O O O 0

Maximum vertical acceleration of load on
second span . . . . . . . . . . . . . .

Permanent deflection of first mid-span under
repeated loading . . . . . . . . . . . .

Permanent deflection of second mid-span under
repeated loading . . . . . . . . . . . .

Variation of permanent curvature within
"hinge length" . . . . . . . . . . . . .

Dynamic test record for sprung case:
a = 0.05160. B = 1.2 . . . . . . . . .

Dynamic test record for sprung case:
a = 0.04313. B = 1.6 . . . . .

Dynamic test record for sprung case:
a = 0.05795. B = 1.6 . . . . . . . . . .

Maximum deflection of first mid-span (sprung
case) . . . . . . . . . . . . . . . . . .

Maximum deflection of second mid-span (sprung
case) . . . . . . . . . . . . . . . . .

Location of load for maximum deflection of
first mid—span (sprung case) . . . .

Location of load for maximum deflection of
second mid-span (sprung case) . . . . .

Maximum compressive spring force on first
span 0 O O O I O O O O O I O O O O O O 0

Maximum compressive spring force on second
span 0 O O O O C O O O O O O O O O O O 0

vii

Page

57

58

59

6O

61

62

63

64

65

66

67

68

69

70

71

.72

Figure Page

37. Location of maximum compressive spring force
on first span . . . . . . . . . . . . . . 73

38. Location of maximum compressive spring force

on second span . . . . . . . . . . . . . 74

viii

I. INTRODUCTION

1.1. Object and Scope

 

Since the end of World War II. structural design
based on the concept of ultimate strength has increasingly
gained recognition. This is evidenced by the appearance
in design codes of practices based on the ultimate strength
theory for reinforced concrete members. and the "plastic
theory" for steel structures. Meanwhile. in the technical
literature considerable analytical and experimental research
work has been reported on fundamental studies of the behavior
of structures beyond the elastic range. Published works
in this area. however. generally deal with "stationary
loads.” i.e.. loads that do not change their positions on
the structure.

For many decades the problem of moving loads has
attracted a great deal of attention because of its obvious
relation to bridge engineering. However. most published
works on this subject are confined to elastic behavior.
except the tWo analytical studies reported by Parkes (l)
and Symonds and Neal (2). Both works considered the response
of simply supported ”rigid plastic? beams subjected to an
unsprung moving mass. Except for the studies by Hills {3).
there has been no reported experimental work on the

subject. The investigation of Hills was concerned with

the inelastic behavior of simply supported beams subjected
to moving unsprung or sprung loads. Reported herein is an
experimental study of the inelastic behavior of two~span
continuous beams.

The experiments performed in this study consist of
subjecting two-span symmetrical continuous beams to unsprung
and sprung loads moving across the length of the beam. A
total of 20 beams were tested in this manner. Each beam
is 6 ft. long (3 ft. each span). made of mild steel. and
has a rectangular cross-section of l" x 1/4". It was loaded
in the weaker direction. The parameters considered in the
study are the weight and the speed of the load. The
weight of the load was varied from 0.56 to 1.8 times the
minimum value of a static (but movable) load that would
first cause yielding in the beam. The load speed used was
varied from 0.2 fps to 15.3 fps.

Quantities measured in the experiments were deflections
at the centers of the two spans of the beam. and the reaction
between the moving load and the beam. To aid the planning
and interpretation of the dynamic tests certain static
tests were also performed. These tests as well as other
related to the experiment program are described in Chapter
II. The results of the investigation are presented in
Chapter III which is followed by a summary. and concluding

remarks in Chapter IV.

1.2. Notation

The symbols used herein are defined in the text

where they first appear. For convenience. the mosr

important ones are summarized here in alphabetical order.

E

Young's modulus

gravitational acceleration

moment of inertia of the beam section
length of one span of the two—span beam
mass per unit length of beam

bending moment

ultimate moment capacity of the beam section
yield moment = 2/3 Mu

concentrated load

(minimum) collapse load. (see Fig. 8)
(minimum) yield load. (see Fig. 8)

fundamental period of the two—span continuous
beam

time

horizontal velocity of the load

weight of sprung part of empty carriage
weight of load

distance from the first or second support

static deflection at (first or second}
mid-span

static deflection at x/L = 0.4323 due to Py
le/2L. speed parameter

W/Py. weight parameter

II . EXPERIMENTAL WORK

2.1. Experiment Set-up

2.1.1. General. The set-up of the experiments is essentially
the same as that used for a previous experimental study of
simply supported beams by Hills (3). For the sake of com—
pleteness. however. the set-up will be briefly described
herein along with certain necessary modifications in order

to accommodate the testing of the continuous span. An

overall view of the set-up except the "load arrester" is shown

in Fig. 1.

2.1.2. Test Beams. (See Fig. 3.) Each test beam is
6'-0" long (3'-0" each span) having a uniform rectangular
cross-section of 1” x 1/4" and loaded in the weaker
direction. The specimens were cut from hot rolled mild
steel (C 1095) bars (each about 20 ft. in length) rolled
specially for this investigation by the Bethlehem Steel
Company. The total weight of the test beam is about

5.1 lb. The elastic fundamental natural frequency of
vibration is 17.7 cps. The yield stress of the material
which averages about 27.500 psi will be discussed in
detail in Sectibn 2.3.1. The static behavior of the

continuous beam is discussed in Section 2.4.

U‘!

To facilitate adjustment for a smooth juncricn between

31

the test beam and the ”approach span, each end of the test
beam (as well as the approach span) was beveled so that

it makes an angle of 600 with the top face of the beam-

At the supports. the beam is connected by set screws to
”support seats" which are. in turn. attached to the
"abutments" in a manner described in Ref. 3. The length
of the support seat is 19 for the end support. and 1/2"
for the middle support. There are elongated slots in the
seats of the approach support and the middle support in
order to allow horizontal movements; the seat at the third

support is pinned and not allowed to undergo any trans-

lation.

2.1.3. Load Carriage. (See Fig. 2.) The load carriage
consists of a load box supported by two leaf springs which
are attached to a chassis bar. The springs have a com—
bined stiffness of 181 lb. per in. One end of the bar

is attached to a wheel. the other to a stabiliZing bar’which
is connected to a monorail by rollers. The load box is
restrained from horizontal motion relative to the chassis

by a rod and sleeve connection. the rod being anchored at
the chassis and the sleeve attached to the box. As a

safety measure (in case the test beam should deflect
excessively under the carriage wheel). a vertical flat

bar hinged to the chassis is pin-connected to another verti-

cal bar attached to the monorail behind the main stabilizing

 

bar. I
slot so
them if

slot.

number <
weight c

to 12.6

bar. The pin between the two bars lies in an elongated
slot so that there exists no vertical constraint between
them if the pin is not in contact with either end of the
slot.

The weight of the load carriage depends on the
number of load plates placed in the load box. The "unsprung"
weight of the carriage is constant in all tests and equal

to 12.6 lb.

2.1.4. Driving System. (See Fig. 4.) The system consists

 

of a 12 ft. tower, at the top of which is supported a weight.
The weight is fastened to a wire rope which runs down out-
side the tower and then horizontally through several pulleys.
During test. the weight is released and drops down from

the tower top. An attachment on the horizontal wire rope
pushes the load carriage. thus transforming most of the
potential energy of the dropping weight into the kinetic

energy of the carriage.

2.1.5. Load Arrester. The load arrester (see Fig. 5)
consists of a pneumatic cylinder with its piston attached
to a wire rope. The rope, guided by four pulleys. extends

across the path of the carriage near the end of the track.

2.1.6. Abutments, Accelerating; Decelerating Tracks,
Monorails and Guide Plates. (See Fig. l.) The abutments
are made of concrete blocks and short pieces of steel

wide flange sections. The tracks and test beams are attached

to small brackets tied to steel plates which, in turn, are
connected to the top of the wide flange steel sections.
The accelerating and decelerating tracks are made of
1" x 1" square bars. The monorail is made of 1 1/2" x 1”
aluminum rectangular bars supported by steel "arms" which
are anchored to the web of the wide flange steel sections.
In order to ensure that the load carriage would
stay on the test beam, a "guide plate" is placed along the
beam (see Fig. 3). A fork—like attachment to one side of
the load carriage straddles the plate as the load moves on
the test beam. Rollers at the end of "fork" minimize the
friction between the plate and the carriage in case they

touch each other. (However, in tests they rarely did.)

2.2. Measurements and Instrumentation

 

The main equipment for the experimental study is

a four channel Sanborn 150 Recording System (Fig. 1).

2.2.1. Beam Deflections. The deflection histories of
both mid-spans are recorded by use of two Schaevitz Linear
variable Differential Transformer, type No. 4000 XS—B, with

a range of.: 4".

2.2.2. Carriage Reaction. To estimate the dynamic force

 

between the carriage and the test beam the vertical
acceleration of the carriage is measured (when the leaf
springs are blocked, i.e., the entire carriage behaves like

an unsprung load) by use of an accelerometer (see Fig. 2).

 

When 1
spring

record

2.2.3.
the m1
the 83.
are clz
the pa:

can be

2.3.

lrn

2.3.1.
the bea
about 2
StreSs
Cut fro
SPEClme

Prepare

When the springs are allowed to act, the strains in the
springs are measured. In either case the response is

recorded by the Sanborn Recording System mentioned above.

2.2.3. Load Position. The times when the load is over
the middle support and the third support are marked on
the Sanborn recording paper by use of microswitches that
are closed by small pieces of steel wire extending across
the path of the load at the supports. The time of entry

can be detected from the record of other transducer outputs.

2.3. Stress-Strain and Moment—Curvature Relations

2.3.1. Stress~Strain Relation. As mentioned previously.
the beams used in this study were prepared from steel bars
about 20 ft. long- In order to determine the yield
stress of the material. three tension test specimens were
out from each bar. The average yield stress of these
specimens is used as the yield stress for the test beams
prepared from the bar.

The values of the yield stress of the specimens
do not vary appreciably. They average around 27,500 psi.
For each bar, the difference of the yield stress of any
specimen from the average value of the three specimens
from the same bar is less than 4.2%; generally, it is on
the order of 2%.

A total of 76 tension specimens were tested for

this investigation. The specimens are 10 1/2" long and.

for convenience, have the same constant cross-section as
the bar, i.e., l" x 1/4". Some of the results obtained by
use of these specimens were compared with those obtained by
use of specimens having the same shape as the ASTM standard
tension coupon. Four pairs of specimens. each prepared
from the same bar were tested for this purpose. The yield
stress indicated by the specimens with the constant cross-
sectional area was found to be a little lower than that
recorded for the standard specimens. The difference.
however, is small and less than 1.5%.

The yield stress was taken to be the apparent stress
at which the load on the test specimen first began to drop.
The speed of the moving part of testing machine was
4 x 10-4 in./sec. Shown in Fig. 6 is a typical stress—

strain diagram for the material used in this study.

2.3.2. Moment-Curvature Relation. In Fig. 7 is shown the

 

moment-curvature diagram obtained from a simple beam test
as-illustrated in the figure. The moment was computed

from the loading on the beam. The curvature was calculated
from the strains at the top and bottom faces of the beam
measured by means of SR-4 strain gages.

The yield moment was computed from the loading at
which the strains of the beam began to "flow." For the
case shown in the figure. it is equal to 301 in.-lb..
while computed from the usual beam theory and using the

yield stress from the tension test. the yield moment is

10

292 in.-lb. The ultimate moment was computed from the
load at which the deflection of the beam began to increase
rapidly. It is equal to 448 in.—lb. as against the theo—
retical value of 438 in.—lb. based on the usual theory of
plasticity using the experimental yield stress.

The "simple beam" test depicted above was repeated
using a different beam. The results obtained are not

appreciably different from those shown in Fig. 7.
2.4. Static Behavior of Continuous Beam

2.4.1. Theoretical Analysis. It seems natural to consider
the static behavior of the structure as a basis for dis—
cussion of its dynamic behavior. In this section, certain
important properties of the static response of a two-span
symmetrical and continuous uniform beam to a single

movable concentrated load is discussed. The beam is assumed
to be made of material having an ideal elasto—perfectly
plastic stress-strain relation. The cross-section is
rectangular. The analysis is based on the usual theories

of elastic and plastic analysis of structures.

In Fig. 8 are plotted the "yield load” and "collapse
load" of the beam as functions of the position of the load
(due to symmetry, only one span is shown in the figure).

The "yield load” is the load at which the maximum stress
in the beam just reaches the yield stress.

The maximum stress or bending moment occurs at

the point where the load is applied. The yield load is

11

thus obtained by setting the (elastic) bending moment at
the section under the load equal to the yield moment.

My =-% Mu. where M.u is the ultimate moment of the cross-
section. The "collapse load" is the load at which the beam
becomes a mechanism with one plastic hinge under the load
and another over the middle support.

It is seen that the yield load is a minimum when
the load is at x/L = 0.4323. in which x is the distance
measured from the first support and L is the length of one
span. Subsequently this minimum load. which will be denoted
by PY and has a magnitude of 3.2148 Mu/L. will be simply
referred to as the "yield load." and used as a measure of
the weight of the moving load. Similarly. the minimum
collapse load which is equal to 5.827 Mh/L and occurs at
x/L = 0.4142. will be simply referred to as the "collapse
load" and denoted by the symbol PC. The value of the ratio
of the collapse load to the yield load is 1.8125.

Fig. 9 shows a plot of the elastic deflection at
the point under the load as a function of the position of
the load. The maximum value is equal to 0.01509 %%3 with
the load at x/L = 0.4620. At x/L = 0.4323. and x/L = 0.5,
the deflections are equal to 0.01493 and 0.01497. respective—
ly. times %%3. Subsequently. the quantity of yy, the
deflection at x/L = 0.4323 for P = Py’ will be used as a

measure of the dynamic deflections. For the beams used in

the tests. the average value of yy is 0.684 in.

12

2.4.2. Experimental Behavior. A static loading test of a

 

continuous beam is depicted in Fig. 10. The load. applied
gradually at x/L ~= 0.4323. is plotted against the maximum
strains recorded for the section under the load and that
over the middle support.

The observed yield load corresponds to the load
at which the observed strains began to increase rapidly.
The "theoretical" values were computed as explained in the
preceding section.

It is of interest to note that while the theoretical
PY is only slightly larger than the observed Py (35.2 lb.
versus 34.8 1b.). the observed PC is appreciably larger
than the theoretical PC (76 1b. versus 63.71b.). The
latter difference can be attributed to the strain-hardening
behavior of the real moment-curvature relation Versus
the ideal elasto—perfectly plastic relation used in com-
puting the theoretical value. The strain-hardening behavior
is also evident in the load-strain curve for the point
under the load.

The above test was repeated for another beam: the
results obtained showed no significant difference from

those described in the preceding.
2.5. Test Program

2.5.1. Parameters of Study. In this investigation there
are essentially two parameters: the weight (W), and the

speed (v) of the load. In the subsequent description of

 

Paramet

Springs

13

the test program and the presentation of the test results.
these parameters are made dimensionless. The corresponding

dimensionless parameters are:

le
a = fi— = 0.00941 v (1)
a = 1'— s 717 W/o (2)
PY y

in which T1 denotes the fundamental elastic period of the
beam. and 0y is the yield stress of the beam material. In
the last expression in Eq. (1). v should be in fps; in
the last expression of Eq. (2) 0y should be in psi and
W in lb.

Note that for the sprung load tests. no additional
parameter presents itself as there was only one set of

springs used in the tests involving sprung loads.

2.5.2. Unsprung Load Tests. Nine values of B wens used
ranging from 0.56 to 1.8. Values of a used ranged from
0.002 to 0.1442 (corresponding to speeds from 0.2 fps to
15.3 fps). The number of speeds used for each value of
5 generally decreases with an increase in the value of
6--from 7 for 0 = 0.56. to 2 for B = 1.8. Also. because
of the ”jump“ phenomenon occurred in the test and to be
discussed later. the maximum value of the load speed
used in the test is generally smaller for larger values

of B.

14

2.5.3. Sprung Load Tests. Four values of B were used ranging
from 0.6 to 1.6. The a values used ranged from 0.002 to

0.1315 (corresponding to speeds from 0.2 fps to 14.0 fps).

III. RESULTS OF INVESTIGATION

3.1. Unsprung Loads

3.1.1. Typical Patterns of Behavior. From the test data
obtained. there were seen four distinct patterns of behavior
of the load-beam system as affected by the speed parameter a
and the weight parameter 6. These patterns are discussed

below.

3.1.1.1. Behavior withqflMedium:_Value of a and "Medium"

 

value ofgfi. In Fig. 11 is shown the test records for the
case of a ”medium" value of a (= 0.05623) and of B (= 1.0).
It should be noted that the terms: ”medium.” "small." and
"large” are used here and subsequently only in a qualitative
sense to facilitate the presentation.

The three vertical lines in Fig. 9 mark the times
when the load was at the first. second. and last support.
going from left to right. The average speed of the
load on the first and second spans. computed from these
times. are 6.12 fps and 5.83 fsp. respectively. The top
trace represents the time history of the vertical acceler—
ation of the moving load; a downward acceleration is above
the datum line. The reaction on the beam is. of course.
directly proportional to this acceleration. Hence. the

acceleration trace may be regarded also as the reaction trace.

15

16

It is seen that there is little variation in the acceler-
ation except when the load is near the second and the third
supports. At these locations, the deformations of the
beam are relatively insensitive to variations in the inter-
action force. The maximum acceleration were 0.641 g and
1.396 g for the first and second span. respectively.

The second trace represents the deflection history
of the mid-point of the first span. The maximum static
deflection of this point. computed on the basis of perfect
elasticity and the load applied statically at the same
point. is marked by a horizontal line in the figure. It
is seen that the maximum dynamic deflection is approxi-
mately 30% larger than the static. After the load has
passed the test beam. this point exhibits a fairly
regular harmonic and slightly damped motion in the first
natural mode. There was no permanent deflection.

The third trace represents the deflection history
of the mid-point of the second span. In this case. the
maximum dynamic deflection is also about 30% larger than
the maximum static. No permanent deformation could be

detected in this test.

3.1.1.2. Behavior with "Small"Value of a and ”Large”

 

Value of Q. In Fig. 12 are shown the test records for the
case a = 0.04468 and B = 1.6. The general patterns of
the response curves are similar to those shown in Fig- 11.

In particular. it may be noted that the reaction on the

l7

beam. as seen from the acceleration trace. did not vary
substantially (except near the supports). The behavior
in this case. however. differs from the preceding one in
the occurrence of permanent sets or deformations. It is
of interest to note that the permanent set of the first
span is 0.053 in. while that of the second span is 0.018
in. only. These sets are actually relatively sma11-—less
than 8% of the yield deflection yy. This difference is
thought to be due to two reasons: (1) the influence of
residual bending moments. and (2) the influence of decreased
load speed in the second span.

For this particular case. the beam may be thought
of as being subjected to a heavy load first at the center
portion of the first span; the load was then taken off and
placed on the center portion of the second span. (Note
that from the acceleration records. the magnitudes of these
loads were about the same.) Because of the large magnitude
of the load. when the load is at the first span. inelastic
action takes place in the beam. and the second span would
be called upon to take on a larger negative bending moment
than that would be required if the response were completely
elastic. (See Fig. l3(a)). Upon unloading—-the load moves
toward the middle support--a distribution of negative
"residual” bending moment would exist in the beam as shown
in Fig. 13(b).

As the load goes near the center portion of the

second span. it causes positive bending moment in this

18

span. In order for inelastic action to take place. the
residual negative moment would have to be overcome first.
Hence. the beam would appear stronger in the second span.
This would explain. at least in part. why the permanent
deformation in the second span is smaller. The other
reason for this smaller damage is the fact that when the
load came on the second span. its speed had been decreased.
and small speeds seem to cause smaller dynamic effects.
A difference in the magnitude of accelerations. though
small. can be seen from the top trace of the record.

For this type of behavior. the shape of the final
deformed beam could appear as that depicted either by
Fig . 14(a) or (c). depending upon how large B was. In
Fig. 14. a "hinge” represents a noticeable region of
permanent deformation or curvature. The "hinge" actually
has the shape of a smooth ”bow.” and its characteristics
are considered in Section 3.1.7. for several typical
cases. It should be noted also that the "hinge" over the
middle support usually represents an yielding of appreciably
smaller magnitude as compared to the yielding of the

mid-span points. In some cases. it is hardly noticeable.

3.1.1.3. Behavior with "Medium" Value of a and "Large"
Value of Q. In Fig. 15 are shown the test record for the
case a = 0.06108 and B = 1.6. The behavior in this case
is quite different from the preceding ones. As the load

enters the beam there is a downward acceleration, signifying

19

a smaller reactionary force than the static. But the
acceleration rapidly becomes upward with a maximum value of
0.612 g (signifying that the reactionary force is 1.612
times the static load) that occurs when the load is approxi—
mately at the two thirds point of the first span. Then.

it suddenly turns downward with a magnitude of one g.

This indicates that the moving load has lost contact with
the surface of the beam or has "jumped." The period during
which the acceleration remains constant at one 9 is the
duration of the load ”in the air." This period covers

the interval from approximately 1/5 of the span length to
the left of the middle support to approximately 1/10 of the
span length to the right of the middle support. As the
load lands on the second span there would. of course. be

an impact as evidenced by the large and rapid variations

of the acceleration. After the load stabilizes itself.

the pattern of acceleration becomes similar to that when
the load was on the first span.

The phenomenon described in the preceding is also
reflected in the deflection traces in Fig. 15. It is seen
that when the load is in the air. the beam vibrates freely.
In contrast to the previous case. the permanent set in the
second span is larger than that in the first span (0.109 in.
versus 0.035 in.). This may be explained by the “jump"
phenomenon and the associated impact effects.

The maximum dynamic deflection in this case is

larger than the maximum static deflection by 47%. and 46%.

20

respectively. for the first and second spans.
The shape of the final deformed beam may appear

either as that shown in Fig. 14(b) or 14(c).

3.1.1.4. Behavior with fLarge" Value of a and VMedium"

 

value ofgfl. In this case the jump phenomenon as described
in the above section also takes place. A permanent dis—
placement is usually seen in the second span. and the

final shape of the deformed beam appears as that shown in
Fig. 14(b). The hinge at the middle support may or may not

be formed.

3.1.2. "Jump” Phenomenon. The jump phenomenon brought out

 

in the preceding section has been noted also in test records
for other combinations of the values of a and 5. Although
the phenomenon appears interesting. to study the behavior
of the system involving such a phenomenon would seem to
have questionable practical significance. as it is a moot
point. for example. whether a vehicle should be knowingly
allowed to jump on a bridge and cause an impact. Neverthe-
less. it is deemed significant to mark the critical combina—
tions of the values of the parameters at which a jump would
first occur. This information is shown in Fig. 16. It
is seen that jump would occur when the value of either
a or 0. or both. are large.

It may be noted that in order to verify the
occurrence of a jump. in some tests a piece of thin white

paper was placed over a piece of thin carbon paper which was

21

attached to the beam. As the load moves on the beam. the
reaction marks on the paper an impression with an intensity
varying according to that of the pressure. The jump
phenomenon is seen by noting a gradual fading and then a
complete disappearance of any trace of carbon on the paper.
The subsequent landing impact of the load after the jump is
manifested by the intense coloring of the white paper.
Although the phenomenon can be observed. it does
not seem feasible to explain it fully. If the load is in

contact with the beam. then the following relation holds:

£1 =-%% + v ax (3)

t
in w 1c 18 t e vertica ve oc1t o t e oa .
'h'hg-Y' h 'll'yfhld

-%% is the (local) vertical velocity of the beam. and-%§
the instantaneous slope of the beam. For a given time the
value of the right-hand side of the equation is fixed.
The actual value of 3% depends on the load trajectory which
of course is influenced by all the variables that enter
into the problem. A jump occurs when the value of g%
is larger than that of the right-hand side of the
equation.

In Fig. 17 are plotted. for various values of B.
the location at which the load leaves the (first) span as
a function of a. It is seen that. for B < 1.2. this

location in general moves closer to the middle support

with increasing values of a. However. such is not the case

22

for large values of B. It may be noted that the range
within which this location varies is not large: 0.83 L to

0.98 L.

3.1.3. Maximum Mid—Span Dynamic Deflections. The maximum
dynamic (downward) deflection of the first mid—span is
plotted in Fig. 18. as a function of a for various values
of 5. Almost all curves indicate a trend of increasing
deflections with an increase in the value of 0. Furthermore.
the larger the value of B. the greater the rate of the
increase. The exception concerns the case with the heaviest
load: 5 = 1.8. In this case the relative large deflection
at a very low speed is thought to be due to the longer time
available for the inelastic deformation to take place.

The ”jump line" as depicted in Fig. 16 is reproduced
in this figure also. indicating that the results to the
left of the line correspond to tests that did not involve
a jump. and those to the right correspond to tests in which
the jump phenomenon was observed. However. it should be
noted that in this case the maximum deflection plotted
generally occurred before the jump did take place. Such
is not the case for the maximum mid—span deflections in
the second span which are plotted in Fig. 19.

The data in Fig. 19 show the same general trends as
those in Fig. 18. However. the increases in the deflections
involving the jump phenomenon. as expected. are much larger.

since a jump affects the second span response to a much

23

greater extent than the first span response. It is also
noted that. unlike the case for the first mid-span de-
flection. the second mid—span deflection for B = 1.8 at
the lowest speed is smaller than that at the next higher
speed. This is thought to be due to the existence

of the residual bending moments set up by the yielding in
the first span as expLained in Section 3.1.1.3.

In Fig. 20 is plotted the distance between the
location of the moving load and the first support when the
maximum first mid-span deflection occurred. This distance
varies within a region from 0.4 tho 0.75 L. For the case
of a single static load and a perfectly elastic beam. it
is 0.4620 L. (See Fig. 9.) In general. an increase in
a results in a larger value for this distance.

Similarly the distance from the second support
to the position of the load at which the maximum second
midespan deflections occurred is plotted in Fig. 21.

The trend is similar to that shown in Fig. 20. But for
large values of a. the results become somewhat erratic

due to the occurrence of the jump phenomenon.

3.1.4. Mid-Span Permanent Deflections. The permanent

 

deflections of the first and second mid-spans are plotted
in Fig. 22. Downward deflections are shown positive. It
should be noted that these permanent deformations are
taken after the load has crossed both spans. Because of

the anti—symmetrical mode of the deflection curve. the

 

per-mane

to offs

small. 1
there is
span at
discussi
longer t.
heavy 105
was cause
span yiel
Second Sp-
Strehgthe
previouSl:
A-
Um dYnam:
reSul ting

Span and

24

permanent deformations at the first and second spans tend
to offset each other.

Therefore. except for the heaviest load considered.
6 = 1.8. the magnitudes of the permanent sets are quite
small. being only a small fraction of yy. For B = 1.8.
there is a relatively large permanent set in the first
span at the lowest load speed. As mentioned in the previous
discussion of maximum deflections. this is because of the
longer time available for the beam to yield under the
heavy load. The negative permanent set of the second span
was caused by the uplifting that occurred when the first
span yielded under the load; the subsequent loading of the
second span did not offset this uplift on account of the
strengthening effect of the residual moment as described
previously.

At higher speeds and with the occurrence of jumps.
the dynamic effects in the second span became dominant
resulting in large positive permanent sets in the second

span and smaller negative sets in the first span.

3.1.5. Maximum Acceleration of Load. The maximum vertical
acceleration of the moving mass on the first span is plotted
in Fig. 23. The acceleration is scaled by the gravitational
acceleration 9. Thus. for a given 5 value. the vertical
ordinate represents the maximum increment of the dynamic
load over the static load in terms of the static load.

It is seen that. for tests that did not involve a

25

jump. this increment is less than unity. For those in—
volving a jump (at higher values of a) the increment can
attain values as high as two. However. it should be noted
that. in general. these maximum accelerations actually
occurred before the jumps took place. The point with an
acceleration of 2.6 g at a = 0.0775 is an exception.

In this case the load jumped and landed also on the first
span; this resulted in the relatively large values of
acceleration.

Similarly shown in Fig. 24 is the maximum acceleration
of the load on the second span. The values of the acceler-
ation are generally larger than those in the first span.

For cases involving no jump. the maximum value is 2 g:

for cases involving a jump. the maximum value reaches

4.6 g. In the latter cases. of course. the high values were
associated with the impact on the second span after the

jump.

3.1.6. Deformation Under Repeated Loading. In Fig. 25 are
plotted for a number of test beams the accumulated permanent
set of the first mid-span under repeated loading. In

each case the static magnitude and the speed of the load
were kept the same.

It is observed that the permanent setseither in-
crease or decrease monotonically with the number of load
applications. The case of increase is associated with large
values of 5. while that of decrease with smaller values of

B and usually at high values of d.

26

This can be explained by the observation already
made in Section 3.1.1. In the case of large values of
B. the first span suffers a greater amount of damage than
the second span. For smaller values of B and at high
speeds. the negative permanent sets are really caused by
the downward permanent deformation in the second span re-
sulting usually from the jump phenomenon.

Similar data are shown in Fig. 26 for the accumulated
permanent set in the second mid—span. In this case. the
deformations all increase with the number of applications
of the loading.

It is noted from both Fig. 25 and Fig. 26 that as
the number of applicationsincreases. there is no sign that
the beam would ”stabilize" in the sense that the increase
in deformation per application would become smaller. Hew-
ever. it must be pointed out that the number of applications
considered in the data is very small. with six as the maximum.
Such stabilisation might take place with a larger number
of applications for cases having smaller values of B.

For the case of repeated loading. measurements were
also recorded of the maximum mid—span deflections and the
slopes of the beam at the two end supports. These data
indicate similar trends to those mentioned above for the

permanent sets: hence. they are not presented here.

3.1.7. Permanent Curvature and ”Hinge Length." In this

section the permanent curvatures of the "hinges" in six

2?

beam specimens are studied. The term ”hinge length”
denotes the distance interval within which permanent curva—
tures are noted. (An end of a ”hinge" is marked by the
point at which the beam begins to deviate from a straight
line.)

In Fig. 27 is plotted the distributions of the
permanent curvature along the hinge length for different
specimens. (The curvatures were obtained from the displace-
ments by use of finite difference expressions for which
a difference interval of 1/7 of the hinge length was used.)
Although the distribution is not symmetrical. in each case
there is only one maximum peak value. It may be noted that
by comparing with the bending moment—curvature graph shown
in Fig 7. the numerical value of these maximum curvature
would appear to be quite reasonable.

In Table l are listed additional data on the ”hinges"
for the six specimens. In this table the term ”angle change"
represents the angle between the tangents at the two
ends of the hinge. and "average curvature"is obtained by
dividing the angle change by the hinge length.

It is seen that generally the hinge length is shorter
in specimens used in dynamic tests than in static tests.

The shortest hinge (specimen 8-22—2) corresponds to a
sprung load case which is to be discussed in the next
section. It is of interest to note that. for this case
the value of a is the largest of the cases considered. and

the distribution of curvature shows the sharpest and highest

28

 

 

mmo.~ mmma.o oom.v u- .. =m\auo Aumoa oflumumv «gem
ooo.m mmoa.o mno.m mcsHchp a =m\Hum mooeo.o m.H muaaum
oav.a ammo.o mna.a manuamcp a =w\muo mmvno.o ¢.H Nunaym
omm.m moaa.o ooo.OH magnum a =e\m:m ovmmo.o ~.H «-mmum
ooo.m omma.o m~>.m mcsummca N =m\aum mnemo.o ~.H muoum
mum.m «maa.o oom.m magnumca a .oa\maum oeomo.o o.H mumaam

A OH xv wwmmmw A OH xv mmmu mcoaumoaamm¢

wmmum>usu mamcm ommum>usu mcsnmmcb . Md :nwmmwm o u coawwmmm

mmmnm>¢ Hmuoe .xmz no masmmm mo quEsz. . = .

.mnsum>u50 ucmcmfinmm 0cm :numcma macaw:

.H magma

29

peak among the curves in Fig. 27. In terms of span length.

the hinge length varies from 1/10 to 1/5. The maximum

2 to 10 x 10‘2 rad./in-

and the average curvature varies from 1.40 x 10“2 to 3.60 x

10’2 rad./in.

curvature varies from 1.775 x 10-

3.2. Sprung Loads

 

3.2.1. Typical Patterns of Behavior. In order to gain some

 

insight into the behavior of the beam—load system consxdered.
three typical test records are presented and discussed in
the following. It should be noted that in all the sprung
load tests. included in the value of B there is a constant
fraction approximately equal to 0.32 that corresponds to the
unsprung portion of the load carriage. The natural
frequency of the sprung load. of course. varies with the
value of B. It is on the order of 6 cps. or about 1/3

of that of the unloaded beam. Damping in the load system

is not significant.

3.2.1.1. Behavior with "Medium" Value of a and “Nediug;

 

Value of_§. Shown in Fig. 28 are the test records for the

 

case: a = 0.0516 and 6 = 1.2. The top trace represents

the vertical acceleration of the unsprung portion of the
load. In contrast to the unsprung case. this acceleration
curve oscillates a great deal. However. if a mean curve

is passed through the oscillations. a trend can be discerned

that is similar to the unsprung case shown in Fig. 11.

30

The magnitude of the acceleration is rather small. on

the order of one third of g corresponding approximately to
0.1 Py' Hence. its influence on the total reactionary
force is relatively small.

The second and third traces. as before. represent
the deflection histories of the mid—points of the first and
second spans. The maximum dynamic deflections are 38%
and 21% greater than the static ones for the first and
second spans. respectively.

The trace at the bottom represents the variation
of the total force in the springs of the carriage. The
datum line represents the static value. An upward excursion
of the trace denotes a decrease in the reaction on the beam.
As expected. as the load first enters the beam. the down-
ward displacement of the beam causes a decrease in the
spring force. The subsequent behavior of the spring force
is governed by the complex interactions of the two mechanical
systems: the beam and the carriage. Near the first mid-
span there is a relative maximum increase of sprung force
of 12.6 lb. (approximately l/3 Py). A similar relative
maximum in the second span amounts to about 1/4 Py only.
This is in agreement with the observation that the maximum
mid-span deflection of the first span is appreciably
larger than that of the second span.

No permanent set or distortion is seen for this test.

31

3.2.1.2. Behavior withyflSmall" Value of a and "Large"

 

Value ofyfi. Figure 29 shows the test records for the case

 

d = 0.04313 and B = 1.6. Qualitatively. these records are
similar to those shown in Fig. 28. The distinguishing
feature of the test recorded in this figure is that there
is a permanent set in the first span. though it is small in
magnitude. There is no permanent set in the second span.
The maximum mid-span deflections are 49.7% and 19.6% larger
than the static values for the first and second spans.
respectively. The maximum increases in spring force near
the mid-span correspond approximately to 0.45 Py and 0.33 Py’
respectively. for the first and second span.

For this type of behavior. the final shape of the
deformed beam is similar to that in the unsprung case. and

is illustrated in Fig. 14(a) or (c).

3.2.1.3. Behavior with ”Medium" Value of a and "Large"

Value of 9. This case is illustrated in Fig. 30 for

 

a = 0.05795 and B I 1.6. The feature of this result is

that the carriage has ”jumped.' or lost contact with the
surface of the beam. in the vicinity of the middle support.
This may be seen by noting the free vibration in that inter-
val as indicated by the two beam deflection traces. Unlike
the case involving the jump phenomenon for an unsprung

load. the acceleration trace does not remain constant but

varies somewhat. This is probably due to the presence of

the spring force. The spring.force. in this case. does

 

not var
contras
vicinit

was inv

first 5
displac
load cr
the equ
interva
permane
(about

of the

load.

32

not vary appreciably in the interval of jump; this is in
contrast with the large variation of spring force in the
vicinity of the middle support for cases in which no jump
was involved.

The maximum dynamic mid-span displacement in the
first span is 60% larger than the static. The permanent
displacement had been 0.296” (about 0.43 yy) before the
load crossed the second span. This value was estimated from
the equilibrium position of the free vibrations in the jump
interval. After the load crossed the second span. the
permanent set of the first span decreased to 0.166"

(about 0.24 yé). This. of course. is due to the uplifting
of the first span when the second span yielded under the
load.

There is no net permanent deformation in the second
span. The reason lies in the upward permanent deflection
and negative residual moments in the second span caused by
the substantial yielding of the first span before the load
enters the second span. A similar situation has been
explained in Section 3.1.1.2.

The end of the jump. coming shortly after the load
passed over the middle support. is marked by the violent
oscillations in the acceleration trace. It is of interest
to note that. in this case in contrast to the case of
unsprung loads. the impact did not result in large response
of the second span. This is thought to be due mainly to

the shook-absorbing influence of the carriage springs.

33

The final shape of the deformed beam may appear as
that shown in Fig. 14(c) with or without the hinge over the

middle support.

3.2.1.4. Behavior with 9Large” Value of d and ”Medium"
Value of 6. In this case the jump phenomenon as described
in the unsprung case and above section also occurs. The
final shape of the beam appears as that shown in Fig. 14(b)

with or without the hinge over the middle support.

3.2.2. ."JumpiLine." As was done for the case of the

 

unsprung load tests. a "jump line" indicating the combinations
of the values of a and B at which jump‘would first take
place is plotted also in Fig. 16. It is seen that this

line lies very close to that for the unsprung load tests.

3.2.3. Maximum Mid—gpanggynamic Deflections. The maximum
deflection of the first mid-span is plotted in Fig. 31.
The trend seen from these curves is similar to that noted
in the case of the unsprung load tests; that is. the deflec-
tion increases with an increase in a. However. at higher
values of a. the rate of increase becomes smaller. even
suggesting a tendency of decreasing deflections had the
values of u been increased further.

Similar data for the second mid—span are shown
in Fig. 32. The large deflections to the right of the jump
line are. of course. caused by the impact resulting from the

jump. The location of the moving load when the maximum

34

deflection occurs is plotted in Fig. 33 and 34 for the two
mid-spans. Likewise. the data are similar to those described

previously for the unsprung load tests.

3.2.4. Maximum Spring_Force. The maximum compressive

 

spring force. in terms of ws. which denotes the static
weight of the sprung part of the carriage for B = 0.56.

is plotted as a function of a for various values of B in
Figs. 35 and 36 for the first and second span. respectively.
The data indicate that the spring force increases as d
increases. For larger a values. the maximum spring force
at the second span increases faster than that at the

first span with an increase in d.

Fig. 37 shows the location of the maximum spring
force at the first span. The location in general moves
toward the middle support with an increase in the value of
either a or B. For small values of a. the location is near
the point x = 0.46 L for all the data presented. An exception
to the above trend occurs for values of a close to 0.05
and larger values of B. In this case. the maximum spring
force occurs at the middle support. The location of the
maximum spring force in the second span is shown in Fig. 38.
The data appear erratic on account of the jumps of the

carriage involved.

IV. SUMMARY AND CONCLUSION

4.1. Summary

An experimental study of the elasto-inelastic

behavior of two-span continuous beams subjected to moving
loads has been reported. The variables of the study have
been the speed parameter a and the weight parameter 6.
In the study the value of a was varied from 0.002 to 0.1442
(0.2 fps to 15.3 fps). and the value of 5 from 0.56 to 1.80
(0.56 to 1.8 times Py’ the minimum yield load). The typical
patterns of behavior observed are summarized as follows.

(1) For medium values of a and B.-the load moved smoothly
over the beam. Although the dynamic deflection was
appreciably larger than the static deflection. there was
no permanent set.

(2) For a small value of d and a large value of B. the
load also moved smoothly over the beam. Permanent de—
flections were observed. The permanent deflection of the
first span was larger than that of the second span. This
is thought to be due largely to the negative residual
moment set up in the beam as a result of the yielding in
the first span.

(3) For a medium value of a and a large value of B.
the load "jumped" off the beam as it approached the middle

support. Then it "landed" on the second span at a short

35

36

distance beyond the middle support. The resulting impact
on the second span caused relatively large deflections and
permanent sets. particularly at the (second) mid—span.

(4) For a large value of a and a medium value of B.
the behavior is similar to the preceding case. but usually
there was no permanent set in the first span.

The maximum permanent deflections observed were
actually small in magnitude. If no jump had occurred in
the test. they were on the order of 0.1 to 0.3 of yy. If
jump had occurred. the permanent deflection could be much
larger. However. for each load weight. the value of the
maximum load speed used was limited by the jump phenomenon
in the sense that the maximum speed used was only slightly
larger than the speed at which jump was first observed.

A limited study on the effects of repeated loading
showed that there was no stabilization of the permanent
deflections after a few cycles of loading. A study of
the length of the region of yielding. or the "plastic hinge
length." indicates that it may vary from 1/5 to 1/10 times
the span length. Except for the mitigatory effect of
the load springs on the impact after a jump. for the data
obtained. there is no marked difference in the general
trends of behavior between the tests using sprung loads

and unsprung loads.

37

4.2. Discussion and Conclusion

 

Perhaps the most prominent phenomenon observed in
the experiments is the "jump." Because of the impact
involved in the phenomenon. it changes the nature of the
problem. As mentioned previously. the significance of a
study of the beam response involving an impact of this type
seems of questionable engineering significance. as it is
a moot point. for example. whether a vehicle should be
knowingly allowed to jump on a bridge.

On the other hand. it seems worthwhile to know the
combinations of the parameters at which such a jump would
occur. In this connection. it may be noted that a jump
would occur at either a heavy load or a high speed. As the
load is made heavier. the speed at which a jump first
takes place decreases. Thus. if the jump is not allowed
to happen. then it would seem that the maximum load that
could pass the beam could not be much larger than the static
collapse load. However, it is noted that at B = 1.8 (the
static collapse load corresponds to 6 = 1.8125). the load
can still cross the beam with a permanent set less than yy.

As possible topics of future investigation, it is
suggested that. first. an analytical study be made to
compare with the experimental work reported herein. and
secondly. the problem of multiple axle loads be considered.
It could happen that due to a distribution of the total

load. higher speeds might be attainable without the

38

occurrence of a jump. This. in turn. could raise the
upper limit of the total weight of the load that can cross

the beam.

LIST OF REFERENCES

(l) Parkes, E- W., "How to Cross an Unsafe Bridge.”
Engineering, V01- 186. November. 1958.

(2) Symonds, P. 8.. Neal. B. G., "Travelling Loads on
Rigid—Plastic Beams." Journal of Engineering
Mechanics Division. ASCE. Vol. 86. No. EMl,
January. 1960.

(3) Hills. R. E.. "Inelastic Behavior of Simple Beams
Subjected to Moving Loads." M.S. thesis (in
preparation). Department of Civil Engineering.
Michigan State University.

39

FIGURES

41

7..
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Figure 1--Overa11 view of test set-up.

 

Figure 2--Load carriage.

42

 

 

Figure 3--Test beam and differential transformers.

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43

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Moment (in.-1b.)

44

     

32000 .
27500 psi

ASTM Standard coupon
Gage length 2"

24000 Pi

 

1.7 x 10‘2 in./in.

 

 

 

 

 

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350 b /1Theoretical value = 438 in.—lb.)
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45

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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15
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Y Y (for 0y=27500 psi)
PC = 1.8125 Py a = 1.8125 Pc = 69.5493 lb.
(for 0y=27500 psi)

Figure 8--Curves of yield load and collapse load.

\

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m.o

q/(peoq pue aloddns puooes ueemqeq eouensrq)

.cmmm umuaw so mouom mcflumm w>HmmmHmEoo EDEmezIImm musmmm

d nmumamumm oommm

omH.o mNH.o OOH.O mho.o omo.o mNo.o

 

{mafia mesh; IIIIIYI/

 

 

 

 

 

71

 

 

 

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mmmo masmmm . \\ /

 

 

 

 

 

 

 

72

omH.o

 

mNH.o

.cmmm Ugoumm co mouom mcflumm ®>HmmemEoo ESEmeZIImm mnsmem

o umpwEmnmm owmmm

OOH.O

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mho.o

omo.o

mNo.o o

 

OQOD

 

1I|II03

 

 

 

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smcfiq QEDU:IIII\\/

 

 

mmmo magnum

 

 

/\
/

 

 

 

 

M/GDIOJ Burxds

S

73

.cmmm umuflm so mouom mQHHQm m>flmmeQEoo Esfiflxmfi mo coaumooquhm musmflm

o umumfimumm Ummmm

omH.o mNH.O ooa.o mno.o omo.o mNo.o o

 

 

\OQNKO
OOHr-i
OGOD

um

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mmmu mcsumm \

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1/(peoq pue qxoddns qsxra ueemqeq eoueqsra)

74

omH.o

.cmmm ocoomm co wouom mcfiumm m>flmmmumfioo ESEmeE mo coflgmUOAIImm wusmflm

mNH.o

OOH.O

o Mmumfimumm commm

mno.o

omo.o

 

ommu mcsumm

\\m

mNo.o

 

 

... wan-HA QED” .- V”.

 

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N.o

q/(peoq pue qxoddns puooes ueemqeq eoueqsrq)