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EXPERIMENTAL ANALYSIS OF
INDETERMINATE STRUCTURES AS
APPLIED TO RIGID FRAMES

Thesis for the Degree of M. S.
MICHIGAN STATE COLLEGE

Henry L. Thompson
I940

was?

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EAPERIMLNTAL ANALYSIS OF INDLTERMINATE
STRUCTURLS AS APPLIED

TO RIGID FRAMES

by
HENRY LOREN THOMPSON

A THESIS

Submitted to the Graduate School of Michigan
State College of Agriculture and Applied
Science in partial fulfillment of the
requirements for the degree of

MASTER OF SCIENCE IN CIVIL ENuINEERIHu

Department of Civil Engineering

1940

CONTENTS

Material
1. Introduction _______________
2. Furpose of Thesis ____________
5. Theory of Model Analysis —————————
4. History of Model Analysis ————————
5. Design and Operation of the
Deformefer Cages ________
6. Construction of theliodel --------
7. Single Span Frame Bent —————————
8. Analysis of the Rigid Frame
Bridge Models ————————
(a) Single Spen Model ..........
(b) Double Span.Model ——————————
(0) Triple Spen.Model ..........
9. Conclusions ................
10. Appendix .................
11. Picture of Deformeters and Models - - - -

(i)

EXPERIMENTAL ANALYSIS OF INLETERMINATE
STRUCTURES AS APPLILD

TO RIuID FRAMES

by
HENRY LOREN THOMSON

INTRODUC ION

.vw

During the past fifteen years there have been more statically
indeterminate structures built than in any other like period in the
history of construction. Such construction has been.made possible to
a great extent by the research worker experimenting with structural
models. Engineers have long recognized the economic and aesthetic
advantages of the continuous structure. The tie that has held the
designer to the old type of structure, consisting of abutments and
simple spans, is the simplicity of the mathematical theory. In the
highly complicated designs the designing organization is confronted
with two main problems-developing or recruiting designers capable
of handling the more complicated theory and the high cost of the de-
sign.

In order to aid in the solution of the problems of giving the
public the "aesthetic structure" the laboratory analysis of the stati-
cally indeterminate structure has been deve10ped. Model analysis is
a distinct method of approach to the solution of statically indetermi—
nate structures and can be used alone or in conjunction with the theo-

retical analysis as a check.

(1)

PURPOSE

 

The research to be described in this thesis has been carried out
with the following definite objectives in mind: (1) the design and the
construction of a sample deformeter gage that will be sufficiently accu-
rate for model analysis and one that can be made cheaply; (2) the test-
ing of the gages as designed with a direct comparison of results obtained
with the deformeter gages and a theoretical analysis of a statically
indeterminate structure; and (5) the experimental analysis of single,
double, and triple span rigid frame bridges to determine the variation
in the moment at the left knee due to a variation of the depth of the

haunch.

THEORY.Q§.MODELI£EALYSIS

The laws laid down by Clapeyron, Castigliano, Maxwell, Hooke, and
others are basic and some knowledge of them is, therefore, necessary
for a complete understanding of the mechanical solution. ,These under-
lying laws are not superseded by the laboratory methods but modern con-
ditions have simply created a demand for a mechanical short-cut in
applying them.

Assume the fixed arch as in Fig. 1 and let it be required to find
the reaction produced at point 1 along the line X—X by a load "P" at
point 2. First with the arch free of any load, suppose the point 1 to
be removed from its anchorage and moved a distance [SIX in the direction
X—X only. To accomplish this it will be necessary to apply not only
the distorting force X0, but, also, a restraining moment 20 to
prevent rotation of the arch about the point 1, and a linear restrain—

ing force To to prevent movement at right angles to the line X—X. As

(2)

 

 

 

(2A)

 

 

 

\ Hg. 4

no Z or Y movement takes place, no work is performed by these restraine
ing forces. Therefore, the external work done or the elastic strain
energy is given by the following equation:

U1 : Sxo‘g‘lx = W'

where W' is the internal work.

Now with the rib in its deflected position add the load "P" at
point 2 and its consequent reactions le, Ylp, and 21p. Since these
reactions perform no work, the second increment of work is

. U2 - %PA 2p = w"
where W" is the internal work. 102p is the corresponding deflection
at point 2 produced by the load "P" and Zle is the correSponding de-
flection at point 1 produced by X. All deflections in this discussion
have a similar meaning. By corresponding deflection it is meant the
measured deflection in the direction of the load or reaction at the
point considered. The total work performed on the arch is

I w = w' 4- W" = ég-onlx +' gPAgp

The same deflected position may be attained with the same exp
penditure of energy by applying X0, le, and "P" simultaneously,
in which case the external work is

U = %XoAh 4" SfxlpAlx +23}, (A 2p’A2x)
Since the external work is equal to the internal work, this equation
becomes

- - L‘ l. .1. 1

Since equation II is the expression for the total internal work, it is

(5)

equal to equation I. Equating I and II and cancelling out similar

terms gives

le 1x = PA 2x
or
le= P 4.2.3..
A lx
This last expression shows that any two related forces are
inversely proportional to the coincident displacement of their points
of application in the direction of the forces or inversely prOportional
to their corresponding displacements.
Similarly, by giving the reaction point 1 a-Y movement and
restraining it from both X and Z movement, the Y reaction may be ex-

pressed as
1x

The same method of attack is used in the determination of the
angular reactions or moment at the supports. Any moment Z may be
expressed as a couple as P times d (Fig. 5). If under the application
of this moment, point I of Fig. 5 moves to l? and point 2 to 2', the

total work done by the moment is
2

As P is equal to Z divided by d the external work becomes

V

 

U = 3d (41+Az) =

iZAzz

(4)

where A zz is the angular rotation expressed in radians. Therefore,
the external or internal work done by a couple is the moment of that
vcouple times the angular rotation of the coincident point divided by
two, provided this rotation is small, otherwise, (A1* A 2) divided
by "d" would not equal A no ‘

Now consider Fig. 4 which is an arch. First assume no load

on the rib and apply a moment Z at the point 1. The total work is
Ul = W' " IZoAlz

With this moment acting apply the load "P" at point 2 restraining point
1 from turning. In order to do this a new moment Z (moment at 1
due to the load "P") must be applied but no work is done by this new

moment. The total work done by applying this load is
02 = ePAzp= w"
The total internal work done by this body with these 'loads is
.. III II = W' + w" = fi-zodlz + éjPAgp

This same deflected position can be attained by applying all

of the loads and reactions simultaneously. The internal work is then

a I l 4. i

w zzoAlzi- 2271131513 3P(A2p .. A22)
- i A ,_ .1. A 4. I A l. A

IV W " 2Z0 12 :3le 12 2P 2p " BP 22

Where Agz is the deflection at point 2 due to the angular rotation at

point 1. Equating equations III and IV, since they are equal, le the

(5)

moment at the support is given by the following expression after
proper cancellation of terms:
Z - A21,
lp - P
12
For this work equation to be truce, the dimensions d, A1, and
A 2 of Fig. 5 must be measured in the same units as 422 of Fig. 4.
Therefore, if A12 radians distortion at point 1 produces A 22 inches
deflection at the load point 2, the moment at point 1 due to a unit load

at point 2 is AizL pound-inches. If the displacement A22 be
A 12

measured in feet, then the moment reaction for a unit load is A 22

pound-feet. lz

HISTORY g; MODEL ANALYSIS

 

The first practical use of model analysis of indeterminate
structures was made by Mr. James R. Griffith, then Assistant Professor
of Civil Engineering, Armour Institute of Technology, in the design of
the catenary structures and signal bridges for the Illinois Central
Railroad's Chicago electrification in 1924. Mr. Griffith was con-
fronted with a three-column structure which was indeterminate by
twenty-one variables. The prospective cost of designing these and
more complicated structures by any analytical method presented a
serious problem.

Apparently the only possibility of avoiding this tedious mathe-
matical analysis was by a method of measuring the deflection in cellup

loid or cardboard models. This method was only a laboratory experiment

(6)

at that time, which had been devised by the late Professor George E.
Beggs of Princeton University and described in a paper presented before
the American Concrete Institute, in 1922. Professor Beggs loaned his
original equipment to Mr. Griffith and by the use of this equipment
Mr. Griffith demonstrated that approximately two—thousand dollars per
year could be saved in the cost of design alone. The Illinois Central
Railroad then purchased one of the first commercial deformeters and
used it in their design. By the use of the deformeters and models a
considerable saving was obtained in the designing cost and in the actual
cost of the structures. ‘

Model analysis started with the late Professor George E. Beggs
in the summer of 1916 when his attention was given to the problem of
the deflections and stresses of a triple span continuous truss bridge
of the B. & L. E. R. R. over the Allegheny River. He found it advan-
tageous to construct the influence lines for the trusses. After having
plotted these lines Professor Beggs noticed that they resembled the
elastic curves of continuous beams fastened at three supports and de-
flected a unit distance at the remaining support for whose reaction the
influence line was draw . To determine the accuracy of this observa-
tion, a tenefoot pole of uniform section and three nails were used. To
draw the influence line for one reaction the pole was restrained by
single nails at the other three reactions and deflected a distance to
scale of one-thousand pounds at the first reaction. The ordinates of
this elastic curve were measured. The values obtained by this crude
mechanical apparatus and by the exact theory showed very close agree-

ment.

In January 1920, Professor Beggs checked the theory of mechanical
analysis (Maxwell's Theorem) by means of a wood model of a single span
bent hinged at the supports and with a horizontal member half the length
of the deck placed as shown in Fig. 5. The members were made of wood
strips five—eights by seven—sights of an inch in cross-section and the
columns and main horizontal member were fifty inches in length center
to center of connections. Metal gusset plates and screws fastened the
bars together. No loads were applied while the deflections were being
measured. The horizontal and vertical reactions were found by the
use of Maxwell's Theorem and then these loads were applied at the three
supports as shown and the pins removed. The supports did not move
which proved the correctness of this theory and application. These
reactions as found by measuring the deflections are seen to satisfy

very well the three laws of static equilibrium.

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(8)

After feeling definitely that this method could be applied to
statically indeterminate structures, Professor Beggs designed some
«deformeter gages to be used in connection with micrOSCOpes to measure the
deflections. The gages produced a known displacement and the micro-
scopes measured the deflections at the assumed points of application
of the loads and in the direction of the loads. These gages as original-
1y designed were somewhat crude in comparison with the gages sold by
the late Professor Beggs during the past few years. This equipment is
probably the most accurate yet devised but is quite expensive. A set
of six gages and three microscopes costs about eight-hundred dollars.

Although Professor Beggs has probably done most for the analysis
of structures by models, there are others who have worked independent-
ly and have used similar or different methods based on.Maxwell's
Theorem. Among these is Mr. Otto Gottschalk who has developed the
”Continostat Gottschalk". His method and apparatus has been described
in various technical literature, the first being a paper entitled
"Mechanical Calculation of Elastic Systems" given before the Franklin
Institute and appeared in the £91m]. pf, the Franklin Institute in
July 1926. One of Mr. Gottschalk's latest publications appeared in the
Medina .91 the. American fascism 9f Qiril Engineers for January 1957,
under the title of ”Structural Analysis Based on Unloaded Models".

The Continostat Gottschalk consists of various mechanical sup—
ports and very flexible splines used as the elastic structure. Known
deflections are produced at the point at which the reactions are sought
and the flexible splines forming the members of the structure bend in

the shape of the elastic curve of the member. The ordinates from the

(9)

original position to the deflected position are measured by means of
the ordinary scale. The reactions at the different points are then
found by the ratio of the deflections. The deflection at the point
where the reaction is sought is usually made equal to one unit, which
makes the elastic curve represent the influence line for that reaction.

An article appeared in the "Epgineeringlflgggfifigggpd", Volume 99,
Number 25, December 8, 1927, by Mr. Anders Bull in which he explains a
method of analysis using wire models. As a partial fulfillment of the
requirements for the Master of Science Degree in Civil Engineering at
the University of Idaho, Mr. Lyman G. Youngs prepared a thesis on the
subject of "Stress Determination in Statically Indeterminate Struc-
tures by Means of Brass Wire Models". This thesis represents a very
good study of the possibilities of wire models. Wire models are use-
ful provided the moment of inertia of each member is constant, but if
the members have a variable depth it is almost impossible to obtain a
model that truly represents the structure. Therefore, it seems that this
method can be used only for the more simple structures.

In order to get around the high cost of the Beggs Deformeters
with the microscopes there have been various other gages devised and
among these is the "home made" deformeter designed by Mr. T. Werner,
Engineer for the Arthur G. McKee Company of Cleveland, Ohio. This
deformeter was developed in the Structural Models Laboratory of the Case
School of Applied Science. These gages are inexpensive to make and
have proved quite satisfactory for most work and particularly for
demonstration.work for structural classes in model analysis. A redesign

of these gages was used in the preparation of this thesis. The same

(10)

fundamental idea was used but an improved method of holding the gages
together and still permitting the desired motions was incorporated in-

to the original design.

DESIUN AND OPERATION 9;: THE DE1*'OI@[ETE1{ ULUES

 

The redesign of the Werner Deformeter is shown on the print,
page 12. The original gage only had two springs which held the bars
together and the wire passing through the springs passed through the
bars at an angle which caused a push of the small bar against the verti—
cal glass plate of the long bent bar. The three springs and the new
method of mounting the springs makes a much stronger gage and one that
is more sure in operation. A set of four gages as shown on the print
(page 12) was made under the author‘s direction at the Michigan State
College of Agriculture and Applied Science and another set made in
the Physics' Machine Shop at the University of Idaho. Both of these
sets worked satisfactorily.

The two bars are held in contact by means of the three springs
and the cylinders "a", "b", and "c". These cylinders were soldered
with extreme care in order that they would be exactly parallel to each
other when the bars are placed together since they must be in contact
with the glass plates of the opposite bar for their entire length.

By inserting a filler plate of uniform thickness between the
cylinders "a" and "b" and the glass plate a definite movement takes
place. When a filler plate is inserted between the cylinder "c“ and
the glass plate another movement will take place at right angles to the

first. If a filler plate is inserted only between cylinder "a" and

(11)

 

 

 

 

 

 

 

 

 

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(12)

the glass plate an angular movement takes place. According to Max—'
well's Theorem this movement and the deflections at the points of
application of the assumed loads is all that is needed for the solution
of the structure.

One gage is necessary for each support of the structure. In
setting up the equipment all the reaction lines and the lines of action
of the loads are drawn as accurately as possible on a sheet of drawing
paper either tacked to a table or drawing board. The long glass plate
of the gage must be set exactly at right angles to the center line of
the model at the support. This is usually the vertical reaction line.
The cylinder "b" is placed directly over the intersection of the two
reaction lines. The straight bar (one with the cylinders) is screwed to
the table or drawing board.

The model is usually made of celluloid and is clamped under the
one-half times five—sixteenths of an inch bar on top of the long bent
bar. The model should be clamped*accurately so the reaction lines on
the model will correspond exactly in position with those drawn on the
drawing board. To keep the model in a horizontal plane it is necessary
to support it by either glass or celluloid plates at different points.
Between these plates and the model a small steel ball about one—sixteenth
of an inch in diameter is placed. The small steel balls reduce the
friction at these points to a negligible quantity. The means of measur-
ing the deflections at the several points are a mirror, mirror bar,
focusing flashlight, and a screen.

The mirror consists of a half razor blade with a needle soldered

on the back of the blade and a small mirror glued on the front. The

(15)

height of the razor blade from the sharp edge of the blade to the top
of the needle point should be about one-half of an inch. This will
give about the best size deflection on the screen. The mirror bar
used is a one-quarter inch copper tube about twelve inches long. In
one end of the bar a needle is soldered at right angles to the longi-
tudinal axis of the bar. The other end is flattened for a couple of
inches from the end and a piece of sand paper is glued to the same side
as the needle. The needle of the mirror bar is placed lightly into
the model at a load point and the other end with the sand paper is
placed on top of the needle of the razor blade. The center line of the
mirror bar should coincide with the action line of the load. A slight
deflection of the load point will tip the mirror and cause a very large
deflection on the screen.

A focusing flashlight of.good quality is used with a small house
bell transformer instead of batteries. This gives a permanent light
’ of the same quality. The flashlight is placed about fifteen feet from
the mirror and a screen is placed at twenty or more feet in front of
the model. The screen consists of a cross-sectional paper about sixty
or more inches high and should be at least ten inches wide. The light
should be focused on the mirror until a good image appears on the screen.

With the model in a normal position (with no dietortion) a read-
ing of the initial position of the image on the screen is taken. A
filler plate is inserted in each support giving the whole model a
deflection equal to the thickness of the filler plate. A second reading
is taken and the difference between the two readings gives the filler
thickness as measured on the screen. The filler plates are hen re-

moved and one plate is inserted in the gage for which the reaction is

(14)

desired. The deflection is again measured on the screen in the same
manner. The ratio of these deflections gives the reaction at the

support considered for a unit load at the load point, since the ratio
of the deflections times the load gives the desired reaction as dis-

cussed in the theory of model analysis.

Let D filler plate thickness as measured on

the screen

d = deflection of the load point as measured
on the screen

H, M, and V Z reactions

‘ Load at load point (considered as

"U
I

unity)

From the basic theory of Model analysis the reactions are given as

follows:

v:__g.__ H=__%... ; M=_.g...K

‘0

K is the distance between the cylinders "a" and "b" measured to the
same scale as the model either in inches or feet giving a result in
pound-inches or pound-feet.

In measuring the deflections the needle of the mirror bar at the
load point moves a very small distance at right angles to the line of
action of the load as well as in the vertical direction (the latter is
the displacement desired). In making this movement the mirror will
be slightly tipped giving a deflection on the screen that is not due
to any vertical movement of the load point. If this movement is large
it will be necessary to correct for this error, which is given ap-

proximately by the following equation:

(15)

e : d2
213k
In this equation "d" is the deflection on the screen due to this move—
ment at right angles to the line of action of the load, "b" is the
length of the mirror bar, and "k" is the number of times the apparatus
is multiplying the actual deflection of the model. ("k" is about 1030.)

The diagram of Fig. 6 is a proof of this error.

 

 

 

0’
k?
b:2 — d2=(b - e)2
b2 - d2 = b:2 - 2be + e2 (e213 an error of
higher order)
e :2 d2
2bk
Fig. 6

CONSTRUCTION gr; THE 2.5 in:

 

The model should be made of well seasoned celluloid of uniform
thickness. However, it can be made of other material but celluloid is
much better than card board or any other material. As the thickness of

the celluloid is constant the depth of the section of the model should

(l6)

be proportional to the cube root of the moment of inertia of the par-
ticular cestion of the actual structure. This will make the model

structural similar to the actual structure.

de‘lB/f
The proportionally factor should be so chosen as to make the model
flexible. The scale of the model should be chosen so the model is of
convenient size.

After the scale and "k" have been chosen the centerline of the
model is laid out on the celluloid sheet. The outline is drawn and
the model is cut out with a coping or band saw. The celluloid is
planed down to near the proper dimensions and theh sanded by means of
sand paper to the exact dimensions. The thickness of the section of
the model should be to the nearest 0.001 of an inch of the proper
dimension. An error in the thickness of the section will cause an

appreciable error in the results.

SINuLE saw FEM g; BENT

The first structure tested or analyzed by this equipment was a
single span frame bent as shown in Fig. 7. The moment of inertia of
the bent was assumed constant for the legs and the deck and assumed to

be one-thousand units to the fourth power. The height was twenty-five

feet and the span twenty feet. The influence ordinates were calculated

by means of formulas from "Stresses in Frame Structures" by H001 and

Kinne. A unit load was considered throughout the analysis.

(17)

 

 

n WK

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The mathematical formulas used to analyze this bent with the loads

5mm: 5 PAN BENT

Fig. 7

on top of the deck were:

(18)

 

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K : I :looo=_5__
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n : Kh =50x_§=l.25
I 1000

After the moments were calculated the vertical and horizontal reactions
were calculated by means of statics. The following tables show the
results as obtained from the mathematical analysis and the results as

obtained from the deformeter analysis.

Moment at the Left Support

 

 

Load Mathematical Deformeter

Eoint Talus _L_ Value
10 0.000 0.000
11 0.240 0.248
12 0.467 0.474
15 0.652 0.646
14 0.769 0.774
15 0.790 0.790
16 0.687 0.674
17 0.453 0.427
18 0.000 0.000

 

(19)

.Mertical.Bsaetion.ai.thsflLaft.Euuaart

 

Load Mathematical Deformeter

Eoint Value Value
10 1.000 1.000
11 0.884 0.884
12 0.761 0.756
15 0.652 0.656
14 0.500 0.499
15 0.568 0.567
16 0.259 0.258
17 0.117 0.120
18 0.000 0.000

Ihzziaantalpfieaciian ai.ihejhef .finpaent

 

 

Load Mathematical Deformeter
_LPoint Value Value
10 0.000 0.000
11 0.040 0.041
12 0.069 0.069
15 0.086 0.087
14 0.092 0.092
15 0.086 0.087
16 0.069 0.069
17 0.040 0.041
18 0.000 0.000

 

From the above tables and the curves plotted to represent this
data on pages 21 and 22, the close agreement of the deformeter values
to those calculated mathematically can be seen. With careful handling
of the gages little trouble was eXperienced in obtaining preper values.
Each time the filler plates were inserted in the gages care was taken to
see that the two bars came together exactly the same manner. A little

experience with the gages created confidence in the accuracy of the work.

(20)

 

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From the experience with the single Span frame bent it was
Satisfactory to assume that the equipment as designed is capable of
accurate work. At least, the error will be small and the value one
obtains from an experimental analysis approaches very closely to the

actual value of the stress in the structure. The mathematical analysis

 

is based on some assumptions and it is doubtful whether or not the

~ m...

values thus derived approach any closer to the actual values in the

structure.

an.“

waitress 913 THE RIGID lfléfi BRIDgg pangs.

The data for the bridge constants for the single and double span
bridges were taken from "The Rigid Frame Bridge" by Mr. A. G. Hayden,
Ch. VI, pp 45-58, and Ch. Ix, pp 107—126, respectively. The data for
the triple span bridge were assumed. The depth of sections as given
in the above text or those assumed were considered as the normal depth.
In the first analysis of each bridge model a depth of ten per cent
greater than normal was used at the load point next to the abutments or
piers. This increased depth varied as the offsets of a parabolic curve
to zero at the center of the Span. For the second analysis the normal
depth of section was used. For the third analysis a depth of section
that was ten per cent less than normal was used at the load point next
to the abutments or piers. The abutments and piers were of the same
section thickness for all three analyses.

The purpose of the variation of the depth of the haunches was
to find out how much and in what direction the moment at the left

knee varied for this situation. In the design of single span rigid frame

(25)

bridges the depth of section next to the abutments and at the center
are usually assumed to be one—fifteenth and one—fortieth of the span
respectively. This has proved best from experience in the design of
such bridges.

The models were made of transparent celluloid and only one bridge

 

model was used for all three analyses in each case. The second and

“if
1

third analyses were made after the models were sanded down to their
proper depth of section each time. In this way more comparative re-
sults were obtained and any errors in making the models were the same y
for all three analyses. All of the tabulations and the curves showing f
the results are in the appendix.
After the values of the vertical, horizontal, and moment re-
actions were found from the deformeter method they were plotted and
smooth curves drawn through the plotted points. Necessary values to

compute the moment at the left knees by statics were taken from the

 

curves and not from the tables of values for these forces given in the

appendix.

Single Span.Model

The curves for the moment at the left knee for fixed supports
show a definite increase as the depth of haunch is reduced. The maxi-
mum moment occurs at the left knee when a unit load is at a point fortys
one-hundredths of the span from the left support. The increase of the
second analysis was 0.65 pound-feet or 7.8 per cent over the first at
the maximum and the third analysis was 0.58 pound-feet or 4.2 per cent
increase over the second analysis. This shows that the increase in

moment is not according to a straight line variation.

(24)

 

For hinged supports a definite increase was shown and the maximum
value occurs with a concentrated load at the center. The increase of
the third analysis over the second was 5.8 per cent. No data for
hinged supports were taken for the first analysis as the deformeters
were not arranged at that time to analyze the models for hinged sup-

ports.

Double Span.Model

The curves for the moment at the left knee for the fixed supports
show an increase in the value of the moment near the maximum as the
depth of the haunch decreases. The percentage increases are 2.9 and 2.6
per cent respectively for the second and third analyses. The maximum
value for the moment at the left knee occurs when a concentrated load is
at a point thirtybeight-hundredths of the Span from the left support.

The results for the hinged supports gave little difference
between the values of the moment at the left knee for the three dif-

ferent analyses. The slight difference could be attributed to experi~

mental error. The maximum value occurs at a point thirty-eight—hundredths

of the span from the left support, the same as for fixed supports.

Tgiple Span.Model
With the triple span model the deformeter analysis showed a
definite decrease in the value of the moment at the left knee. How-
ever, this decrease was small, being 1.6 and 2.5 per cent respectively
for the second and third analyses. For the hinged supports the decrease

was 2.2 and 5.4 per cent respectively.

(25)

‘m— «a 4“.“‘5 . -u

 

CONCLUSIONS

 

In the design of the deformeter gages the principle of the
Werner Deformeter has been retained. The principle on which the gages

work is the use of filler plates to give displacements at the supports

 

between the three cylinders, "a", "b", and "c", and the glass plates.
These displacements can be measured as well as displacements at the

load points. The deformeters as designed for use in this thesis have
an improved method of holding the bars together and give a more sure

action of the gages. The results should be more accurate as it is

1‘40.— .9 ‘r-scm o

easier to get the bars to come together exactly the same each time
which is an important point in the operation of the gages. The gages
are very easy to manipulate.

In the operation of the equipment it is best to have one person
operating the gages (placing the filler plates in the gages and taking

them out as required) and a second person taking the readings at the

 

screen. This saves much time over that required to operate the gages by
one person, as well as the walking back and forth to the screen. When
two persons are operating the equipment the Werner Deformeters are much
faster than the more expensive Beggs Deformeters and just as accurate.
With the Beggs Deformeters a microscope must be focused each time a
reading is desired, while one focus of the flashlight usually is
enough for two to four load points before a change of the position of the
flashlight is necessary.

The results as obtained for the single Span frame bent with the

gages showed a very close agreement with the theoretical analysis.

(26)

Also, the results for the single span rigid frame bridge for the
second analysis (which was for a normal thickness) checks very closely
with the results for the influence lines as given for this same bridge
in the text "The Rigid Frame Bridge" by A. G. Hayden, pp 65 and 66.
Although the influence line for point SL is not shown it would be of
identical shape to that shown for point 6L except that it would start
at zero and go down to a maximum of about 8.8 or 9.0 pound-feet.
These are the only curves that can be checked with the theoretical
analysis. This close check leads one to feel confident that the other
results as obtained with the deformeters are as accurate or will approach
as closely to the actual value of the moments and reactions that exist
in the structure as the theoretical analysis since the latter is based
on some assumptions.

The cost of the four gages to complete a set with the mirror
bar was forty—eight dollars. Therefore, the gages are inexpensive, and
since they do give accurate results, are recommended for use both as
laboratory demonstration work and for research work on statically inde-
terminate structures. However, because of the method necessary to use
to obtain the displacements at the supports, the gages can only be used
for structures with horizontal and vertical reaction lines. At least
the gages must be placed so that the glass plate is either horizontal
or vertical.

No attempt was made to check the experimental results for the
three bridge models except as noted above, nor, account for the results
for the variation of the depth of the haunch from a theoretical stand-

point. The values as given in the tabulations in the appendix were

(27)

those as obtained from the work and all readings were taken at least
twice and some checked more than once. These results were then averaged
and the averaged values plotted to make the curves of the reactions as
shown in the appendix. However, a theoretical analysis would seem
to indicate that the moment at the knee should be reduced as the depth
of the section is reduced as more material at the knee would "draw"
the greater amount of moment from a theoretical fixed-end moment. By
not changing the abutment or pier sections probably had something to
do with the fact that the moment at the knee was not reduced in the
single and double span models (an increase was noted). However, in
the triple span model a decrease in the moment at the left knee was
shown. There is the possibility that a still more accurate method
than yet devised would be necessary for research of this nature where
small changes in the model would show up better in the results. If
this is true, the allowable error in the construction of the model is
greater than thought by those who have worked with model analysis. It
would not be necessary to sand the sections down to the nearest 0.001
of an inch but possibly the nearest 0.01 of an inch is all that is neces-
sary. This would make the models much easier to construct which is one
objection to the use of this type of analysis.

The results show need for further research with the deformeter
analysis with a theoretical analysis as a direct comparison so as to
have a more definite tie—up of the deformeter analysis and the mathe—

matical theory.

(28)

APPEMI

’-

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lluflimammaLinss eve

Single Span.Model Reactions

Fixed Supports ————————————— 55-o5
Hinged Supports ------------ 36-u7

Double Span Model Reactions

Fixed Supports ———————————— 43—46
Hinged Supports ------------ 47-49

Triple Span Model Reactions

Fixed Supports ------------ 58-61
Hinged Supports ------------ 62-64
Tablgg Page
I Single Span Rigid Frame Bridge and
Model Constants --------------- 51
II Section Thickness of Single Span Model ------ 51
III Experimental Results - Single Span
Model (Fixed Supports) + ----------- 58
IV Experimental Results - Single Span
Model (Hinged Supports) --------- - 59
V Double Span Rigid Frame Bridge and
Model Constants --------------- 40
VI Section Thickness of Double Span Model ------ 41

'VII Experimental Results - Double Span
Model (Fixed Supports) ------------ 50

VIII Experimental Results - Double Span
Model (Hinged Supports) ----------- 53

IX Triple Span.Rigid Frame Bridge and
Model Constants - - - - — - - — — - — — - — - 55

(29)

Aprbnnig (gong )

Tables

 

X Section Thickness of Triple Span Model ——————

XI Experimental Results - Triple Span
Model (Fixed Supports) ------------

XII Experimental Results - Triple Span
Model (Hinged Supports) -----------

(50)

69

Siuapg SPAN 61016 66165 661165 ALL MODEL 01:6T16265

 

.—

Table I

 

 

Coordinates of Model**

 

391116 51:51 1 1654) I 1ft) 1. L311 alml..-x_(in.l_
1L 6.00 16.00 1.0 0.4 0.15 0.55
2L 2.65 1.55 4.0 0.5 0.10 1.55
5L 5.21 2.75 8.0 0.0 0.00 2.67
4L 5.65 4.75 12.0 —0.5 .0.10 4.00
5L 4.00 5.55 15.9 0.1 0.05 5.50
6L 5.55 5.75 16.9 5.9 1.50 5.65
7L 2.75 1.75 17.5 7.9 2.65 5.77
6L 2.15 0.65 17.7 11.9 5.96 5.90
9L 1.70 0.41 17.9 15.9 5.50 5.97

10L 1.40 0.25 16.1 19.9 6.65 6.05
llL 1.25 0.16 16.2 25.9 7.96 6.07
116 1.25 0.16 16.2 27.9 9.50 6.07
10R 1.40 0.25 18.1 51.9 10.65 6.05
96 1.70 0.41 17.9 55.9 11.96 5.97
66 2.15 0.65 17.7 59.9 15.50 5.90
76 2.75 1.75 17.5 45.9 14.65 5.77
66 5.55 5.75 16.9 47.9 15.96 5.65
56 4.00 5.55 15.9 51.7 17.25 5.50
4R 5.65 4.75 12.0 52.1 17.56 4.00
56 5.21 2.75 6.0 51.6 17.26 2.67
26 2.65 1.55 4.0 51.5 17.16 1.55
16 6.00 16.00 1.0 51.4 17.15 0.55
*5)?

Scale of Model 1": 5' * Bridge Constan 5 taken from "The Rigid Frrme

Bridge" by Arthur 0. Hayden, Ch. Vi, pp 45—58.

Table‘l;

~7-

SECTIL‘N THICKNESS 01" SINCLLC SPILN RIC—ID r'FJJfiL BlilDuE AICDE'L

 

 

 

 

Point Thickness
First Anglvsis .Sepgnd Analysis___Thirdޣnalysj5

1L 1.575 1.575 1.575
2L 0.694 0.694 0.694
5L 0.842 0.842 0.842
4L 1.009 1.009 1.009
5L 1.048 1.048 1.048
6L 1.024 0.951 0.858
7L 0.782 3.720 3.658
8L 0.602 0.564 0.526
9L 0.465 0.446 0.429
10L 0.575 0.5 8 0.561
11L 0.527 0.526 0-525

 

 

 

 

Deformeters and Models

(51A)

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1' APLPLIIJLIITILLI RELJULTS

 

S NuLE SPIN RIuID FRil L BRIDEQ - FIALD_§UPPORTS

 

 

Results for First Analysis

 

Moment at Left Vertical Reaction Horizontal Ree ction Moment at

EninL_n111_JhuuxnuL......._Le£t_finan:Ll.____..l&£jLSyn“art Ififilgflnee

 

 

 

6L 0.96 0.12 -l.90
7L 0.07 0.92 0.27 —5.92
8L 0.26 0.85 0.40 —5.70
9L 0.81 0.78 0.52 -7.52
10L 2.08 0.68 0.64 -8.56
11L 5.41 0.57 0.71 —8.03
11R 4.41 0.42 0.71 —6.62
10R 5.27 0.50 0.64 -4.95
9R 5.02 0.20 0.52 -5.47
8R 4.05 0.11 0.40 -2.06
7R 2.82 0.07 0.2 -1.45
6R 1.42 0.05 0.12 -0.61
.Eeanlts.£nr.Eecnnd.AnaLxfii§

Moment at Left Vertical Reaction Horizontal Reaction Moment at

.EQ1ndL_..___ihnuxudL_.....__léfl3LiEKanaiL.._____LéfiELihuuxujL_l_._.léfiiLfixuxa
6L 0.96 0.12 -2.06
7L 0.14 0.92 0.28 -4.07
8L 0.2 0.88 0.45 ~6.18
9L 0.85 0.80 0.57 ~8.00
10L 2.01 0.69 0.68 —9.01
11L 5.40 0.57 0.75 —8.64
llR‘ 4.57 0.42 0.75 -7.26
10R 5.50 0.50 0.68 -5.58
9R 5.15 0.21 0.57 ~5.95
8R 4.59 0.12 0.45 -2.54
7R 2.95 0.08 0.28 -l.58
6R 1.49 0.05 0.12 -0.77

 

 

(58)

Table III (ggnt.)

Resultg for Third Analysis

 

Moment at Left Vertical Reaction Horizontal Reaction Moment at

 

 

Egint Supgort Lei: Supoort _~_Leflt Suanort Left Knee
6L 0.29 0.97 0.14 -2.2
7L 0.25 0.95_ 0.50 —4.59
9L 1.12 0.80 0.57 -8.51

10L 1.71 0.69 0.69 —9.52

11L 5.49 0.54 0.77 -8.96

11R 4.50 0.42 0.77 -7.58

10R 5.27 0.50 0.69 -5.89
9R 5.05 0.2 0.57 -4.26
8R 4.29 0.12 0.45 —2.86
7R 2.82 0.07 0.50 —1.90
SE 1.48 0.05 0.14 -0.95

Table l}:
EAPLRINLNTAL RESULTS
SINLLE SPAN RIuID FRIML BRIDqE - HINLLE SUPPORTS
Results for the Second Analysis*
if Vertical Reaction Horizontal Reaction Moment at
Eointo Left Sugport Left_Suonrt Left Knee
6L 0.95 0.10 —l.59
8L 0.79 0.51 -4.95
10L 0.61 0.49 -7.79
11L 0.55 0.54 —8.58
9R 0.50 0.41 —6.52
SE 0.22 0.51 -4.95
7R. 0.12 0.20 —5.18

 

 

 

 

 

 

 

-* The deformeter gages were not constructed so as to use hinge

supports when the first analysis of the single span bridge was

analyzed, and, therefore, only the results for the second and
third analyses are shown.

(59)

Table 3: (cont.)

Results for the Third Analysis

 

 

 

 

 

 

Vertical Reaction Horizontal Reaction Moment at
Point Left_Supgort Left Supoort Left Knee

6L 0.92 0.11 —1.85

7L 0.88 0.25 -5.66

8L 0.79 0.54 -5.40

9L 0.69 0.45 -7.00

10L 0.61 0.52 —8.50

11L 0.55 0.57 -9.06

11R 0.47 0.57 -9.06

10R 0.58 0.52 -8.50

9R 0.29 0.45 -7.00

8R 0.18 0.54 -5.40

7R 0.15 0.25 -5.66

6R 0.07 0.11 —1.85

Table 1
DOUBLQ SPfiN RIUID FRLNL BRIDeL 1ND LODLL CONSTANTS*
' e s' s N '

_Boint thtll I Lft4) LY Lit) X LfIJ x (in.) y lin.) Depth
1L 2.5 1.50 1.8 0.0 0.00 0.600 0.962
2L 2.6 1.81 6.8 0.0 0.00 2.267 1.024
5L 5.5 5.58 11.8 0.0 0.00 5.955 1.286
4L 5.7 4.96 16.7 0.0 0.00 5.567 1.455
5L 2.8 2.25 18.2 5.0 1.67 6.067 1.098
6L 2.0 0.85 19.2 10.0 5.55 6.400 0.796
7L 1.5 0.59 19.9 15.0 5.00 6.655 0.614
8L 1.1 0.15 20.4 20.0 6.67 6.800 0.446
9L 1.0 0.11 20.5 25.0 8.55 6.855 0.405
10L 1.2 0.20 20.4 50.0 10.00 6.800 0.491
11L 1.7 0.56 20.2 55.0 11.67 6.755 0.692
12L 2.6 1.87 19.7 40.0 15.55 6.567 1.055
15L 4.0 6.59 19.1 45.0 15.00 6.567 1.559
14 5.7 17.71 18.1 50.0 16.67 6.055 2.190
15 2.0 0.79 15.1 50.0 16.67 4.567 01777
16 2.0 0.79 8.1 50.0 16.67 2.700 0.777
17 2.0 0.79 5.1 50.0 16.67 1.055 0.777

 

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(40)

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7R
6R
5R
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Dimensions of N l

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18.55
20.00
21.67
25.55
25.00
26.67
28.55
50.00
55.55
55.55
55.55
55.55
55.55

6.567
6.557
6.755
6.800
6.855
6.830
6.655
6.403
6.067
5.567
5.955
2.267
0.600

1.559
1.055
0.692
0.491
0.405
0.446
0.614
0.796
1.098
1.455
1.286
1.024
0.962

- --u—o—ac- -.-.-———-_..— -.—. -“*.-*-.H

* Bridge constants taken from "The RL id Frane Bridg e" by Arthur

G. Hayden, Ch. IX, pp 107-126.

SECTION THICKNLSS 0F

 

Tab1e_yl

DOUBLL SP4 ”N RIGID rRAJE BRIDuE

-_‘ «._.—1. _-..“

 

 

 

_ -thichneas
Point

5L 1.207 1.098
6L 0.858 0.796
7L 0.641 0.614
8L 0.455 0.446

_ 9L 0.405 0.405
10L 0.501 0.491
11L 0.751 0.692
12L 1.125 1.055
15L 1.715 1.559
15R 1.715 1.559
12R 1.125 1.055
11R 0.751 0.692
10R 0.501 0.491
9R 0.405 0.405
8R 0.455 0.446
7R 0.641 0.614
6R 0.858 0.796
5R 1.207 1.098

(41)

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1.405

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0.947
0.655
0.482
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0.459
0.586
0.754
0.988

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- Point VL VC VB. HL HO H R
5L -+0.950 '+0.025 +0.025 +0.20 —0.15 —0.07
6L, +0.890 -+0.07O +0.040 +0.56 -O.26 -0.ll
7L -+0.800 '+O.160 '+0.040 '+0.50 40.58 —0.15
8L +0.660 -+0.550 40.010 '+0.59 -0.51 -0.08
9L +0.480 +0.560 -0.050 ‘*0.54 ~0.58 +0.04
10L -+O.500 «+0.810 -0.110 +0.45 —0.58 +0.15
11L -+0.180 -+0.950 -O.100 +0.28 —O.46 +0.17
12L +0.110 40.970 -0.080 +0.17 -0.52 +0.15
13L +0.050 +0.990 -0.040 +0.09 -0.16 +0.07
15R -0.040 +0.990 «+0.050 -0.07 '+0.16 -0.09
12R —0.080 +0.970 1+0.110 -0.15 ‘*O.52 -0.l7
11R -0.100 +0.950 +0.180 -0.17 +0.46 -O.28
10R —O.110 +0.810 +0.500 -O.15 ‘+0.58 -0.45
9R -0.050 +0.560 +0.480 -0.04 +0.58 -O.54
8R ‘+0.010 ‘+0.550 +0.660 +0.08 +0.51 -O.59
7R +0.040 -¥0.l60 +0.800 +0.15 +0.58 -0.50
GB +0.040 1+0.O7O +0.890 +0.11 +0.26 -0.56
SE +0.025 +0.025 +0.950 +0.07 +0.15 -O.20
.Mgmgniaizg;myaiiunuuflmi Mgmgnta_a:_tn§Juz£ngxaa
Point ML MC MR MK
5L +0.15 —0.98 -0.91 —5.19
6L 'P0.51 -l.88 -1.58 -5.71
7L +0.70 -2.76 -1.92 -7.65
81. +1.67 -5.44 -l.51 —8.25
9L +2.18 -5.65 -O.57 -7.15
10L +2.42 —5.16 +0.85 -4.79
11L +1.79 -2.45 +1.16 --2.81
12L +1.29 -l.54 +0.97 -1.57
15L +0.66 -0.79 +0.57 -—O.68
15R —0.57 +0.78 -0.66 +0.60
12R -0.97 +1.54 -1.29 +1.20
11R --1.16 +2.45 -1.79 +1.68
lOR -O.85 +3.16 -2.42 +1.66
9R +0.57 +5.65 -2.18 +1.02
8R +1.51 “5.44 -1.67 +0.27
7R +1.92 +2.76 --0.70 -0.25
SE +1.58 +1.88 -0.51 -0.24
Sfi +0.91 ‘+0.98 -0.15 -0.27

 

(50)

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(51)

Point VL V0 V3 HI. HG HR
5L +0.96 +0.05 +0.02 +0.20 —0.12 -0.08
6L +0.91 +0.06 +0.05 +0.57 -0.2 —0.15
7L +0.80 +0.16 +0.05 +0.54 -0.59 —0.14
8L +0.67 +0.55 +0.01 +0.61 —0.52 -0.09
9L +0.47 +0.57 -0.05 +0.57 -0.59 +0.02

10L +0.29 +0.80 -—0.09 +0.44 -0.59 +0.14
11L +0.18 +0.92 -0.10 +0.50 -0.47 +0.17
12L +0.10 +0.97 -0.07 +0.19 -0.52 +0.15
15L +0.05 +0.99 —0.04 +0.09 -0.16 +0.07
15R -0.04 +0.99 +0.05 —-0.07 +0.16 -0.09
12R -0.07 +0.97 +0.10 -0.15 +0.52 -0.19
11R -0.10 +0.92 +0.18 -0.17 +0.47 -0.50
10R -0.09 +0.80 +0.29 -0.14 +0.59 -0.44
9R -0.05 +0.57 +0.47 -0.02 +0.59 -0.57
8R +0.01 +0.55 +0.67 +0.09 +0.52 -0.61
7R +0.05 +0.16 +0.80 +0.14 +0.59 —0.54
GB +0.05 +0.06 +0.91 +0.15 +0.24 -0.57
SE +0.02 +0.05 +0.96 +0.08 +0.12 -0.20
HummufimLa$_Lh§_fiuaaazLa BkmmxuaL§$LLha4LefliJfin§§
5L +0.14 -l.01 -1.07 45.19
6L +0.55 —1.87 -1.82 ~5.85
7L 4'0.79 -2.72 -2.18 -8.05
L +1.70 -5.40 ~1.76 -8.45
91.. +2.46 -5.67 -0.55 -7.07
10L +2.60 -5.19 +0.78 -4.75
11L +2.09 -2.44 +1.50 -2.95
12L +1.45 -1.60 +1.12 -1.80
15L +0.72 -0.81 +0.62 -0.70
15R —0.62 +0.81 -0.72 +0.55
123 -1.12 +1.60 —1.45 +1.04
1.13. -1.50 +2.44 -2.09 +1.54
10R -0.78 +5.19 -2.60 +1.49
9R +0.55 +5.67 -2.46 +1.05
8R +1.76 +5.40 -1.70 +0.50
7R +2.18 +2.72 -0.79 -0.15
6R +1.82 +1.87 -0.55 ~0.57
5R +1.07 +1.01 -0.14 —0.28

Table 111 (cont .)

Results for the Thigg Analysig

.“‘—

 

 

 

 

 

 

 

mmgmflmb WEQL.EQ.§£EQHL
Point VL _ VC VL HL HQ m
5L +0.96 +0.02 +0.02 +0.20 -O.l.2 -0.08
6L -+O.90 -*0.07 +0.05 ‘+0.58 -0.25 -0.15
7L ‘+0.81 -¥0.16 +0.05 I+0.54 -O.59 —0.15
8L .+0.67 -+O.54 —0.0l +0.65 —0.54 -0.09
9L +0.47 +0.57 -0.04 +0.58 -O.62 +0.05
10L '+O.29 ‘+O.78 -0.08 1+0.45 ~0.58 '+0.l5
11L -+O.18 +0.91 —0.09 +0.50 —O.47 +0.17
12L +0.10 +0.97 -0.07 +0.19 -O.52 '+O.15
15L .+0.05 -+O.99 —0.05 +0.09 ~O.17 -t0.08
15R -0.05 +0.99 +0.05 -0.08 +0.17 -0.09
12R -0.07 ‘+0.97 +0.10 -0.15 +0.52 -O.19
11R —0.09 -+0.91 '+0.18 -O.17 +0.47 -0.50
10R -0.08 -+O.78 ‘+O.29 -0.15 '+0.58 -0.45
9R -0.04 '50.57 +0.47 -0.05 -+0.62 -0.58
8R —0.01 -+0.54 ‘+O.67 '+0.09 +0.55 ~0.62
7R +0.05 «r0.16 +0.81 ‘+0.15 ‘+O.59 —O.54
6R -+0.05 '+0.07 +0.90 ‘+O.15 '+0.25 -O.58
SR -+0.02 +0.02 +0.96 l+0.08 +0.1? ~O.20
MQMMWLS Mammgtihaieitim
Point ML MC MR MK
5L 1'0.16 -1.01 -1.08 v5.18
6L {'0.47 -1.94 -1.92 —5.95
7L §-0.87 -2.91 -2.52 -8.29
8L §-l.82 -5.61 —l.75 ~8.65
9L ' -+2.59 -5.71 -0.48 -7.09
10L 1-2.8l -5.50 +0.88 -4.70
11L -r2.27 -2.46 +1.56 -2.71
12L -r1.56 -l.70 '+1.22 -l.65
15L -k0.78 —O.89 '+0.69 -0.82
15R -O.69 -+0.89 ~0.78 ‘+0.47
12R —1.22 -rl.70 -1.56 +0.95
11R -l.56 'f2.46 -2.27 '+1.50
108 -0.88 +5.50 -2.81 +1.27
9R +0.48 +5.71 -—2.59 +0.98
8R .+1.75 '+5.61 -1.82 -+O.15
7R +2.52 '+2.91 -0.87 —0.19
6H +1~92 -+l.94 -O.47 -0.27
SE +1.08 -+l.01 -0.16 ~O.28

 

(52)

Tablg VIII
EAPQRIWENTAL RESULTS

DOUBLE SPAN RIQID IRALJE BRIDGE _ HINulfl_I)_ SUPPORTS

 

Results for the First Analvsigg

_..M -.. *-

‘

 

 

 

 

 

 

 

 

 

5L +0.94 +0.05 +0.05 +0.16 --0.15 -0.04 -2.67
6L +0.86 +0.08 +0.06 +0.51 —0.26 -0.08 —5.18
7L +0.75 +0.17 +0.07 +0.44 —0.57 -0.09 -7.55
81. +0.65 +0.55 +0.05 +0.47 -0.44 -0.04 -7.85
9L +0.44 +0.59 -0.04 +0.42 -0.47 +0.05 -7.01
101. +0.29 +0.79 -0.10 +0.51 -0.45 +0.11 -5.18
11L +0.18 +0.95 -0.11 +0.21 -0.54 +0.15 -5.51
12L +0.11 +0.97 -0.09 +0.15 -0.25 +0.10 --2.17
15L +0.06 +0.99 -0.05 +0.06 --0.11 +0.05 -1.00
15R —0.05 +0.99 +0.06 -0.05 +0.11 -0.06 +0.84
12R -0.09 +0.97 +0.11 ~0.10 *0.25 -0.15 +1.67
11R -0.11 +0.95 +0.18 -0.15 +0.54 -0.21 +2.17
10R -0.10 +0.79 +0.29 -0.11 +0.45 -0.50 +1.84
9R -0.04 +0.59 +0.44 -0.05 +0.47 -0.42 +0.84
8R +0.05 +0.55 +0.65 +0.04 +0.44 -0.47 -0.66
7R +0.07 +0.17 +0.75 +0.09 +0.57 -0.44 —1.50
6R +0.06 +0.08 +0.86 +0.08 +0.26 -0.51 —l.55
5R +0.05 +0.05 +0.94 +0.04 +0.15 -0.16 -0.66
Results for 32112 Second fill-5.1.5..
¥ Verffigfieac—t—IOQL Egrfzofielfieactions 115:1ent atTeftm
Point VL V0 V3 H1, H0 HR MK
5L +0.95 +0.05 +0.04 +0.17 -0.15 -0.05 -2.84
6L +0.86 +0.08 +0.06 +0.52 -0.24 -0.08 ~5.55
7L +0.76 +0.18 +0.06 +0.44 ~0.55 -0.08 —7.55
8L +0.62 +0.54 +0.05 +0.47 -0.45 -0.04 -7.85
9L +0.45 +0.59 -0.04 +0.44 -0.49 +0.04 ~7.01
10L +0.51 +0.79 -0.10 +0.51 -0.42 +0.11 -5.18
11L " +0.18 +0.95 —0.11 +0.20 -0.55 +0.12 -5.51
121. +0.11 +0.97 -0.08 +0.15 —0.22 +0.09 -2.17
15L +0.05 +0.99 ~0.05 +0.06 -0.11 +0.05 -1.00

 

(55)

TabIg VIII (EgntJ

mmmmmsm (09.33.)

 

 

. - u

 

 

 

 

Point VL Vb Vfi HL_u HG HR “K _
151?. -0.05 +0.99 +0.05 -0.05 +0.11 -0.06 +0.84
12R -0.08 +0.97 +0.11 —0.09 +0.22 -0.15 +1.50
118. -0.11 +0.95 +0.18 -—O.l2 +0.55 -O.2O +2.17
10R -0.10 +0.79 +0.51 -0.11 +0.42 -0.51 +1.84
9R -0.04 +0.59 +0.45 —0.04 +0.49 -0.44 +0.66
8R +0.05 +0.54 +0.62 +0.04 +0.45 -0.47 —0.66
7R +0.06 +0.18 +0.76 +0.08 +0.55 -0.44 -—l.55
6R +0.06 +0.08 +0.86 +0.08 +0.24 -0.52 -1.55
5R +0.04 +0.05 +0.95 +0.05 +0.15 -0.17 —0.84

mnnammmq Anglican

Point 17L V3 V3 HL HG EB. ML '
5L +0.95 +0.05 +0.04 +0.17 -0.15 -0.05 --2.84
6L +0.85 +0.08 +0.07 +0.55 -0.24 -0.08 ~5.51
7L +0.74 +0.19 +0.06 +0.45 -0.56 -0.08 —7.55
8L +0.61 +0.56 +0.05 +0.47 -0.45 —0.05 —7.85
9L +0.45 +0.59 -0.04 +0.41 -0.46 +0.04 -6.85
10L +0.50 +0.79 -0.09 +0.50 -0.40 +0.10 --5.01
11L +0.19 +0.92 -0.11 +0.20 -O.55 +0.12 -5.51
12L +0.11 +0.97 -0.08 +0.15 -0.22 +0.09 42.17
15L +0.06 +0.99 -0.05 +0.06 -0.11 +0.06 -1.00
15R -0.05 +0.99 +0.06 -0.06 +0.11 -0.06 +0.84
12R -0.08 +0.97 +0.11 -0.09 +0.22 -0.15 +1.50
11R —0.11 + 0.92 +0.19 -0.12 +0.55 -0.?0 +2.17
10R -0.09 +0.79 +0.50 -0.10 +0.40 -0.50 +1.67
9R —0.04 +0.59 +0.45 —0.04 +0.46 —0.41 +0.66
8R +0.05 +0.56 +0.61 +0.05 +0.45 -0.47 -0.66
7R +0.06 +0.18 +0.74 +0.08 +0.56 -0.45 —1.55
6H +0.07 +0.08 +0.85 +0.08 +0.24 -0.55 ~1.55
5R +0.04 +0.05 +0.95 +0.05 +0.15 -0.17 -—0.84

-—.0--—-_-

 

.—-—-.—

 

 

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TRIPLE_SPAN RIGID EEAME BRIDUE 1ND LODEL CUIT MTJNTS

 

 

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Point 51:21) I (£14) 5L581ngt),_“_-_~11n. "1._pepth
1L 2.5 1.50 0.0 1.8 0.000 0.40 0. 550
2L 2.8 1.81 0.0 8.8 0.000 1.51 0. 585
5L 5.5 5.58 0.0 11.8 0.000 2.82 0. 755
4L 5.7 4.98 0.0 18.7 0.000 5.71 0. 819
5L 2.8 2.25 5.0 18.2 1.111 4.04 0. 827
8L 2.0 0.85 10.0 19.2 2 2'; 4.27 0. 455
7L 1.5 0.59 15.0 19.9 5.555 4.42 0.551
8L 1.1 0.15 20.0 20.4 4.444 4.55 0.255
9L 1.0 0.11 25.0 20.5 5.558 4.55 0.250
10L 1.2 0.20 50.0 20.4 8.887 4.57 0.281
11L 1.7 0.58 55.0 20.2 7.778 4.55 0.598
12L 2.8 1.87 40.0 19.7 8.889 4.49 0.591
15L 4.0 8.59 45.0 19.1 10.000 4.57 0.891
14L 5.7 17.71 50.0 18.1 11.111 4.22 1.251
15L 2.0 0.79 50.0 15.1 11.111 2.91 0.444
18L 2.0 0.79 50.0 8.1 11.111 1.80 0.444
17L 2.0 0.79 50.0 5.1 11.111 0.89 0.444
18 4.0 8.59 55.0 19.4 12.222 4.42 0.891
19 2.8 1.87 80.0 20.5 15.555 4.59 0.591
20 1.7 0.58 85.0 21.0 14.444 4.70 0.598
21 1.2 0.20 70.0 21.4 15.558 4.78 0.281
22 1.0 0.11 75.0 21.5 18.887 4.79 0.250
25 1.2 0.20 80.0 21.4 17.778 4.78 0.281
24 1.7 0.58 85.0 21.0 18.889 4.70 0.598
25 2.8 1.87 90.0 20.5 20.000 4.59 0.591
28 4.0 8.59 95.0 19.4 21.111 4.42 0.891
178 2.0 0.79 100.0 5.1 22.222 0.89 0.444
188 2.0 0.79 100.0 8.1 22.222 1.80 0.444
158 2.0 0.79 100.0 15.1 22.222 2.91 0.444
148 5.7 17.71 100.0 18.1 22.222 4.22 1.251
158 4.0 8.59 105.0 19.1 25.555 4.57 0.891
128 2.8 1.87 110.0 19.7 24.444 4.49 0.591
118 1.7 0.58 115.0 20.2 25.558 4.55 0.598
108 1.2 0.20 120.0 20.4 28.887 4.57 0.281
98 1.0 0.11 125.0 20.5 27.778 4.55 0.250

88 1.1 0.15 150.0 20.4 28.889 4.55 0.255

78 1.5 0.59 155.0 19.9 50.000 4.42 0.551

88 2.0 0.85 140.0 19.2 51.111 4.27 0.455

58 2.8 2.25 145.0 18.2 52.222 4.04 0.827

48 5.7 4.98 150.0 18.7 55.555 5.71 0.819

58 5.5 5.58 150.0 11.8 55.555 2.82 0.755

28 2.8 1.81 150.0 8.8 55.555 1.51 0.585

18 2.5 1.50 150.0 1.8 55.555 0.40 0.550

 

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12.19.18 14.

5802191; 1810888 5 01 T88 TRIPLE 52.4.8 810113 R1188. 13111ng 80081

 

 

 

 

 

 

Thickness
.Egint First Analysis 1§8g9nd Analysis .ibird Analysis
5L 0.690 0.627 0.564
6L 0.490 0.455 0.419
7L 0.566 0.551 0.555
8L 0.259 0.255 0.251
9L 0.250 0.250 0.250
10L 0.286 0.281 0.275
11L 0.418 0.596 0.575
12L 0.642 0.591 0.541
15L 0.980 0.891 0.802
18 0. 980 0.891 0.802
19 0.642 0.591 0.541
20 0.418 0.596 0.575
21 0. 286 0.281 0.275
22 0. 250 0.250 0.250
25 0. 286 0.281 0.275
24 0.418 0.596 0.575
25 0.642 0.591 0.541
26 0.980 0.891 0.802
158 0.980 0.891 0.802
12R 0.642 0.591 0.541
11R 0.418 0.596 0.575
10R 0.286 0.281 0.275
QR 0.250 0.250 0.250
BR 0.259 0.255 0.251
7R 0.566 0.551 0.555
6R 0.490 0.455 0.419
5R 0.690 0.627 0.564

 

(56)

 

 

 

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9L +0.48 +0.58 —O.ll +0.04 +0.58 —3.55 +0.05 —0.08
101. +0.29 +0.82 -0.15 +0.02 +0.45 -0.54 +0.12 —0.01
11L +0.18 +0.95 -0.11 0.00 +0.28 -0.45 +0.15 +0.02.
12L +0.11 +0.97 -0.08 0.00 +0.17 —0.51 +0.12 +0.02
15L +0.05 +0.99 -0.04 0.00 +0.09 -0.16 +0.06 +0.01
18 -0.04 +0.99 +0.05 0.00 -0.07 +0.16 -0.07 —0.02
19 -0.08 +0.97 +0.10 0.00 -0.15 +0.51 -0.15 -0.05
20 -0.10 +0.92 +0.19 -0.0l -0.17 +0.45 -O.25 —0.05
21 -0.09 +0.80 +0.52 -0.05 -O.15 +0.52 -O.57 0.00
22 -0.06 +0.56 +0.56 -0.06 -0.08 +0.49 -O.49 +0.08
25 —0.05 +0.52 +0.80 -0.09 0.00 +0.57 -0.52 +0.15
24 -0.0l +0.19 +0.92 -0.10 +0.05 +0.25 -0.45 +0.17
25 0.00 +0.10 +0.97 -0.08 +0.05 +0.15 -0.51 +0.15
26 0.00 +0.05 +0.99 -0.04 +0.02 +0.07 -0.16 +0.07
15R 0.00 -0.04 +0.99 +0.05 -0.01 —0.06 +0.16 -0.09
12R 0.00 -0.08 +0.97 +0.11 --0.02 -0.12 +0.51 -0.17
118. 0.00 -O.11 +0.95 +0.18 —0.02 -O.15 +0.45 -0.28
10R +0.02 -O.15 +0.82 +0.29 +0.01 -0.12 +0.54 -O.45
9R +0.04 —0.11 +0.58 +0.48 . +0.08 -0.05 +0.55 -0.58
SE +0.05 —0.08 +0.54 +0.69 +0.14 +0.06 +0.44 -0.64
7R +0.05 -0.05 +0.17 +0.85 +0.15 +0.11 +0.52 -—0.58
6R +0.04 -0.05 +0.08 +0.91 +0.11 40.10 +0.20 «0.41
SR +0.02 -0.01 +0.02 +0.97 +0.06 +0.06 +0.09 -0.21
WWW __ W7 Le ' $251199
Point M1 Mg M 5 M4 M}:
5L +0.18 —0.72 —0.50 —0.70 .5.54
6L +0.52 -1.45 -0.92 -1.55 —6.55
7L +1.00 -2.24 -l.28 —l.65 -8.52
8L +1.70 -2.85 -O.67 -1.49 -8.96
9L +2.16 ~5.15 +0.09 -O.81 -7.56
10L +1.98 -5.08 +0.77 -0.05 -5.18
11L +1.61 -2.44 +1.02 +0.44 -5.10
12L +1.04 -1.74 +0.85 +0.42 -l.77
15L +0.55 -O.95 +0.42 +0.26 -0.81

 

Table LL ( cont.)

Results for the Find“. Analysis (cont.)

 

 

 

 

 

 

M e s m 0 93! t a e“ K 53;:
Point. M1 M2 M5 M4 Mk
18 -0.46 +0.75 -0 48 --0.28 +0.84
19 —0.84 +1.55 4 .03 -O.50 +1.51
20 -0.97 +2.42 -1 55 -0.62 +1.88
21 -0.75 +2.76 --2 19 -0.54 +1.74
22 -0.07 +2.70 -2 70 +0.07 *1.29
25 +0.54 +2.19 -2 76 +0.75 +0.52
24 +0.62 +1.55 -2 42 +0.97 +0.11
2.5 +0.50 +1.00 -1 55 +0.84 0.01)
26 +0.28 +0.48 -—0 6.1 +0.46 0.00
15R -0.26 -0.42 +0 95 -0.5'5 -0.08
12R -0.4.‘<’ -0.85 +1 74 -1.04 —O.14
11R —-0.44 -1.02 +2 44 —1.61 ~J.15
138. -0.05 ~0.77 +5 08 -1.98 -0.22
9R +0.81 -0.09 +5 15 -2.16 —0.5:2
. BR +1.49 +0.67 +2 85 -1.70 -0.86
TB +1.65 +1.28 +2.24 -1.00 -0.88
GB +1.55 +0.92 +1 45 -—0.52 -0.68
53- +0.70 +0.50 +0 72 ~0.18 -0.?
___..lixztiqaLfleggtimfi... MWOnL..-
P01nt_ V1 V2 V5 Y4..-__ E1..- 9. ‘12.---“ “.32.. {1. _
5L +0.96 +0.05 -0.01 +0.02 +0.22 -0.09 -J.06 -0.~;)'7
6L +0.91 +0.08 -0.05 +0.04 +0.41 -0.20 -—O.10 —0.12
7L {-0.82 +0.18 -0.05 +0.05 +0.56 —0.52 -0.10 43.14
8L +0.67 +0.55 —0.08 +0.05 *0.65 -0.44 —0.06 -0.15
9L +0.47 +0.59 -0.11 +0.05 +0.57 -0.54 +0.05 -0.07
10L +0.29 +0.81 -0.12 +0.02 +0.41 -0.55 +0.15 ~0.L)1
11L +0.17 +0.91 -0.11 +0.01 +0.27 —0.44 +0.15 +0.02
12L +0.10 +0.97 -0.08 0.00 +0.17 -0.51 +0.12 +0.05
15L +0.05 +0.98 -0.04 0.00 +0.08 -0.16 +0.06 +0.02
18 -0.04 +0.99 +0.05 0.00 -0.07 +0.16 -0.07 -0.02
19 -0.07 +0.97 +0.10 0.06) -0.15 +0.51 -O.15 —0.05
20 —0.10 +0.91 +0.20 -0.01 -0.16 +0.45 -0.25 -0.05
21 -—0.10 +0.79 +0.54 —0.05 -0.15 +0.52 -0.57 0.00
22 -0.07 +0.57 +0.57 -0.07 -0.07 +0.48 -0.48 +0.07
25 —0.05 +0.54 +0.79 —0.10 0.00 +0.57 -0.52 +0.15
24 —0.01 "0.20 +0.91 -0.10 +0.05 +0.25 -0.45 +0.16
25 0.00 +0.10 +0.97 -0.07 +0.05 +0.15 -0.51 +0.15
26 0.00 +0.05 +0.99 -0.04 .+0.02 +0.07 -0.16 +0.07

 

m

(66)

«~‘-—.- ‘-----. - --..--“—‘

 

 

 

 

Table; 1; (cont;)

 

 

 

 

 

 

 

 

.325011§.£0141h218a00nd.Aaallais CQQnL.)
___.lezittiqfl 130.49.01.an 40212100141154an ...
Point __4VJ jg» __475--__-'V ~ .”__H1 -_-H2 -_;25----.331.
15R ~+o.00 -0.04 -+0.98 +0.05 —0.02 —0.06 +0.16 —0.08
123 -+0.00 -0.08 1+0.97 +0.10 -0.05 —0.12 1+O.51 -0.17
118 +0.01 —0.11 +0.91 +0.17 -0.02 -0.15 +0.44 -0.27
10R +0.02 -0.12 *0.81 +0.2 +0.01 -0.15 +0.55 -0.41
9R +0.05 -0.11 +0.59 +0.47 +0.07 -0.05 +0.54 -0.57
SE +0.05 -—0.08 +0.55 +0.67 +0.15 +0.06 +0.44 -0.65
7R +0.05 -0.05 +0.18 +0.82 +0.14 +0.10 +0.52 -0.56
6R +0.04 -0.05 +0.08 +0.91 +0.12 +0.10 +0.20 -0.41
SE +0.02 -0.01 +0.05 +0.96 +0.07 +0.06 +0.09 -0.22
_qugnt 5.51; fig 51109921. 5 . 1m 3.110.211 .11th Jim-2.9.
Point _ 21--..-1E3 05 M4g» ’ Mk _-
5L +0.17 —0.77 ~0.54 —0.77 -5.54
6L -+0.46 —1.55 -O.91 -1.56 —6.37
7L 1+1.00 ~2.26 -0.99 -1.62 -8.57
8L +1.65 -2.99 -—0.65 -1.48 -8.81
9L +2.19 --5.54 +0.07 -0.75 -7.54
101. +2.15 -5.07 +0.78 +0.09 -5.08
11.1. +1.61 -2.15 +0.98 +0.45 -—5.04
12L +1.09 -1.8’) +0.81 +0.46 -1.81
15L +0.50 -0.94 '+0.41 +0.25 -0.84
18 -0.50 +0.72 -0.49 —O.52 +0.84
19 -0.86 +1.60 -1.00 —0.55 +1.50
20 -1.00 +2.54 -1.67 —0.71 +1.67
21 -0.64 +2.91 -2.50 -0.56 +1.74
22 —0.05 +2.84 -2.84 +0.05 +1.12
25 +0.56 +2.50 -2.91 +0.64 +0.55
24 +0.71 +1.67 --2.54 +1.00 +0.19
25 +0.55 +1.00 -l.60 +0.86 +0.05
26 +0.52 +0.49 -o.72 +0.50 0.00
158. -0.25 -0.41 +0.94 -0.50 -0.08
12R -0.46 —0.81 +1.80 --1.09 -0.14
11R -0.45 -0.98 +2.51 -1.61 -0.15
103 -0.09 -0.78 +5.07 -2.13 -o.22
9R +0.75 -0.07 +5.54 -2.19 -O.52
8R +1.48 +0.65 +2.99 --l.65 -0.69
7R +1.62 +0.99 +2.26 -l.00 -0.88
6R +1.56 +0.91 +1.55 -0.46 -0.68
SE +0.77 +0.54 +0.77 -0.17 —O.27

 

(67)

 

 

 

 

 

...............

..............

.........

...............

Table 23; (0001:.)

he Third Analysis

 

 

 

 

 

“JMLWQDS.” __1142116554152444195L._
Point 71 v2 v35 74 H1 H2 H5 H4
5:. +0.97 +0.02 -0.01 +0.02 +0.21 -0.09 -0.06 —0.06
6L +0.90 +0.07 -0.02 +0.05 +0.40 -0.20 -0.10 —0.11
7L +0.82 +0.16 -0.05 +0.04 +0.55 .054 —0.10 -0.14
8L +0.67 +0.56 ~0.08 +0.04 +0.60 -0.44 -0.05 —0.12
9L +0.47 +0.60 -0.10 +0.05 +0.56 -0.54 +0.05 .0.07
10L +0.29 +0.80 .0.11 +0.01 +0.41 -0.54 +0.15 -0.01
11L +0.18 +0.90 —o.10 0.00 +0.28 -o.45 +0.14 +0.05
12L +0.10 +0.95 .0.07 0.00 +0.17 .0.55 +0.12 +0.05
15L +0.04 +0.97 .0.04 0.00 +0.08 -0.16 +0.06 +0.02
18 -0.04 +0.97 +0.04 0.00 -0.07 +0.16 —0.07 -0.02
19 -0.07 +0.96 +0.07 0.00 -0.12 +0.52 —0.15 —o.04
20 -0.09 +0.91 +0.19 -0.01 -o.15 +0.44 —0.25 .0.04
21 -o.09 +0.76 +0.52 .0.05 —0.15 +0.51 —o.56 .0.01
22 -—0.06 +0.57 +0.57 -0.06 -0.06 +0.50 —0.50 +0.06
25 —0.05 +0.52 +0.75 —0.09 +0.01 +0.56 —0.51 +0.15
24 -0.01 +0.19 +0.88 —0.09 +0.04 +0.25 —0.44 +0.15
25 0.00 +0.07 +0.96 .0.07 +0.04 +0.15 41.52 +0.12
26 0.00 +0.04 +0.97 —0.04 +0.02 +0.07 -0.16 +0.07
15R 0.00 .0.04 +0.97 +0.04 —0.02 -0.06 +0.16 —0.08
12R 0.00 -—o.07 +0.94 +0.10 -0.05 -0.12 +0.55 —0.17
112 0.00 -0.10 +0.90 +0.18 -o.05 —0.14 +0.45 -0.28
102 +0.01 -0.11 +0.80 +0.54 +0.01 -0.15 +0.54 -0.41
92 +0.05 -o.10 +0.60 +0.47 +0.07 —0.05 +0.54 -0.56
SE +0.04 -0.08 +0.56 +0.67 +0.12 +0.05 +0.44 -O.60
7R +0.04 -0.05 +0.18 +0.82 +0.14 +0.10 +0.54 -o.55
6R +0.05 -o.02 +0.07 +0.90 +0.11 +0.10 +0.2 -0.40
52 +0.02 .0.01 +0.02 +0.97 +0.06 +0.06 +0.09 -0.21
___Mggents at. 3116 5110;291:113 MQJZEDLEJ: Left. Kngg
Point M1 M2 M5 M4 Mk
5L +0.12 -0.77 -0.56 —0.75 -5.54
6L +0.56 -1.59 -0.90 —1.55 -6.42
7L +0.92 -2.42 —1.05 -l.67 43,27
8L +1.65 -5.05 -o.67 -1.45 -8.55
9L +2.26 -5.55 +0.12 -0.68 -6.95
10L +2.19 —5.25 +0.78 +0.14 -4.66
11L +1.75 -2.57 +1.02 +0.55 -2.95
12L +1.17 —1.88 +0.85 +0.54 -1.64
15L +0.57 -0.95 +0.45 +0.52 -0.74

 

(6

8)

 

15.21.2221 (£01111)

Resglfig for the Third Analzglg (cont.)

 

 

 

 

 

 

 

 

 

 

 

 

~10“ Immunemmg r35... M om 5:11.251ng

Point M1 M2 M5 M4 Mk

18 -0.51 +0.79 --0.51 .0.55 +0.67

19 -o.90 +1.75 -1.10 —O.66 +1.10

20 —1.01 +2.57 —1.75 —0.82 +1.48

21 -0.64 +2.86 -2.57 ~0.65 +1.50

22 +0.07 +2.97 -2.97 .0.07 +0.95

25 +0.65 +2.57 -2.86 +0.64 +0.42

24 +0.82 +1.75 -2.57 +1.01 +0.15

25 +0.66 +1.10 -1.75 +0.90 0.00

26 +0.55 +0.51 .0.79 +0.51 0.00

152 -0.52 —0.45 +0.95 -0.57 -O.16

122 -0.54 -0.85 +1.88 -1.17 -0.21

112 -0.55 -1.02 +2.57 -1.75 -0.22

102 -0.14 —0.78 +5.25 -2.19 -0.50

92 +0.68 —0.12 +5.55 -2.26 -0.45

82 +1.45 +0.67 +5.05 -1.65 -0.64

72 +1.67 +1.05 +2.42 .0.92 -O.67

62 +1.55 +0.90 +1.59 -o.56 -0.62

SE +0.75 +0.56 +0.77 -O.12 -0.22

2561:5111;
22213311512204}. MSULTS
TRIPLE. 5212 21.12 222.11; 221062 _ 212020 50220225
Resultg for the First Analxsig
Moment at

P01nt V1 V2 V5 V4 H1 H2 H5 H4 Mk
5L +0.97 +0.04 -0.05 +0.04 +0.19 -0.08 -0.06 -0.05- -5.01
6L +0.87 +0.11 -0.05 +0.07 +0.54 ~0.16 .0.09 -0.08 —5.68
7L +0.78 +0.21 -0.09. +0.09 +0.47 —o.27 —0.10 -0.11 -7.85
8L +0.64 +0.59 -0.15 +0.10 +0.51 .0.52 -0.07 -0.11 -8.69
9L +0.47 +0.62 -0.17 +0.08 +0.45 -—0.57 +0.01 -0.09 -7.69
10L +0.50 +0.82 -0.16 +0.04 +0.52 .0.55 +0.08 .0.05 -5.55
ML +0.19 +0.95 —0.14 +0.02 +0.21 -0.29 +0.10 —0.02 -5.51
121. +0.11 +0.97 .o.10 +0.01 +0.15 .0.20 +0.08 -0.01 —2.17
15L +0.05 +0.98 —0.05 0.00 +0.06 —0.11 +0.05 0.00 —1.00
x - , 1 ___ ..... -- ”___, _

 

 

 

 

Table XII (cont.)

Results for the First Analysis (cont.)

...-

 

 

 

 

 

 

 

 

 

 

Moment at

____fl§xtinglufi§8 ’0 ...lkujgxuugfl.89891102511. léilnflnga
P011113 V1 V2 V5 V4 H1 H2 H5 H4 MK
18 —0.05 +0.98 +0.05 0.00 -0.06 +0.11 +9.05 +0.00 +1.00
19 —-0.09 +0.97 +0.15 -0.01 -0.11 +0.20 —0.10 +0.01 +1.85
20 -O.l2 +0.95 +0.21 -0.02 —O.15 +0.29 —0.16 +0.02 +2.51
21 —0.15 +0.80 +0.56 —0.05 -0.15 +0.55 —0.2 *0.05 +2.51
2?. —O.10 +0.59 +0.59 -O.10 —O.11 +0.50 —0.50 +0.11 +1.85
2 -0.05 +0.56 +0.80 -O.15 -0.05 +0.25 -0.55 +0.15 +1.00
24 -—0.02 +0.21 +0.95 -—0.12 -0.02 +0.16 -0.29 +0.15 +0.55
25 —0.01 +0.15 +0.97 -0.09 -0.01 +0.10 —0.20 +0.11 +0.17
26 0.00 +0.05 +0.98 -0.05 0.00 +0.05 -0.11 ‘0.06 0.00
15?. 0.00 -0.05 +0.98 +0.05 0.00 -0.05 +0.11 -0.06 0.00
12R +0.01 --0.10 +0.97 +0.11 +0.01 -0.08 +0.20 -O.15 0.00
11R +0.02 -0.14 +0.95 +0.19 +0.02 -0.10 +0.29 -0.21 -0.55
10R +0.04 -0.16 +0.82 +0.50 +0.05 -0.08 +0.55 -O.52 —0.84
9R +0.08 -O.17 +0.62 +0.47 +0.09 -0.01 +0.57 -—0.45 -1.50
8R +0.10 -—0.15 +0.59 +0.64 +0.11 +0.07 +0.52 -0.51 +1.85
7R +0.09 +0.09 +0.21 +0.78 +0.11 +0.10 +0.27 -O.47 -l.85
6R +0.07 -0.05 +0.11 +0.87 +0.08 +0.09 +0.16 —O.54 -l.54
5R +0.04 -0.05 +0.04 +0.97 +0.05 +0.06 +0.08 -0.19 -0.84

§§g1__§,fbr the Second Ana1ysis

‘ Moment 8t

.___Jkujdigfl4ikaxdfisuuL1._. ‘ ec *' ._.. L2£I_Kn§§
Point 1E1 V27 75 74_ H1_ H2, H5 H4» ME;
L 4-0.98 -+0.05 -0.02 '*0.05 '+0.18 -0.09 -0.05 -0.04 +5.01
6L '*0.86 -+0.11 -0.05 +0.07 '+0.54 —O.16 ~0.09 -0.08 -5.68
7L +0.78 +0.21 -0.09 +0.08 +0.46 -0.25 -0.10 -O.10 +7.69
8L +0.65 +0.59 -0.15 +0.09 +0.51 -—O.52 —0.07 -0.10 -8.52
9L +0.47 +0.64 -0.17 +0.07 +0.44 -0.57 +0.02 -0.08 -7.55
10L +0.50 +0.82 -0.16 +0.04 +0.51 -O.55 +0.08 -0.04 -5.55
1.11. +0.19 +0.92 —0.15 +0.02 +0.21 —O.29 +0.10 -0.02 -5.51
12L +0.11 +0.98 -0.10 +0.01 +0.15 -O.21 +0.08 0.00 +2.17
15L +0.05 +0.99 -0.05 0.00 +0.06 -0.10 +0.04 0.00 -l.00
18 -0.04 +0.99 +0.05 0.00 —0.05 +0.10 -0.05 0.00 +1.00
19 -0.09 +0.98 +0.11 0.00 —0.10 +0.20 -0.10 0.00 +1.67
20 —O.l2 +0.92 +0.21 -—0.02 -0.14 +0.28 -0.16 +0.01 +2.54
21 —0.15 +0.80 +0.57 -0.05 +0.14 +0.55 -0.24 +0.05 +2.54
22 -0.09 +0.59 +0.59 -0.09 -O.10 +0.51 —0.51 +0.10 +1.67
25 -0.05 +0.57 +0.08 --0.15 —0.05 +0.24 -O.55 +0.14 +0.84
24 ~0.02 +0.21 +0.92 —0.12 -0.01 +0.16 -O.28 +0.14 +0.55
25 0.00 +0.11 +0.98 -0.09 0.00 +0.10 -0.20 +0.10 +0.17
26 0.00 +0.05 +0.99 —0.04 0.00 +0.05 -0.10 +0.05 0.00

(70)

Table £1; (cont.)

Secord, Anslvsis (cont .)

 

 

 

 

 

 

 

 

 

 

Moment at
YerticaltReactions ___jkujixuflgfllEfiasfidsumL_.- Léfiieineg
Point V1 V2 V5 V4 H1 Ho H5 H4 MK
15R +0.00 +0.05 +0.99 +0.05 0.00 +0.04 +0.10 +0.06 0.00
12R +0.01 +0.10 +0.98 +0.11 0.00 +0.08 +0.21 +0.15 0.00
11R +0.02 +0.15 +0.92 +0.19 +0.02 +0.10 +0.29 +0.2 +0.55
10R +0.04 +0.16 +0.82 +0.50 +0.04 .0.08 +0.55 +0.51 +0.67
9R +0.07 +0.17 +0.64 +0.47 +0.08 +0.02 +0.57 +0.44 +1.54
8R +0.09 +0.15 +0.59 +0.65 +0.11 +0.07 +0.52 +0.51 +1.85
7R +0.08 +0.09 +0.21 +0.78 +0.10 +0.10 +0.25 +0.46 +1.85
6R +0.07 +0.05 +0.11 +0.86 +0.08 +0.09 +0.16 +0.54 +1.54
5R +0.05 +0.02 +0.05 +0.98 +0.04 +0.05 +0.09 +0.18 +0.84
Results for the Thirg Analysis
Moment 9t
Veggies; Reactions fiozizgntal fleegtigns Le?t Knee
Point V1 V2 V3 V4 H1 H2 H5 H4 MK

5L +0.960 +0.055 +0.02O +0.055 +0.19 +0.09 -0.05 +0.04 +5.01
6L . +0.880 +0.100 +0.050 +0.060 +0.54 +0.16 +0.09 +0.08 +5.68
7L +0.780 +0.220 +0.080 +0.080 +0.46 +0.25 +0.10 +0.09 +7.52
8L +0.64O +0.59O +0.12O +0.09O +0.48 +0.52 +0.06 +0.09 +8.02
9L +0.460 +0.62O +0.140 +0.060 +0.41 +0.55 +0.01 +0.07 +7.02
lOL +0.5OO +0.820 +0.150 +0.050 +0.50 +0.54 +0.08 +0.05 +5.01
11L +0.190 +0.920 +0.150 +0.020 +0.20 +0.28 +0.10 +0.01 +5.18
12L +0.110 +0.97O +0.090 +0.010 +0.12 +0.19 +0.08 0.00 +2.00
15L +0.050 +0.990 +0.04O 0.000 +0.05 +0.10 +0.05 0.00 +1.00
18 +0.040 40.990 +0.050 0.000 +0.05 +0.11 +0.05 0.00 +0.84
19 +0.09O +0.960 +0.120 0.000 +0.09 +0.20 +0.10 0.00 +1.50
2O +0.110 +0.92O +0.210 +0.020 -0.15 +0.28 +0.17 +0.01 +2.17
21 +0.120 +0.800 +0.57O +0.050 +0.12 +0.55 +0.24 +0.04 +2.17
2 +0.090 +0.580 +0.580 +0.09O +0.09 +0.50 +0.50 +0.09 +1.50
25 +0.050 +0.570 +0.8OO +0.120 +0.04 +0.24 +0.55 +0.12 +0.84
2.4 +0.020 +0.210 +0.92O +0.110 +0.01 +0.17 +0.28 +0.15 +0.55
25 0.000 +0.120 40.960 +0.090 0.00 +0.05 +0.11 +0.05 +0.1?
26 0.000 +0.050 +0.990 +0.040 0.00 +0.05 +0.11 +0.05 0.00
15R 0.000 +0.040 +0.990 +0.050 0.00 +0.05 +0.10 +0.05 0.00
12R +0.010 +0.090 +0.97O +0.110 0.00 +0.08 +0.19 +0.12 0.00
11R +0.020 +0.150 40.920 40.190 +0.01 +0.10 +0.28 +0.20 +0.17
10R +0.050 +0.150 +0.820 +0.500 +0.05 +0.08 +0.54 +0.50 +0.50
9R +0.060 +0.14O +0.62O +0.460 +0.07 +0.01 +0.55 +0.41 +1.17
8R + 0.090 +0.120 +0.59O +0.64O +0.09 +0.06 +0.52 +0.48 +1.50
7R +0.080 +0.080 «0.220 +0.780 +0.09 +0.10 +0.25 +0.46 +1.67
6R +0.060 +0.050 +0.100 +0.880 +0.08 +0.09 +0.16 +0.54 +1.17
5R +0.055 +0.020 +0.055 +0.960 +0.08 +0.05 +0.09 +0.19 +0.84

 

(71)

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Beggs, George 12., "Discussion on Design of‘ a Multiple-Arch System",

Griffith, James R., "Design Cost of Highway Bridges Reduced by {
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Gottschclk, Otto, Esq., "Mechanical Calculations of El? stic Systems",
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Wilson, Wilbur M., "Laboratory Tests of Multiple-Span Reinforced
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