estudo sobre distorção em soldagem
DESCRIPTION
Masubuchi, 1965TRANSCRIPT
NPS ARCHIVE1969NGUYPI N 1 IP! N KM
LIBRAKYmVAL P o ^M,:-,!jATE SCHOOiMONTTEREr . 93940^
NAVAL P0ST3RADUATE SCHOOLMONTFREY " ' 93940
A STUDY ON SHRINKAGE
DISTORTION OF BUTT WELD
DUDLEY KNOX LIBRARYNAVAL POSTGRADUATE SCHOOLMONTEREY, CA 93943-5101
by
Nguyen-Tien- Ich//
S.B., Naval Academy of Brest (FRANCE); (1957)B.A., Saigon University (VIET-NAM) ; (1962)
Submitted in partial fulfillment of the requirements
for the Degree of Master of Science and Naval Engineer
at the
Massachusetts Institute of Technology
June, 1969
\HY'^ ^\. " -%^^^^<ir^'^'\^
^^6u^,^iv i-\c+H
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11
r.
laVAL f ...' ^/:TE SCHOOL
A STUDY ON SHRINKAGE DISTORTION OF BUTT WELD
by
Nguyen-tien-Ich
Submitted to the Department of Naval Architecture and
Marine Engineering on May 23, 1969 in partial fulfillment
of the requirement for the degree of
Master of Science and Naval Engineer
ABSTRACT
In a welded structure, shrinkage and external constraintare closely related. Shrinkage decreases as Degree of Con-straint increases
.
In the first part, this paper presents:(1) a formula that can be used to find the transverse shrin-
kage of a free joint in the case of a butt weld,(2) the definition of the degree of constraint of restrained
structures
,
(3) the empirical correlation between transverse shrinkageand degree of constraint.
In the second part, this paper presents:(1) analytical formulae for the degree of constraint of some
simple joint configurations,(2) the numerical method for the degree of constraint of
other joint configurations and the tabulation of the re-sults obtained.
In the last part, the use of the degree of constraintto determine the cracking susceptibility of a welded struc-ture is suggested and a method of experimentation to verifythe numerical results obtained in the second part is proposed.
Thesis Supervisor: Dr. Koichi Masubuchi
Title: Associate Professor ofNaval ArchitectureMassachusetts Instituteof Technology
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ACKNOWLEDGEMENTS
The author expresses his profound gratitude to Prof.
Koichi Masubuchi whose constant guidance and encouragement
are most valuable; to Prof. Sherman C. Reed for his discreet
solicitude; to Prof. Jerome J. Connor and Mr. George T. Will
of the Civil Engineering Department whose help in the com-
puter programming are most crucial; to Prof. Norman Jones
and Prof. Alaa E. Mansour whose offices are always widely
open for free consultations; to Prof. Theodore H. Plan of
the Aeronautics and Astronautics Department for his gene-
rous and enlightening assistance.
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IV
Table of Contents '
Page
Section I ABSTRACT ii
Section II SUMMARY 1
Section III INTRODUCTION 9
1. General 9
2. Technical Background 10
3. Purpose of the Study 26
Section IV PROCEDURE 29
1. Mathematical Approach 29
2. Choice of Appropriate Numerical Method 30
Section V RESULTS 43
1. Presentation of the Results 4,3
2. Discussion of the Results 64
3. The Problem of Experimentation 73
Section VI CONCLUSION 78
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Appendices
Transverse Shrinkage Computations
lA. Elliptic Slit - "PSR" and "CSTG" Type 1
IB. Elliptic Slit - "CSTG" Type - Fine Gridwork 31
2. Elliptic Slit - "LST" Type 76
3. Straight Slit 122
4. Straight Slit with Circled Ends - "CSTG" Type 140
5. Straight Slit with Circled Ends - "LST" Type 164
6. H-Slit 219
7A. Lehigh Test Specimen, No Sawcut 248
7B. Lehigh Test Specimen, With Sawcut 272
8. H-Slit, Aluminum Plate 301
List of Illustrations
Table
1 Node Displacements, Elliptic Slit, "PSR" and"CSTG" Type 36
2 Values of K, Elliptic Slit, "PSR" and "CSTG"Type 37
3 Node Displacements, Elliptic Slit, "CSTG" Type,Finer Gridwork 38
4 Values of K, Elliptic Slit, "CSTG" Type, FinerGridwork 39
5 Node Displacements, Elliptic Slit, "LST" Type 40
6 Values of K, Elliptic Slit, "LST" Type 41
7 Node Displacements, Straight Slit 45
8 Values of K, Straight Slit'
46
9 Node Displacements, Straight Slit with CircledEnds, "CSTG" Type 49
10 Values of K, Straight Slit with Circled Ends,"CSTG" Type 50
11 Node Displacements, Straight Slit with C: rcledEnds, "LST" Type 51
12 Values of K^ Straight Slit with Circled Ends,"LST" Type 52
13 Node Displacements, H-Slit 55
14 Values of K, H-Slit 56
VI
15 Results of Lehigh Test Specimen, Without Sawcut 59
16 Results of Lehigh Test Specimen, With Sawcut 61
17 Results of H-Slit, Aluminum Plate 63
Figure
1 Schematic Representation of Changes of Temperatureand Stress During Welding 14
2 Non-Dimensionalized Degree of Constraint as Func-tion of Ratio Weld Length/Slit Length in anInfinite Plate 21
3 Relationship Between Degree of Constraint andTransverse Shrinkage in a Slit-Type Specimen 24
4 Effect of External Constraint on the TransverseShrinkage of Butt-Welded Joints 25
5 Non-Dimensionalized Degree of Constraint as Func-tion of Ratio Weld Length/Slit Length, EllipticSlit, Finite Plate 42
6 Non-Dimensionalized Degree of Constraint as Func-tion of Ratio Weld Length/Slit Length, StraightSlit, Finite Plate 47
7 Non-Dimensionalized Degree of Constraint as Func-tion of Ratio Weld Length/Slit Length, CircledEnds, Finite Plate 53
8 Non-Dimensionalized Degree of Constraint as Func-tion of Ratio Weld Length/Slit Length, H-Slit,Finite Plate 57
9 Non-Dimensionalized Degree of Constraint as Func-tion of Ratio Weld Length/Slit Length, All Typesof Slits Studied Previously 65
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SUMMARY
Distortions and residual stresses unavoidably accompany
every welded structure. They are not independent of each
other. In the case of an unrestrained structure, the shrin-
kage distortion is determined by the welding heat and can be
determined. However, practical structures are seldom com-
pletely free and welding is performed more or less under ex-
ternal constraint, therefore it seems that if factors rela-
ting to welding heat are definite, the ratio of shrinkage
distortion under external constraint to the one in a free
welded joint is uniquely decided by the degree of external
constraint to which weldments are subjected, or:
(shrinkage under external constraint, S ) = (shrinkage
in free welded joint, S ^) x F
where F is a function depending on the degree of constraint,
S ^ is the transverse shrinkage of the unrestricted joint or
free joint. For a butt joint, S ^ has the following expres-
sion:
\f = ^1 r2 l°^e W7 * ^2 (72)h ° h
where
:
A = sectional area of the groove of the butt joint,
h = plate thickness,
W = weight of deposited metal per unit weld length.
Wo = weight of deposited metal per unit weld length perwelding of each pass, and
C, and C„ are coefficients depending on arc voltage, arc
current, arc efficiency, filler metal diameter, and
melted weight of rod per unit current and unit time.
For example with the Ilumenite type filler rod of 3.2 mm
diameter, with weld current of 120A and welding speed of 0.3 cm/s
,
it has been found that in the C.S.G. unit system:
C^ = 0.0960, C^ = 0.0416
In this order of idea, Masubuchi has defined the degree of
constraint as:Uniform transverse stress, a
YoDegree of Constraint, K = —— (1)
Average Transverse Displacement [v]
„
where [v] . is the average of the transverse displacement taken a-
long the weld length i, and has found a graphical correlation
between transverse shrinkage and degree of constraint K (refer-
ences 1,2).
Later, Watanabe and Satoh, after analyzing the experimental
results obtained by Masubuchi and others, have found a correlation
of the form:
^t 1^ ^(2)
^tf 1+ 0.086K°*^^
therefore, in order to determine the shrinkage distortion, the
value of K must be known.
On the other hand external constraint promotes stress. In
fact, the level of stress can be found by combining (1) and (2)
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and for a butt weld, S - [v] for an actual weld. Therefore
a is the average reactiono = S ^ X—py ^° stress along the weld due^° 1+0.086K to constraint
If the external constraint is too high, residual stress can reach
a dangerous level where cracking can occur. Here again, it is es-
sential to know K.
In a few cases, K can be found analytically. But in most
practical cases, it has to be computed numerically.
The object of this study is to compute numerically the values
of K for some test specimens and to propose a method of experimen-
tation to verify the computed results. Also the degree of con-
straint in the case of patch weld has been determined analytically
The numerical method used is the finite element method in
which the plate structure is divided into small elements in each
of which static equilibrium and geometrical compatibility have to
be satisfied. In this case, for each specimen the values of K
are computed as a function of R = — where I is the weld lengthJu
(in this case, it is the length upon which the uniform stress a
acts) and L is the length of the slit. The results are presented
as the non-dimensionalized degree of constraint K = , > > (E is
the modulus of elasticity) curves versus R = £/L in the attached
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diagram (Figure 9) .
One exception is made in the case of the Lehigh test spe-
cimen where the weld is made all along the length of the slit
'
1
T 1 p—
I
1
1"
c -—
o
without saw cut with saw cut
#
(£=L, R=1.0). In these cases, experimental data are available
for comparison:
-«•
K *computed measured
K —Ko computed measured
computed
No Sawcut 45.3 Kg/mra^-mm 44 Kg/mm^-mm 2.87
With Sawcut i28.9 Kg/mm^-mm 27 Kg/mm ^ -mm j 6.70
Assuming E = 19.9 x 10^ Kg/mm^
The object of the Lehigh test specimen is to find the critical
value of K or range of values of the degree of constraint above
which weld cracking may develop, below which it may not. The ana-
lytical results obtained in the case of a patch weld ,which consists
of a circular disc welded to a plate, are for the case of a plate
with large dimensions (infinite plate)
:
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Patch Weld
(1) If the weld can be made along the whole circle in one pass,
then:
KuR
1 E2 R
R is the radius of the plate,
o is the radial stress of the perimeter of the disc, and
u„ is the radial displacement at the weld.
(2) If the weld can be made within a sector 2a, then:
•
K = R
f^R^2a
HE2aR 2-f (2a)
X
where [u ] „ is the average radial displacement over the weldec^ sec-
tor 2a and f (2a) is a series function of 2a with f (0)=f (n)=0 and
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the series converges as —
.
Because of the linearity of the relations in elastic defor-
mation, the displacement is inversely proportional to the modulus
of elasticity E, and K, which is defined as a stress divided by
the corresponding average displacement, is proportional to E.
Therefore a new factor, called specific degree of constraint K'
,
and defined as:
K' = KE
is independent of the linear elastic material and dependent only
on the geometry of the weld.
Therefore experiments to determine the values of K to verify
the computed results can be conducted on any linear elastic ma-
terials, preferably the ones that have small E and are not expen-
sive. Plastic materials are proposed, and a mechanical way to
stress the specimens is to drive two wedges to make an assembly
r - - - - -1
c^--ij^--1
r
; '-l->-"^-L. ->:
wedge shape
with parallel faces into the slit. Each specimen will have
several sets of wedges with different widths I so that K' can be
•found as function of R = — . Strain gauges put near the edges of
•
the slit and visible marks made at the immediate vicinity of the
edges for optical measurements by microscope or comparator are
used to determine stresses and displacements. In these experi-
ments the material at the loading zone is in compression instead
of tension as in the welding. It is of no consequence as far as
the material is linear elastic in compression. It is actually,
and the value of E is between 3 to 5 x 10^ psi (in compression).
The problem of shrinkage control could be conceived as a
problem of choice between two alternatives, based on the degree
of constraint K:
aYo
K =
^-hand the Watanabe-Satoh relation:
iv], S^
•
^tf ^tf 1+0.086 K°*^^
^tf ^^ ^^® transverse shrinkage of a free joint and for a butt
weld, S^ - [v]^ at the weld. These alternatives are either to
limit the shrinkage by increasing the external constraint, and
thus accept a higher value of K and hence a higher level of re-
sidual stress or to reduce the residual stress level by relaxing
the external constraint thus reducing the value of K and to accept
a larger value of shrinkage distortion.
In the problem of cracking control, it is essential that the
external constraint will result in a value of K below the range
of critical degree of constraint K , characteristic of each ma-
terial and each weld type.
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In these two problems, the important factor is the degree
of constraint K.
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9
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INTRODUCTION
I. General
Welding is an efficient method of fabricating struc-
ture. Various products ranging from miniature electronic
components to bridges, ship hulls, rocket motor cases have
been made by welding.
- One of the troublesome problems that accompany the con-
struction of welded structures is shrinkage distortion. The
more complex the structure is, the more involved the problem
becomes. Shrinkage distortion can cause mismatch of joints
^^ which leads to the possibility of welding defects.
The correction of weld distortion is costly and in
some cases impossible. It is therefore desirable to develop
some techniques to predict somehow the approximate values
of the shrinkage distortion in order that palliative mea- '
sures can be devised that will neutralize or reduce the
effect of shrinkage distortion.
Another problem associated with welding is cracking.
Cracks may form as a result of the welding operation and can
occur within the weld metal or the base metal in the heat-
affected zone.
There are two types of cracking that are experienced in
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welding: hot cracking and cold cracking. Hot cracking is be-
lieved to take place during the solidification of the weld-
ment and can occur in the weld metal or in the heat affected
zone. Hot cracks are of intergranular type. Cold cracking
occurs at much lower temperatures than does hot cracking,
in steels at temperatures below the start of the austenite-
martensite transforiaation. Cold cracks may be formed during
or after the welding operation; cold cracks are of trans-
granular type. Cracking, hot or cold, depends on many fac-
tors such as chemical composition of the base and filler
metals, mechanical restraints, welding condition (heat, input)
.
For given welding conditions, chemical compositions of the
base and filler metals, it is desirable to develop a way of
determining quantitatively the degree of restraint that may
promote cracking.
II. Technical Background
Distortion in a welded structure may be determined as
a function of structural parameters, material parameters and
fabrication parameters.
The structural parameters include the geometry of the
structure, the shape of the joining boundary and the type of
the joint.
The material parameters are the nature of the base and
filler materials.
The fabrication parameters include the welding process,
the heating procedure, the welding sequence, the degree of
t
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constraint.
The degree of constraint, which will be the principal
object of this study, is a parameter which characterizes the
external mechanical constraint to which a weld is subjected.
For example, the degree of constraint of a weld made through-
out along the median line of a rectangular plate, case (a).
ra) (b)
is different from that of a weld made only partially along
this line, case (b) . In fact, in case (a), the weld can be
considered to be free while in case (b) , it is not. And in
case (a) the shrinkage is larger than in case (b) and we can
imagine that there should be a relation between shrinkage and
degree of constraint.
According to Masubuchi (reference 2) , to analyze weld
distortion, it is necessary to establish analytical relation-
ships among these three sets of parameters and distortion.
For a simple butt weld as is the case of this study, the
dimensional changes produced in the structure by each weld are
to be determined. This can be done by:
(1) Analyzing the heat flow,
(2) Analyzing the thermal stresses during welding to
determine incompatible strains, and
(3) Determining the dimensional changes.
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In fusion welding, a weldment is locally heated by the
welding heat sources. During the thermal cycle, the weldment
is subjected to thermal stresses. When the weld is completed,
incompatible strains are created in the region near the weld.
Incompatible strains, including dimensional changes associated
with solidification of the weld metal, metallurgical transfor-
mations, and plastic deformations, are the sources of residual
stresses and distortion. When welding processes and parameters
are changed, the heat flow pattern is also changed causing a
change in the distortion of incompatible strains, hence in
shrinkage and distortion.
The problem of determining the distribution of incompatible
strains is extremely difficult. When a material undergoes plas-
tic deformation, the stress strain relationship is not linear.
Furthermore, plastic properties of the material change with
temperature.
When the incompatible strains are known, theoretically or ex-
perimentally, the problem of determination of dimensional changes
can be handled analytically. Moriguchi has developed a funda-
mental theory of stresses caused by incompatible strains, and
Masubuchi has applied Moriguchi ' s theory to the study of resi-
dual stresses and distortion due to welding.
Assuming that the dimensional changes in welds are determined,
either analytically or experimentally, the next step is to deter-
mine the distortion induced in the structure by these dimensional
changes. Although plastic deformation is produced in small areas
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near the weld, most of the remaining material in the structure
is elastic. Consequently, the induced distortion can be ana-
lyzed by using the elastic assumption. Hence the relations used
to determine the induced distortion are independent of fabrica-
tion parameters and depend only on well-established material
parameters
.
A. Changes of Temperature and Stress During Welding (from refer-ence 2)
Figure 1 shows schematically how residual stresses are formed
in a weld. Figure la shows a bead-on-plate weld in which a weld
bead is being laid at a speed v. 0-xy is the coordinate axis;
the origin, 0, is on the surface underneath the welding arc, and
the X direction lies in the direction of welding.
Figure 1 shows temperature distribution along several cross
sections. Along Section A-A, which is ahead of the welding arc,
the temperature change due to welding, AT, is almost zero (Figure
lb-1) . Along Section B-B, which crosses the welding arc, the
temperature distribution is very steep (Figure lb-2). Along
Section C-C , which is some distance behind the welding arc, the
distribution of temperature change is as shown in Figure lb-3.
Along Section D-D, which is very far from the welding arc, the
temperature change due to welding again diminishes (Figure lb-4)
.
Figure Ic shov;s the distribution of stresses along these
sections in the x direction, a . Stress in the y direction, a ,X -^
y
and shearing stress, t , also exist in a two-dimensional stress^ ' xy
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0. V/eld
AT «
I. Seclion A-
A
Stress ^0
E
1
.1
2. Seclion B-Bo:eoo
AQp'Ap^'
3. Seclion C-C
Residual
stress
\/
==iBiy Siirro?
/J. Seclion D-D
Mb. Ternpcrolure Chongc c. Stress a
A-57275
IWUR}-: 4. SCHEMATIC REPRESENTATION OF CHANGES OF TEMPERATURE AND
STRESSES DURING WELDING. .' •
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*field (Figure la)
.
Along Section A-A, thermal stresses due to welding are al-
most zero (Figure lc-1) . The stress distribution along Section
B-B is shown in Figure lc-2. Stresses in areas underneath the
welding arc are close to zero, because molten metal does not
support loads. Stresses in areas somewhat away from the arc are
compressive, because the expansion of these areas is restrained
by surrounding areas that are heated to lower temperatures. Since
the temperatures of these areas are quite high and the yield
strength of the material is low, stresses in these areas are as
high as the yield strength of the material at corresponding tem-
peratures. The amount of compressive stress increases with in-
creasing distance from the weld or with decreasing temperature.
However, stresses in areas away from the weld are tensile and
balance with compressive stresses in areas near the weld. In
other words
,
So • dy = (a)
*-k .
across Section BB. Thus, the stress distribution along Section
BB is as shown in Figure lc-2.
Stresses are distributed along Section C-C as shown in
*In a general three-dimensional stress field, six stress compo-nents, 0,0,0,1 , T , T exist.
X y z xy zy zx
**Equation (a) neglects the effect of a and x on the equili-brium condition. ^ ^
•
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Figure lc-3. Since the weld-metal and base-metal regions near
the weld have cooled, they try to shrink causing tensile stresses
in areas close to the weld. As the distance from the weld in-
creases, the stresses first change to compressive and then be-
come tensile.
Figure lc-4 shows the stress distribution along Section D-D.
High tensile stresses are produced in areas near the weld, while
compressive stresses are produced in areas away from the weld.
The distribution of residual stresses that remain after welding
is completed are shown in the figure.
The cross-hatched area, MM', in Figure la shows the region
where plastic deformation occurs during the welding thermal cycle
The cross-hatched area near the origin indicates the region
where the metal is melted. The region outside the cross-hatched
area remains elastic during the entire welding thermal cycle.
Because of the difficulty in determining the distribution
of incompatible strains, no analysis has yet been developed to
trace the change of two-dimensional thermal stresses during
welding and to determine distributions of three residual-stress
components, 0,0, and t . In other words, no analysis has
been made in which both heat flow and stress fields are treated
as two-dimensional problems. In all studies conducted so far,
the problem has been simplified in some way.
B. Shrinkage Distortion
Shrinkage distortion in welding can be considered as the
17
result of the combination of two groups of factors:
(1) Factors related to welding arc, arc voltage, welding
current, welding speed, type and size of electrode,
(2) Factors related to external constraint.
1 - Effect of Welding Heat
The effect of welding heat on the transverse shrinkage in
the case of butt weld free of external constraint is known in
the form:
where
S £ is the transverse shrinkage of the free joint,
A is the sectional area of groove of the butt joint,
b is the thickness of the plate,
W is the weight of deposited metal per unit length ofthe weld.
Wo is the weight of deposited metal per unit length ofthe weld in one pass, and
C^ and C^ are constants depending on the arc voltage, arc
intensity, heat efficiency of the welding arc, size and
type of the filler metal and the melted weight of the
rod per unit current and unit time.
2 - Effect of External Constraint
In practical work, welding is performed more or less under
external constraint. In this case, shrinkage distortion in weld
metal is depending also on the mechanical constraints and smaller
/ 18
than in free joints. In their work, Kihara and Masubuchi at-
tempted to determine quantitatively the degree of constraint and,
in a slit weld, to find a relation between transverse shrinkage
and degree of constraint.
is the centerof the plate
Accordingly, the degree of constraint K is defined as:
aK =
[y]
(2)
where I is the length of the weld in a slit of length L, a is
the uniform stress applied along the weld length Z and [v] . is
the corresponding mean value of transverse displacement over the
portion of the slit where the load is applied. The physical
meaning of K is that when uniform transverse stress a is ap-
plied along the part of the slit between x=x, and x=x^,
(Ix^-x, I = i) displacement v will occur along the slit. The mean
value of the transverse displacement [v] . defined by:
x^
["^h^I Ivdx is related to a by:
^yo ^""^""h
In the case of a slit in an infinite plate, and using the analogy
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19
between the residual stress in this case and the vortex theory
in Hydrodynamics, they have shown that (see references 1 and 2)
CiZim^&r- ^,
^— {—>,L
V = - - - -2 L L F
(3)
with
where
a 02F = Z [ / sin0sini:ed0]
n=l 01
£ - X2 x^, X2>x^
X.
0^ = cos"^ (-^!—
)
^L/2
-1 ^2= cos ( )
L/2
£The value of K as a function of R = :r- has been computed and the
curve K = K TT RX
E/L 2 F (R) has been drawn in Figure 2 for the case
where the weld is symmetric, that is x, + x_ = 0, and 0^ = tt-0-
the encircled dots on the diagram are the value of K = z^rrf found
experimentally. We note that K has the dimension of a stress
divided by a length and therefore K is dimensionless . In the
20
experiments described in reference 1, the following results have
been obtained:
Slit Lengtl Weld Length L K E/LK
L (mm) I (mm) Tj-i ( Kg/mm ^ /mm) (Kg/mm ^ /mm)
513 173 0.337 51.6 41.13 0.905340 164 0.488 56.4 62.06 0.740174 174 1.000 76.8 121.26 0.636523 174 0.330 52.0 40.34 0.915349 175 0.502 55.2 60.46 0.730174 174 1.000 76.8 121.26 0.636510 163 0.320 38.3 41.37 0.930176 176 1.000 76.0 119.89 0.636i171 171 1.000 78.2 123.39 0.636;340 168 0.494 57.0 62.06 0.735!172 172 1.000 77.8 122.67 0.636'337 172 0.510 55.5 62.61 0.725165 165 1.000 81.0 127.88 0.636202 202 1.000 66.2 104.46 0.636224 224 1.000 59.6 94.20 0.636129 129 1.000 103.5 163.57 0.63696 96 1.000 139.1 219.79 0.636
390 149 0.382 45.6 54.10 0.840290 150 0.518 52.2 72.76 0.720
1
In their report, Kihara and Masubuchi did not mention whether the
welds are symmetric or not in the case of x<1.00. But it has
been confirmed by Masubuchi that the welds are made symmetrically
The length of the plate is L^^ = 1200 mm, its width is B = 800 mm,
its thickness is h = 19 mm. The slit detail is shown in the
accompanying figure.
21
(I
aL?^
•73
cV
c^ ^o
s3
\i_
22
1 2 , i^ » •> «A
:Co"
3 v^Ul^x
J..i^Zoo >
The experimental values of K are higher than those given by
the theory, instead of the contrary, since the plates are actually
finite and the shrinkages are expected to be larger consequently.
There are several reasons for this. One is that the weld cannot
be perfectly symmetrical, while the values of K and K are lowest
in the symmetric case. Another is that the methods of measuring
the shrinkages as used by the authors give a lower value than
they are actually, since the width of the slit is about 12.8 mm,
while the shrinkages are measured along parallel lines which are
45 mm. apart and the value of the shrinkage at that distance of,45mm - 12.8mmthe weld (
they are at the weld.
= 6.1mm) is expected to be smaller than
Using this degree of constraint thus defined, Masubuchi has
found a corre2,-^tion between the mean shrinkage and K .in a slit
type specijrien as reproduced in Figure 3.
Expanding this idea, Watanabe and Satoh (reference 3) later
determined the value of K for other configurations such as the
H-type and the circular-ring type. Then they found an empirical
correlation between the transverse shrinkage S and the degree
of constraint K for various types of weld in the form:
• tf 1+0.086K0.87 ^c
•
T?
S being the transverse shrinkage of a free joint which has been
defined in the previous part. And they conclude (reference 3)
that, "It may be said... that these data are approximately repre-
sented by one curve despite the fact that various conditions are
different from one to the other. It can be concluded, therefore,
Stthat the function F (= -^—)...or the ratio of shrinkage dis-
^ ^tftortion under external restraint to the one in unrestrained
welded joint is decided by K and it is independent of the otherSt
conditions." The graph of ^— = f (K) has been reproduced in^tf
Figure 4 with some experimental data for comparison. The value
of S, in the case of butt weld is:
A ^ W^„ /A ,^/2
Log —^C (-h^ W„ ^ h
^tf = ^1-. ^°^e r^S^rr^
A: sectional area of groove of butt joint
h: plate thickness
W: weight of deposited metal per unit length
Wo : weight, of deposited metal per unit length perwelding of each pass.
V- In the C.G.S. unit system and with the iLmenite type, the co-
efficients C, and Cp have the following values:
Welding Current Welding Speed Size of Electrode C, C„(A) (cm/sec) (diameter in cm)
120 0.3 3.2 0.0960 0.0416
150 0.3 4 0.1021 0.0584
210 0.3 5 0.1530 0.0745
260 0.3 6 0.1249 0.0690
0.8
EE
0.6
o>o>o
•c 0.4
V)
co0)
0.2
i/L f^'^f^weld
Second or
third weld
1 o 6
035-0.55 A A
<035 D O
x:Mul1iIayer v/elding sequence
.1 1 ' JL 1 J u J_-«^. j«—a-50 .100 150
Degree of Constraint (K), kg/mnnVmm
24
viriTRE . RELATIONSHIP BETWEEN DEGREE OF CONSTRAINT ANDFIGURE 3. ^^™ ^3^ SHRINKAGE IN A SLIT-TYPE SPECIMEN
•
•
25
'O 20 <0 CO eo lOO 120 KO
Restroint coet!icicrit f 4O . K^/mrr^- mm
FIGURE 4. Effect of ExternalConstraint on the TransverseShrinkage of Butt-Welded Joints
\
16
III Purpose of the Study
The correlation obtained by Watanabe and Satoh presented in
the previous text enables the prediction of the transverse shrin-
kage and the level of transverse residual stress of a plate due
to given welding conditions if the degree of constraint of the
weld is known, provided that this empirical correlation is cor-
rect. The Watanabe-Satoh correlation has been obtained from a
set of experimental data conducted on three types of joints; the
straight slit type, the H-slit type and the circular ring type
specimen. Some values of K given by Watanabe and Satoh are:
esTTT^rzz
-L-^ - n L
(for an infinite plate)
Straight-Slit Type
L.
Ut
1
z:^
K = EB
1+ ( 2L
s -
(No information was given asto the width of the H bran-ches, the length of theplate)
H-Slit Type
27
-7
E 1 ri b b -a'K = TTT ii [log — -
4n b-a ^e a j^2+^;
Circular Ring
The straight slit formula has been found by Masubuchi in refer-
ence (1) or (2) . The circular ring formula can be found by using
the elastic theory (Reference: Theory of Elasticity ^ by Timoshenko,
2nd edition, problem No. 4, page 126). The H-slit formula can be
found by using the elastic theory, according to Watanabe and
Satoh.
In these three cases, the geometrical boundaries are rela-
tively simple and analytical formulae are possible. Unfortunately,
with the probable exception of the ring type, these specimens are
not easy to make. Usually to make a slit, it would be easier to
begin by drilling two relatively large circular holes at the ends
and then cutting the plate along a line joining their centers.
/"
Also, the dimensions of the plate are finite. Then the
region is not single-connected any more and analytical formula-
tion becomes very difficult if not impossible, and numerical
methods should be used. The same remark can be applied to actual
structures.
28
The purpose of this study is to compute the value of the
degree of constraint K for some types of specimens, to propose
a method of experimentation which, when performed, will provide
a means of verifying the numerical method used in the computa-
tion of K.
This numerical method of computing K provides also a method
of quantitatively evaluating the cracking susceptibility of a
weld. In this case, some experimentation methods have been pro-
posed (see reference 4, page 32-39), such as the Lehigh test, in
which the values of K are measured for different specimens. If
the value of K is too high the plate will crack and there should
be a critical value K , or a range of values, of the degree of
constraint K for which:
K below the critical range, the plate will not crack, and
K above the critical range, the plate will crack.
The critical range of K is determined experimentally and
could be considered as a material and weld type proper.ty. Then
if a weld is to be made for a given plate material, its ability
to crack can be predicted by computing the value of K of the
weld to be made and thus its feasibility can be decided quanti-
tatively before the weld is actually made.
•
29
PROCEDURE
I Mathematical Approach
In most cases, a numerical method will be used which is
based on the finite element method of structural analysis de-
veloped recently and a summary of which can be found in reference
(5) . The computer programs for the finite element methods in
structure analysis are made available by the Civil Engineering
Department of M.I.T., whose user's instructions can be found in
reference (7)
.
The basis of the method is to divide the structure into a
finite number of small elements. Within each element, the dis-
placements u and V at each point (x,y) in the x- and y-direction
are assumed to be series functions of x and y of the form:
1 rr3u(x,y) = .^^ a.. X y
1 „:
M
^Vi h,.
v(x,y) = Z E b . . X y1 3 ID
a. . and b. . are functions of the coordinates (x^^, y^,) of the nodes
M limiting the elements. The functions u(x,y) and v(x,y) are such
that displacement compatibility is satisfied along the boundaries
of the elements to ensure the condition of convergence to the
true solution. By the use of the variational method, a system
of linear equations between forces and displacements can be derived
#
30
and can be solved for forces if displacements are given or vice
versa.
The computer programs set up by the Civil Engineering Depart-
ment for plates are of four types:
(1) "CSTG" : constant strain triangle, global formulation in which
the elements are triangles, the strain within each element is
assumed to be constant and one global referential system is
used for the whole plate.
(2) "CSTL" : constant strain triangle, local formulation. The
only difference with the "CSTG" is that each element has its
own individual referential system.
(3) "LST": linear strain triangle. The elements are triangles,
the strain within each element is assumed to vary linearly.
(4) "PSR" : plane stress or plane strain rectangle. The elements
are rectangles.
The finite element methods provide a powerful means of anal-
ysis of structure. Any complicated structure can be investigated
without much difficulty. They have, however, their own weaknesses
The input data need to be carefully set up. The results are ob-
tained in the form of numerical answers to the problem and the
influence of different factors are not evidenced. Therefore,
where it is possible, analytical approach will be used.
II Choice of Appropriate Numerical Method
There are two alternatives in the use of the previously-
Jl
mentioned computer programs, namely the constant strain ele-
ment; "CSTG", "CSTL", "PSR" or the linear strain element "LST."
Therefore the computation of the degree of constraint in
the case of a straight slit in a rectangular plate has been
made first, using both the constant strain element and the linear
strain element programs as means of verifying the adequacy of
the methods themselves and a comparison between the two program
types since analytical results are available (equation [3] , for
an infinite plate) . Also in the case of constant strain, two
types of gridwork are used: one relatively coarse, made of a
combination of triangles and rectangles, "CSTG" and "PSR" types,
with 105 nodes. The other is made with a finer gridwork, of
triangles, "CSTG" type, with 170 nodes. The results are pre-
sented in Tables 1 to 6 , the loading being symmetrical with
respect to the center of the slit. The input and output data
are presented in Appendices la and lb for constant strain and
in Appendix 2 for linear strain. The symbols used are:
R - ^
£: loading length
L: length of the slit
v: local transverse displacement (in the directionperpendicular to the slit)
£ : average transverse displacement over the loadinglength Z
K : degree of constraint computed
aK = -^ '
•
32
K : non-dimensionalized degree of constraint computed:
K E: Young Modulus ofK = :^ Elasticity
^'^ L: Length of the slit
K : non-dimensionalized degree of constraint obtainedoo
,
-
analytically m the case of an infinite plate:
K00
K =E/L
The use of non-dimensionalized K (K or K ) makes the resultsc °°
somehow comparable in the form of curves relating K or K^ to R and
are presented in Figure 5. The difference between K^ and K could
not be considered as a measure of the accuracy of the numerical
method because K^ is the non-dimensionalized degree of constraint
in an infinite plate and for which the only significant dimension
is the slit length L of the weld and K is the non-dimensionalized
degree of constraint in a finite plate and for which the signifi-
cant dimensions include the slit length L, the plate length Lq
and the plate width Wq . But we can expect K to be smaller than
K since the restraint due to a finite plate should be less than00 ^
that due to an infinite plate.
The average transverse displacement [v] , is defined as:
1/2
[v] ^ = -^ / vdx
-%/2
for a symmetric weld, where £ is the weld length.
In order to determine [v] . , the local values of v at var-
ious nodes where loads are applied are plotted as a function of
^
(•
(#
33
the abscisse x of the nodes. Then a curve (C) is faired in
through these points and the area limited by the axis Oy
(which is also the axis of symmetry), the ordinate x = •*^/2/
the curve (C) and the axis Ox is measured graphically by a
planimeter
.
f;.
_ Area= / vdx
^—
A,B,C,D are loaded nodes
^D 2
J 1 ,."» ,' .-^
A » ^ . jj
By dividing the value of this area to Z/2, the average
half transverse displacement v/2 is obtained, since the weld
is also symmetrical with respect to Ox and the transverse
displacement is the distance which both sides of the weld
come in to each other.
From the results in Figure 5 these conclusions could
be formulated:
(1) The linear type gives an unusual swing: the curve of K
is above that of K for small R and becomes smaller thanCO
that of K computed by the "CSTG" method for large va-
lues of R.
(2) The constant strain types "CSTG", "PSR" in the two
gridworks give more consistent results. The K curve
obtained is almost parallel to the K^ curve. Also,
there is not much difference between the two gridworks.
^#
(i
34
one coarse and the other fine. '
Therefore, the constant strain element type with a
relatively coarse gridwork can be used in the computation
of the degree of constraint of various joint configurations
in plate structure.
Ill Units Used
1) Length:
(a) inch
(b) millimeter wherever comparison with Japanese data
is desirable.
2) Force:
(a) Pound
(b) Kilogram wherever comparison with Japanese data
is desirable.
m
(#
(#
ELLIPTIC HOLE
Dimension Characteristics:
35
t
1"
P- :>-'--- VA- - - .- -'
»
«s L-z" ;
~ f
w
— c "Length of Plate, Lo=6
Width of Plate, Wo=4"
Plate Thickness, T=0. 2"
v.-
Length of Slit, L=2"
Maximum Width of Slit,W=0.2"
Large Axis a _ ,
^
Small Axis b~
Mechanical Characteristics:
Force Applied = 2 x 10^ lb/in
a = 10^ psi
Young's Modulus, E = 30 x 10^ psi
^ = 15 x 10^ psi/inL
•EH
-p
e(U
w
-Hfd
>-i
-p
-p
ctd
pen
COU
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M fd c cii e o • • • • • • • • •
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CT> •
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00•
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J3-H2^
37
TABLE 2
Values of the Degree of Constraint K and the Non-Dimensionalized
Degree of Constraint K = K Z(p/j\ for Different Values of R = ="
(Constant Strain Element)
R - ^^ - Lv/2
(10-^ in)
K
(10^ psi/in)
K KCO
0.1 1.797 27.824 1.855 2.079
0.2 3.125 16.000 1.067 1.268
0.3 4.192 11.927 0.795 0.975
0.4 5.039 9.923 0.662 0.823
0.5 5.781 8.649 0.577 0.731
0.6 6.315 7.918 0.528 0.673
0.7 6.730 7.429 0.495 0.637
0.8 6.9 04 7.242 0.483 0.617
0.9 6.866 7.282 0.485 0.614
38
U)
(D
X!UC•H 1
CM1OrH
1C •H•H U—
'
G ^Q)
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5h 1n •HP CO
w 1+J
h^ >H Cm (1)
< cu gEH 4:^ OJ
•p
c•H
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BQ) -p
c(d td
rH +J
a CO
03 C•HQ u<D
TJ
S
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II• • • • • • • • m
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4->
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0) iw U -H in 00 r- in CM r- •^
tT>rH fd CN CM cr\ in CM CM r^ 00
fd fd rH \CN r- nH r> CN in 00 r^
U ffi Cu l> 1• • • • • • • • •
0) CO rH ro ^ in in V£» KO KD KO
> •H nH-< Q ^
int^ •
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r^ •
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•
rH• • •
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in•
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a\ n 00 a\ in •^ 00 r-» CN(N •
• • • • •
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rH CM (3^ CM o^ 00 00 in ^ r^CN •
• • •
C7^
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00•
00•
rH•
CM •^ in in MD r^ r- 00
in KD ^ 00 rH r- t^ "^
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iH •
•
rH•
CO•
CO•
CM• •
in• •
00•
r-i n '^1' in VD VC r>- 00 00
CM ro MD rH CO cr> CM inin <^ CM CN r^ a\ 00 H
rH • 00•
CN• • •
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CTv
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<u --CO CO
CO 0)
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jg -H
(%
39
TABLE 4
Values of the Degree of Constraint K and the Non-Dimensionalized
Degree of Constraint K =, ,ir-r- for Different Values of R = —
(Constant Strain Elements - Finer Gridwork)
R - ^^ - L
v/2
(10~^ in)
K
(10^ psi/in)K K
00
0.1 1.725 28.986 1.932 2.079
0.2 3.098 16.139 1.075 1.268
0.3 4.157 12.028 0.802 0.975
0.4 5.000 10.000 0.667 0.823
0.5 5.725 8.734 0.582 0.731
0.6 6.222 8.035 0.536 0.673
0.7 6.577 7.602 0.507 0.637
0.8 6.804 7.349 0.490 0.617
0.9 6.780 7.375 0.492 0.614
^
it
m(X)
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w
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LD • CN CN VOQ) rH • • •
13 '^ VO r^
s 00 (^c» ro CT> ro '^3'
iH • in in 00• • •
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00 <-\ cr> vx)
CN ro r^ r^ inrH • rH cr> r^.H • • • •
CM "^ «x> 00
ID o\ <j\ a\ ChCTi i-\ in 00 inCNJ • ^ r^iH • • • «
CN in VD 00
in in r^ rHrH rH r^ •^
CTi • r-- rH 00 .H• • t •
CN in vxi 00
in CN roro r- CN r~- roiH • \o r-iH t • • •
CN in vo 00
in in ^ "^UD CTi vo
iH • in 00 VD cr>• • • •
CN "vf «^ r^
0) --U) wU) 0)H Si
Node Absc(inc
e»
(#
(»
40
^IhqCM •^ MD 00
II• • • •
o o O o(^
+J
aog ^-^
0) c0) u •H r^ in CN iHen td rH o O inCi rH CM CM *X) 00 m oH iw Pu \ 1
• • • •
(U rH U) |> O (N •^ VD r-> fO -H rH
<C ffi Q '
or^ o"^ •
rH
LO IT)
CM <T>
CM •
oO
in <T>
•^ •
Oo IT)
.H 00CM •
o <y^
COCO
00•
o •
LO "^H r^ no • r--
CM o •
o •
t^UD r^ iHrn •
o •
lO
IT) inUD KO in00 • 00rH o •
in
i*
TABLE 6
41
Values of the Degree of Constraint K and theKNon-Dimensionalized Degree of Constraint K = . , ,
Zfor Different Values of R = —
j-i
(Linear Strain Elements)
R - i^ - Lv/2
(10"^ in)
K
(10^ psi/in)K Koo
0.2 2.617 19.106 1.274 1.268
0.4 4.805 10.406 0.694 0.823
0.6 6.302 7.934 0.529 0.673
0.8 7.051 7.091 0.473 0.617
42
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43
RESULTS
I Presentation of the Results
A. Straight Slit Type - The results are assembled in tables
7 and 8 and in Figure 6 ; the input and output data are in
Appendix 3
.
B. Straight Slit with Circular Holes at the Ends - The re-
sults are in Tables 9 and 10 for the "CSTG" type and in
Tables 11 and 12 for the "LST" type and are assembled in
Figure 7 for both types . The input and output data are in
Appendices 4 for "CSTG" and 5 for "LST." Here again the
two methods, "CSTG" and "LST" are used in order to compare
them again because the weld configuration is different from
the previous cases.
C. H-Slit Type, Steel Plate - The results are in Tables 13
and 14 and ^^n Figure 8. The input and output data, are in
Appendix 6
.
D. Lehigh Test Specimen - The results are in Tables 15 and
16. The input and output data are in Appendices 7a and 7b.
E. H-Slit Type, Aluminum Plate - The results are in Table 17
The input and output data are in Appendix 8
.
STRAIGHT SLIT HOLE
Dimension Characteristics:
44
w.-KO n v>vv\
_ y
t̂
r
1
/ -.boo VV...M -1
•X
I
1
j
1
-> yc^-pi- ./
->
Length of Plate, Lo=300 mm
Width of Plate, Wo=200 mm
Plate Thickness = 5 mm
Length of Slit, L=120 mm
Width of Slit, W=1.5 mm
Mechanical Characteristics:
Force Applied = 5000 Kg/mm
2a = 1000 Kg/mm
Young's Modulus E = 21,10 Kg/mm
E _ 21,100120
= 175.8 Kg/mm -mm
wpq<
(0
u
<D
e•H
•H
co-HPU(D
U•HQI
>H
Q)
Xi-P
•H
W-P
a)
e0)
ufd
w•HQ(U
O
^Ih^ n rg ro VDII \ \ \ \ rH-
« rH iH CM in
1
0) o UD iH 00 in rotn (d ,—
,
CNJ a\ rH o ;0rd M -P g CTl o CTl ro rHM M-i 0^ C g • • • • •
Q) rH tn (U— ro in in VD VD
> fd -H g l>|fN<: ffi o
in• r^
M< o ro"^ VX) •
rH
^• ro
rH LO rH^ in •
ro
<T\ in• VD rvj
cr> o in CMro m •
ro•
VD r-• o ro
ro in in Oro "vT • •
in
^ r-- VD• •^ (Ti 00
iH O r-{ rH VDCO ^ • •
in
•
in
VD ro <y\• I^ r- ro
00 in 00 r^ CNCN CO • •
in•
VD
nH o r>- ro• ^ o ^ o
w <X) o o ""^ CN r^<u CN m • • • •
e ^ in VD VDfd
o o 00 O• <T\ o CN 00
(U o in in 00 VD oTl CM CN • • • •
"^ in VD r^
O rH r- 00 C3^• (N in o CM r^
CX) o CO cy\ r-i cr> ro"H CM • • • • •
m "j^ VD VD r-
CTi ro VD rH H/ • in rg ^ VD <-i
in in r- CM ro rH VDiH rH • • • • •
CO in VD r^
roCM o "* ro• rH CM CM ro 00
00 o o •^ in ro t^iH rH • • • • •
"^ in in t^ r^
in <T\ CN 00 VDVX> ^ •"^ ^ o^
00 •
in •
in•
VD• •
00•
^ in VD r- r^
CM o rH in roCM o CTi CTi •<d<H •
oCM
• •
VD• • •
•^ in VD r- 1 r-»
<D
wCO
HQ) U —^
'
T3 W 6
gS-S
45
TABLE 8
46
Values of the Degree of Constraint K and the
KNon-Dimensionalized Degree of Constraint K =^^i, /t ^
Zfor Different Values of R = —
j-i
%
R - ^^ " L
v/2(mm)
K2
(Kg/mm -mm)K K
CO
1/3 3.926 127.36 0.724 0.943
1/2 5.091 98.21 0.559 0.731
2/3 5.918 84.49 0.481 0.649
5/6 6.305 79.30 0.451 0.615
1 6.133 81.53 0.464 0.637
•
(^
(•
47
)
.^
J Ms «is• oo
II
T5
-=7
P
TJ
-+-
c5
20='
)
o
I'
I5
o
(•
m
STRAIGHT SLIT, CIRCLED ENDS
Dimension Characteristics:
48
~A
•
W. ' ^
J.L
V A
!1'
;r^
1CH ^-—=(:)
'3).^" ^-:^
"^i 1
^ -^ --L. C.
"->1
— TO"Length of Plate, Lo=12
Width of Plate, Wo=8"
Plate Thickness, T=0.1"
Slit Overall Length, L=6"
Loading Length, LL=5"
End Circle Diameter, D=0. 5"
Width of Slit Along^.^^
1"
Loading Length, 32
Mechanical Characteristics
Force Applied
Yo
= 8 X 10 lb/in
4= 8 X 10 psi
Young's Modulus E = 30 x 10 psi
ELL
= 6 X 10 psi/in
(1
(i
#
49
•
C!
•H
1
OH'wO
dw
•H -pP CU oQ) eU 0)
•H iHQ
1
wa^ >H
•Hw fd
^ ^ MCQ -P •p
< COEH cH -P
0) fd4J -pC wOJ Ce o(D uUcd
H04to
•HQ0)
TJOS
•
^^ 1-q CN '^ <D ro oII
• • • • •
« O o O o rH
1 ^Q) Q) Ctn U -H o ^ O CN ,-{
(d 4-1 fd -P CN CN rH ^ cn <X)
h rH rH C ^rg 00 "^ 00 rH CN(U rd a. (Dl> 1
• • • • •
> K w g o o rH rH CN CN< •H nH
Q ^
ro CN(N O 00*JD LO
•
CN
CN•
in inIT) ^^J 00lO •
CNin
•
o rH 00cr> o CTi CO^ •
CM •
rH
00•
rH
IT) in cr»
r^ r^ '^ in•^ •
•
o•
CN
o *X> ^ O^en in vo cn roro • •^ <Ti CN
i-i •
rH•
rH•
CNIf)
in n 00 <ys
B CN CM in 00 00a ro • VD O COs iH • • •
dj
rH CN CN
TJ O o in cr» (T\
o VD o rH 00 o Os CN • CN r^ CN in
r-i • • • •
rH rH CN CN
in 00 r^ CN (N^ r^ ^ 00 O OCN • ro 00 ro VOo • • • •
t-\ rH CN CN
o o rH C^ CN CNVD in in 00 in r- t-«
nH •
o • • •
ro• •
o f-i H CN CN
in <-t CO in r^ r^CN cn 00 o rH r-i
<T» •
o00
• •
o• • •
o rH CN CN CN
00 O O ,-i CMo KD rH ro ^ -^rH •
o00
•
in•
O• • •
o rH CN CM CN
Q) ^w cn
W 0)
•H jc;
(U o u .
^d w c
gS-i
(•
m
50
TABLE 10
Values of the Degree of Constraint K, the
Non-Dimensionalized Degree of Constraint K =
and the Specific Degree of Constraint K' =
KE/LKE
for Different Values of R =
(Constant Strain Elements)
£
H
^= L
0.2
0.4
0.6
0.8
1.0
v/2
(10~^ in)
0.820
1.414
1.840
2.132
J.261
K
(10^ psi/in)
4.878
2.829
2.174
1.876
1.769
K
0.813
0.472
0.362
0.313
0.295
K
1.268
0.823
0.673
0.617
0.636
K'
(in"^)
0.163
0.0943
0.07247
0.0625
0.p590
(t
m
m
Table 11
Node Displacements in the Y-Direction (
•
t
Node Names
1 147 9 164 16 178 24 193 26 201
NodeAbscisse(inches)
0.0 0.125 0.25 0.375 0.50 0.625 0.75 0.875 1.00 1.125i
1
0.489 0.466 0.457 0.401 0.344
0.718 0.697 0.693 0.653 0.631 0.582 0.556 0.502 0.457
0.896 0.874 0.869 0.829 0.807 0.766 0.754 0.730 0.738 0.726 C
1.051 1.028 1.022 0.980 0.957 0.916 0.909 0.894 0.916 0.923 C
1.174 1.151 1.144 1.100 1.076 1.033 1.028 1.017 1.047 1.064 1
(%
m
#
9
51
iiO
-2in) , Linear Strain Elements
AverageHalf
Displace-ment
v/2
(10"^in)
«432 216 39 300 47 315 49 323 55 3381
62
1.25 1.375 1.50 1.625 1.75 1.875 2.00 2.125 2.25 '2.12502 2.503
0.445 0.2
t 0.620 0.4
> 0.730 0.690 0.645 0.777 0.6
5 0.953 0.948 0.954 0.924 0.903 0.836 0.765 0.944 0.8
\ 1.106 1.112 1.131 1.116 1.115 1.076 1.050 0.990 0.941 0.852 0.774 1.070 1.0
(I
m
52
TABLE 12
Values of the Degree of Constraint K, the
Non-Dimensionalized Degree of Constraint K = -
cific Degree of Constrai
(Linear Strain Elements)
K(E/L)
and the Specific Degree of Constraint K' = K/E
#
--iv/2
dO"^ in)
K
(10^ psi/in)K K
CX)
1
K'
(in"^)
0.2 0.445 8.989 1.498 1.268 0.300
0.4 0.620 6.452 1.075 0.823 0.215
0.6 0.777 5.148 0.858 0.673 0.172
0.8 0.944 4.237 0.706 0.617 0.141
1.0 1.070 3.738 0.623 0.636 0.125
53
r-* 4y\ t.>
H-SLIT HOLE - STEEL PLATE
Dimension Characteristics
54
\U^~ c^OO V-.. Uv
v...
a"!
1—
n
JU ^O^Hv-
.V
6 i
LL=!':
r\
u
3) 1 1v..
Vvvvw --W
L. -c: ' ^JC') I'l "I ---^l
L„ ^ ^00 v ^ -5>
Length of Plate, Lo=300 mm
Width of Plate, Wo=200 mm
Plate TL'ickness, T=5 mm
Slit Overall Length, L=140 mm
Loading Length, LL = 120 mm
H-Branch Width, D = 10 mm
H-Branch OverallHeight,
H = 60 mm
Mechanical Characteristics
Force Applied = 50 Kg/mm
=10 Kg/mm'
Young ' s Modulus E =
E ^LL
21,100 Kg/mm
?,1,100 nr- oo T^ / 2! '^— = 175.83 Kg/mm -mm
#
#TABLE 13
-2Displacements m the Y-Direction (10 mm)
55
20
20
22
25
Node Names
26 28 36 38 42 48 57
AverageHalf
Displa-
30 35 40 45 50 55 60
£R=-
cement LL
v/2I
I
(10~^mm) ,
'
)2 5.130i
5.762 1/3
>4 7.650 7.287 6.709 7.839 1/2
'2 9.710 9.402 9.207 8.566 7.909 ' 9.551 2/3
58 1^489 11.208 10.865 10.493 10.033 9.543 8.896 I 11.03 5/6
)1 13.059 12.792 12.481 12.151 11.781 11.423 11.114 10.825 10.610 12.36
TABLE 13
Node Displacements in the Y-Direction (10
Node Names
13 15 20 22 26 28 36
NodeAbscisse(mm)
10 15 20 25 30 35 40
6.040 5.996 5.823 5.592 5.130 I
8.316 8.282 8.141 7.954' 7.650 7.2871 6.7091
10.308 10.276 10.146 9.972 9.710 9.402 9.207 8.566 7.909
12.061 12.030 11.905 '11.738 11.489 11.208 10.865 10.493 10.0331
'• •
13.616 13.585 13.463 13.301' 13.059 12.792 12.481 12.151 11.781;
•
TABLE 14
56
Values of the Degree of Constraint, theKNon-Dimensionalized Degree of Constraint K = . ) .- >
and the Specific Degree of Constraint K' = K/E
for Different Values of R = £/LL
--k v/2
(lO"^ mm)
K2
(Kg/mm -mm)K K«,
K'
(mm"-^)
1/3 5.762 86.78 0.493 0.943 4.164 X 10"^
1/2 7.839 63.78 0.363 0.731 3.023 X lO"-^
2/3 9.551 52.35 0.298 0.649 2.481 X lO"^
5/6 11.03 45.83 0.258 0.615 2.172 X lO""^
1 12.36 40.45 0.230 0.637 1.917 X lO"^
#
57
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~*'T* ;^~"" "^"
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1.—.—
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V*""^
"/v" r% '
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f /^ r>» «»\ 4» ••' (.1
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^ 3 ^ M ^
'•J I J »* '4 «'
i
LEHIGH SPECIMEN
(No Sawcut)
Dimension Characteristics:
58
^'l
_V_
I I
I I
o -> X
.. .1) I — I
T r, •; I '
<
—
>!
Length of Plate, Lo=300 mm Slit Overall Length, L=150mm
Width of Plate, Wo=200 mm Loading Length, LL = 125.2mm
Plate Thickness, T=10 mm End Circle Diamater ,D=12 . 5mm
Slit Width, W = 2mm
Mechanical Characteristics:
Force Applied =10 Kg/mm
=10^ Kg/mm^
Young's Modulus E = 21,100 Kg/mm'
59
ID
w
Eh
+J
13
O
o
c
eHU<U
a,to
Hxi
vA
0)
+»
mo
(0p
(0
0)
* eT3 1
(1) CMU 6:3 g ^
WW \ •^fd en0) «e —
__^
g1 r>-
CN rH
« 1 00\ "^
en;i4'"-'
1
0) 0)
en ud U-l fd 4-* <N -^ 00h rH rH C \ g nQ) rd Oi (U 1 > E o> ffi w g -- •
<3 -H r-\
Q
*x> 00•vf • r-iH CN IT)
rH KO •
O"^
m O -*o VD VDH •
w
OO
0) <y\ O •^
e 00 in 00fd •
oa>>
^ 00 o 00r^ ^ a^
s •
oo^
(N o CT\
LO ro O•
rH
yo0!^ o v~-
,
'^ <N iH•
<Nr- o (NrH H
•
rH
00ro
H O•
rH
-p
(U QJ 1w u —w fd
•H rH WQJ U ^ QJ fli+JT3 (fl 1 T3, w n
S § --O -H QJ
1 S Q g
QJ
CUId
£14
•HQJ cCJ
W UQJ P
fd fd
fd fd
fd
gou
QJ
C•Hfd
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O
u)
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5-1
OWW0)
m M-l
o
>1
QJ
g
OP
QJ 0)
Sh >J3 -Htn enfd
QJ
g«
i
LEHIGH SPECIMEN
(With Saw Cut)
Dimension Characteristics:
60
' 1 li S M.W1
7. Sv^'v
i^— >
I
'
1. n-=. Co \-\\ vv
«-?', : bOv^»"'-*:. i
I.
1 V/lft'uMM ?
""x
r c>!/ ^ , -vie 9-XL) l'^v^
V•>! C X s \1 .SiiM
j
1 1
1 ., 1 cr ^ ,
t
—
Saw cut width = Imm
Saw cut spacement, s = 2 5 mm
Saw cut length determined by x = 60JTim
Other characteristics remain unchanged.
i
61
W
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g
H-SLIT HOLE, ALUMINUM PLATE
Dimension Characteristics
62
-Mr i
A y
—i'
L L r \0-
A
X
J*-
1
Vy,./M
- y
Length of Plate, Lo=20"Width of Plate, Wo =14"
Plate Thickness, T=0. 5"
Slit Overall Length, L=14"
Loading Length, LL = 12"
H-Branch Width, 0=2"H-Branch Overall Height, H=8"
Mechanical Characteristics:
Force Applied = 5 X 10-" lb/in
a = 10 psiJ o
Young's Modulus E = 10 x 10 psi
t
TABLE 17
Result of the H-Slit, Aluminum Plate
1
Node Names
1 9 13 20 24 32 36 43 47 57 66 81
NodeAbscisse(inches)
0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4
NodeDisplace-ment ^
v/2(10~^in)7.34 7.33 7.31 7.26 7.21 7.13 7.05 6.97 6.88 6.80 6.72 6.6(
TABLE 17
Result of the H-Slit, Aluminum Plate
63
Node Names
24
1.6
32
2.0
36
2.4
43
2.8
47
3.2
57
3.6
66
4.0
81
4.4
90
4.8
102
5.0
AverageHalf
Displace-ment
v/2
(lO'^in)
K
(lO^psi/in)
7.21 7.13 7.05 6.97 6.88 6.80 6.72 6.66 6.61 6.58 7.005 7.14
64
II. Discussion Of Results
A. Interpretation of the Results
The valid results for different joint configurations
are assembled in Figure 9.
As expected, the K value for a finite plate is smaller
than that for an infinite plate. The effect of finiteness
is also evidenced by the difference in the K value between
the elliptic slit and the straight slit. In the first case,
if we refer to page 35 we see that the ratio between the
plate length L^ and the slit length L is:
(r^) = I = 3elliptic hole ^
while the same ratio for the straight slit, page 44 is
(Lo.) = 300 _J_ ^3L straight slit "^^^ "'-
*^
Both plates have the same aspect ratio L^/Wo = 3/2. It can
there-ore be concluded that for straight slit weld config-
uration, (there is not much difference between an ellipse
with an axis ratio of a/b = 10 and a straight slit) , the
value of K depends on the ratio between plate length L, and
slit length L. K is larger for larger ratio L^/L and for
straight slits having the same length, the degree of con-
straint K is larger for the one with larger L^/L. This re-
sult conforms to common sense.
65
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66
In the case of a straight slit with circles at the ends,
the diameter of the circle is larger than the slit width, the
value of K is smaller than that for a straight slit without
large-circled ends. And the lowest value of K is obtained
for an H-slit, where the width of the H-branch is larger than
that of the weld slit (D/W = 10/3 = 3.33).
All these results seem to conform to common sense and
could be considered as an indication of the correctness of
the procedure. Theoretically, it cannot be otherwise since
it can be proved that, in the finite element method, if the
compatibility conditions are satisfied, the results will
converge to the true solution as the gridwork becomes finer
and finer. In the cases of constant strain element, the
compatibility conditions are satisfied.
A full verification of the method is made in the case of
the Lehigh specimen (tables 15 and 16) . In the case "without
sawcut", the results are:
2 2K^-.T 1 4-'^ = 48.17 Kg/mm -mm, K , = 44 Kg/mm -mmcalculated ^' ' measured ^'
and the difference is t—- = .*., _ = 7.17%. In the case "with
sawcut", they are;
K__^T^, , ^,^n = 30.69 Kg/mm -mm, K n = 27 Kg/mm -ramcalculated ^' measured ^
and the difference is =7— = ^-, ' ^c^ = 12.02%.
These values of K have been computed by taking the
3 2Young's Modulus E = 21.1 x 10 Kg/mm , which is a little high
f
67
3 2If we take E = 19.9 x 10 Kg/mm , then we will obtain
K' = ^|-^ X 19.9 = 45.3 Kg/mm^-mm
for the plate without cut and
K' = 2^'^^ X 19.9 = 28.9 Kg/mm^-mm,
|t?rtk pUle ^^t^ s^«/cut.
With these values, the differences between the calculated and
measured values of K become:
"""^l 45.3-44 1.3 ^ Q_o= z . o7%K^ 45.3 45.3
and .
2 ^ 28.9-27 ^ 1.9 ^ . _.„^K2 28.94 28.9 o./uo
With these differences, the values of K calculated seem
to be acceptable if we think of the complicated boundary con-
dition in the case of the plate with sawcut as described on
page 60.
B. Significance of the Results '
Coming back to Figure 9, it can be seen that the value
K increases sharply for a value of x = i/L smaller than
about 0.30. The degree of constraint K is related to the
transverse stress a by a = K[v]„. From the empiricalYo Yo ^
Watanabe-Satoh relation, the transverse shrinkage is related
to K by
S^ = S. ^ X^^ 1+0.086 K^-^"^
68
Along the weld line S = v and for a long straight weld,
[v] p - V since v is almost uniform, except near the ends,
then, by substituting the expression of S in the relation
giving o~~ , we have:
^° ~ ^^ ""
1+0.086 K^*^*^
where a is the average reaction stress along the weld due
to constraint.
The value of ~ /S. ^ as a function of K is plotted below,
<^y.
H
K
69
o is an increasing function of K, and could be consi-Yo
dered as an indication of the residual stress level near the
weld.
Therefore, it can be said that if a straight slit in a
large plate is to be welded, and if the weld is long enough
so that more than one block is necessary, the residual stress
would be lower if each weld is made symmetrically and if the
ratio of the weld length £ to the slit length L is larger
than about 1/3,
Also, the Watanabe-Satoh relation can present a problem
of choice: if the transverse shrinkage is to be limited to
small values, then high degree of constraint and consequently
high level of residual stress will result. On the contrary,
if the residual stress is to be small, then low degree of
constraint is necessary (if the degree of constraint can be
controlled) and high values of transverse shrinkage will
result.
On the other hand, welded structures are subjected to
cracking. Mechanical factor is one of the main factors that
promote cracking. The reason might be attributed to high
residual stress associated with high degree of restraint.
Therefore, it may be important to determine a critical value
of the degree of restraint K necessary to produce cracking.
Experimentation is necessary and a series of tests, known as
the Lehigh test specimen (reference 4, page 32) is proposed
70
to determine K . Basically, it consists of several plates
with a straight weld made along a part of its longitudinal
median line. If the plate is not cut, a high degree of con-
straint can be achieved and cracking may occur. To decrease
the degree of constraint of the weld, several sawcuts are
made along the perimeter of the plate, and the degree of con-
straint decreases as the length of the sawcut increases.
Then there should be some value of the sawcut length for
which cracking begins to disappear, the corresponding value
of the degree of constraint is the critical value K which
can be considered as a material constraint for a given weld
type.
Thus w,ith the value of K determined, we can ^predict
whether a weld will crack or not due to its mechanical re-
straint, since the value of K can be computed for a given
weld configuration. One field of application could be in
the patch weld, which consists of welding a circular disc
,.,vS>*->''"•",•,
/f -^
r'
.
ty' ''
' .
71
in a flat plate- If the dimension of the hole is such that
the disc dimension is small compared to the plate dimension
and the weld can be made in one pass, then we have (reference:
Mathematical Theory of Elasticity, by Sokolnikoff , Chapter 5)
:
S_, = —=;— (for an infinite plate)
a_, is tbe uniform radial stress at the weld,
S_, is ttie radial shrinkage along the weld and is
equal to the sum of the radial displacements of the disc and
of the plate at the radial distance, r = R.
Then:
TC = ^ - 1 ^^Sr
" 2 R
In this formula, we find that the degree of constraint
in the case of a complete patch weld is inversely proportional
to R. Therefore, there should be a critical value R of Rc
below which K > K and the weld will crack.c
>
However, this value of K:
K - ^^ " 2R
is an upper limit, for it is exact only for an infinite plate.
For a finite plate, the value of K will be small. The effect
of relative dimension could be seen more clearly by examining
the expression for the radial stress outside the disc, which is
r2^ = a„ • ^r R 2
r
72
At infinite, a =0. If R/r = 1/10,. then a /a^ = 1/100. AtIT IT K
a distance equal to 10 times the radius of the disc, the
radial stress is one hundredth of the stress at the weld.
This value might be still important in some cases.
If the disc is large so that the weld can only be made
sector by sector, and if the weld is made symmetrically with
respect to the disc center, then with an infinite plate, we
have, for a weld sector of angular value 2a, as in the figure:
K(2a) = R
'"r1
n E2aR 2 - f (2a)
X
2a
where f(2a) is a convergent series (it converges as 1/n ) with
f(n) = f (0) = 0, [u-,] ,. is the mean value of the radius dis-
placement taken along the weld angular distance of 2a.
The form of the expression suggests that for K(2a) to be
smaller than a critical value K :
c
K(2a) < K
then for a given radius R, there is a value 2a for which
K(2a ) = Kc c
and for a < a , K>K . Thus, if a large circular disc is jto
73
be welded to a plate and the weld has to be made sector by
sector, symmetrical around the origin, then there is a cri-
tical value 2a of the angular weld sector below which thec
weld might crack and above which the weld would not crack.
III. The Problem Of Experimentation
In the past, experiments have been conducted by actually
welding the specimens and measuring the transverse shrinkage
and the value of the degree of constraint. Little use has
been made of the elastic assumption and of the resulting
linear relations between stress and displacement.
One inconvenience of the measurement of the strain due
to welding is the high temperature at the weld, and strain
gauges had to be put some distance away from the weld. On
the other hand, because the weld line is not well defined
after the weld is made, the displacements had to be measured
also some distance away from the weld. Furthermore, in some
cases, weld specimens are difficult to make, as the case of
a small slit in a big plate, and consequently expensive.
Taking into account the elastic assumption, it can be
noted that for a given average transverse stress, a , at the
weld line, the local displacement and hence the average dis-
placement is inversely proportional to the Young's Modulus E.
Therefore, the degree of constraint, defined as:
aK
/4
is proportional to E, and the ratio of K and E, which has
been defined as specific degree of constraint K':
KK' =
is independent of the material, provided that it is linear
elastic.
Consequently, there could be a simple method of experi-
mentation in the measurement of the degree of constraint K by
having specimens made of material that has a low modulus of
elasticity E and by not actually performing the weld but by
using some mechanical means to simulate the thermal stress.
One such material could be plastic such as cast phenolic
resins for mechanical and chemical purpose whose stress-
strain curves are reproduced in Figure 10 (reference : "Tech-
nical Data on Plastics'/ by Manufacturing Chemists' Associa-
tion, Inc., page 49) and which have a modulus of elasticity
5 5m compression of 3 x 10 psi to 5 x 10 psi (ibidem, page 52)4
To impose stress in the model, wedges can be driven into
the slit. In this case, two similar prismatic wedges with
iill!
Section A-AWedges
Cross-sectionSample
7^
...,.j,i-»'--.'"—'— --*->•- --^ r»V.l.^U.- 1^^ - I ,fl" t.,*t-'', 'l' t'-itt *.
-i-
^
/MECHAinCAL AI.'D CHEMICAL PU^vPOSE
Sniii • S7R*IK CWVE 'iK COHPKESSIOK
SOOO
•ISOO
«000
3S00
r: 3000
ISOO
2000
ISOO -
1000
t $00
/
' *
i^:;';ij
Figo. 10 — stress - Strain Curve of 'Plastic »
\
i^-
76
elongated rectangle triangle cross section made by cutting
a prismatic wedge with rectangular cross section along the
diagonal plane should be used so that the faces in contact
with the slit edges remain parallel. The sample will be in
compression to minimize the friction between the active faces
of the wedges and the slit edges. Some lubricant, such as
Teflon, graphite or oil could be used.
The width of the wedge in contact with the plate repre-
sents the weld length Z. Several sets of wedges with dif-
ferent widths are necessary to establish a curve giving the
specific degree of constraint K' as a function of the ratio
R between weld length £ and slit length L: R = i/L for each
weld configuration.
To measure the stress, sensible strain gauges can be
used along the edges of the slit on both sides. To measure
the transverse displacement along the slit, visible marks
are made along the edges of the slit and their displacements
can be measured by a microscope, or a comparator. After
measurement of local stresses and displacements along the
load length Z, their averages will be computed and the spe-
cific degree of constraint K' obtained by :
77
1 ^% U
For the verification of the Watanabe-Satoh curve, there
is, however, unfortunately no simple way and full-size experi-
ments have to be performed. Given a specimen, we can first
compute its degree of constraint and deduce its transverse
shrinkage using the Watanabe-Satoh curve. Then actual wel-
ding is performed and experimental data obtained are com-
pared with computing results.
If the Watanabe-Satoh relation is correct, then the
knowledge of the value of K is most important in the trans-
verse shrinkage control since these laboratory results seem
to be applicable to actual structures as has been mentioned
in reference 3. In these situations, if the degree of con-
straint can be computed, then the transverse shrinkage can
be determined and hence the level of residual stress is also
known which will enable one to predict whether weld cracking
due to mechanical constraint could happen and subsequently
to take necessary corrective steps if possible.
78
CONCLUSION
The finite element method, with constant strain ele-
ments, seems to give consistent results. A qualitative
verification of its validity remains somehow in the fact
that the results conform to common sense. Two quantitative
verifications have been made and the results can be consi-
dered to be acceptable, in the case of the Lehigh specimens.
Taking into account the elastic assumption, and the
resulting proportionality between the degree of constraint
K and the modulus of elasticity E, a method of experimen-
tation that enables the determination of K experimentally
has been proposed.
The determination of the transverse shrinkage due to
butt weld of a plate structure can be made, if the degree
of constraint is known, by using the Watanabe-Satoh corre-
lation betv:pen relative transverse shrinkage S /S ,^ and de-
gree of constraint K.
The Watanabe-Satoh correlation has been obtained empiri-
cally through experiments performed in three different spe-
cimens: the straight slit, the H-slit and the circular ring.
Some further experiments on various slit types would be
desirable.
If the numerical method proves to be good and the
Watanabe-Satoh correlation to be valid, then shrinkage dis-
79
tortion control is possible. In this case, there are two
alternatives, deduced from the Watanabe-Satoh curve: either
to accept a high degree of constraint and hence a high level
of residual stress and a low value of transverse shrinkage
or a low value of the degree of constraint and hence a low
level of residual stress and a high value of transverse
shrinkage.
On the other hand, welded structures are subjected to
cracking. External constraint could be one major mechanical
factor contributing to weld cracking. For each material and
weld type, there could be a critical degree of constraint, K ,
above which weld cracking may occur and below which it may
not. The Lehigh test specimens are used for this purpose.
Once the critical value K is determined, it will be possible
to know in advance whether a weld is crack-susceptible and
therefore to decide on its feasibility or corrections if
necessary and possible. One example of application can be
found in patch welding.
In both cases, distortion control or weld cracking sus-
ceptibility verification, it is essential to compute the
value of the degree of constraint K of the weld.
80
REFERENCES
(1) H. KIHARA, K. MASUBUCHI , Y. OGURA and Y. MATSUYAMA
.
"Report No. 24 of transportation technical researchinstitute," The Unyu-Gijutsu Kenkyujo Mejiro, Toshimaku,Tokyo, Japan, 1957.
(2) K. MASUBUCHI. "Interpretative Report on Control Dis-tortion and Shrinkage in Welding." A draft being pre-pared for publication from the Welding ResearchCouncil
.
(3) M. WATANABE and K. SATOH. "Effect of welding condi-tions on the shrinkage distortion in welded structure."Welding Journal Research Supplement, pp. 377s - 384s,August 1961.
(4) "Weldment evaluation methods," DMIC Report 244, DefenseMetals Information Center, Battelle Memorial Insti-tute, August 19 68.
(5) D. C. TOLEFSON and L. BRAND. "Introduction to FiniteElement Methods of Structural Analysis," Marine Tech-nology, pp. 331-346, October 1968.
(6) ZIENKIEWICZ. "The Finite Element Method in Structuraland Continuum Mechanics," McGraw-Hill, 1967.
(7) "Strudl Finite Element User's Manual," Civil EngineeringDepartment, Massachusetts Institute of Technology, 1969.
»
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ELEMeM INCICENCES
1 1 10 2
2 1 9 10
3 9 11 10
4 II 19 10
5 11 18 19
6 18 20 19
7 20 28 19
8 20 27 28
9 2 7 2 9 28
10 29 37 28
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26 28 30 21
27 28 37 30
28 37 39 30
29 37 46 3 9 ..'
3C 4 6 49 ^9
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9 32 5 7 6C 49
33 5 7 59 6C
34 57 56 59
35 5 6 58 59
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37 3 L2 13 4
38 13 12 21 221
39 22 21 30 311
4C 31 30 39 40
41 40 3 9 49 50
42 50 49 60 61
43 61 60 68 69
44 60 59 67 68
45 59 58 66 67
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56 6 7 66 7/-. 75
57 6 '
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60 33 32 ^1 42
61 42 41 51 52
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67 11 76 84 85
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70 7 6 le) 16
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•77 72 71 79 80
78 80 79 87 8 8
79 88 87 95 96
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8 9 44 4 3 5 3 54
9C 54 53 64 65
91 65 6 4 72 73
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94 89 88 9 6 97
95 9 7 9 6 104 1C5
96 96 95 103 1C4
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99 9 3 92 ICC 101
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FLEMEM PROPERTIESf
1 TO 316 TYPE •CSTG' THICKNESS .2
37 TO ICl TYPE 'PSR' THICKNESS .2
CCNSTANTSt
.
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STIFFNESS ANALYSIS
LISr CISPLAClMENIS STRESSES ALL
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RESULTANT JOINT DISPLACEMENTS - SUPPORTS
JOINT'
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X OISP. Y OISP. Z nisp
1 G.O -0.0768688J2 CUD -0 . 7^-14 1 463
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RESULTANT JCINT DISPLACEMENTS - SUPPORTS
JOINT / DISPLACEMENT
X IJISP. Y DISP. Z DISP
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RES U L T A N T J n I N f DIS PL ACE MENTS - FR EE J (1 1 N T S
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31
APPENDIX IB
Elliptic slit , 'CSTG' , fine gridwork .
The following presents :
1- Plate division, the numbers are node names , page 32,
2- Input data, pp 33 - 48 .
3- Output data , pp 49 - 75 •
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100 1.3 0.1
lciT.3 C.3
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103 1.3 0.7
'lC4~T. 3 C.9
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106 1.4 0.2
107 l.A C.A
10 8 1.4 0.6
109 1.4 C.R
110 1.4 1
111 1.4 1.4
112 1.4 1.8
113 1.5 C.C S
] lA 1 .S 0.1
115 1.5 C.3
116 1.5 C.5
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13A 2. C C.C S
135 2.0 0.4
136 2.0 C.8
137 2.0 1.2
128 2.0 1.6
139 2.0 2.0
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141 2.2 0.?(
142 2.2 0.6
143 2.2 l.C
144 2.2 l.'^
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14 6 2.4 C.C S
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148 2.4 O.B
1^9 2. A 1.2
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165 3.0 C.2
166 2.0 C.6
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26 62 77 71
2~7~71
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30 76 75 85 1
31 75 84 85
32 A 3 17
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3A 17 30 23
35 23 30 36
36 30 44 36
37 36 44 50
38 A4 57 50
3S 50 57 63
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41 63 71 78
42 71 86 7R
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48 4 18
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46 85 CC 91
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A9 2^ IR.^.6.
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51 2? 3 6 31
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56 58 6 3 72
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58 72 78 R7
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60 87 92 10?
61 92 101 10?
62 <52 91 101
63 91 ICC ICl
64 91 90 100
65 CC 99 100
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68 IR 31 24
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72 45 58 51
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75 6A 72 79
76 72 87 79
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96 73 79 88
97 79 93 88
98 88 93 103
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100 103 108 117
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102 ICP 1C7 1 16
103 107 1 1 9« 116
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10^ 106 114 115
106 106 105 114
107 105 113 114
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109 8 19 25
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111 25 32 38
112 32 46 38
113 38 46 52
114 46 59 52
115 52 ^-9 6 545
116 59 73 65
117 65 73 80
118 73 88 ar
1 1^1
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121 f;4 1C3 109
122 1C9 1C3 117
123 9 8 20
124 8 2 5 20
125 20 25 33
126 25 ^8 33
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128 3 8 52 4 7
129 47 52 60
130 52 65 60
131 60 65 74
• 132 65 8C 74
133 74 80 89
134 80 9A 89
135 89 944
104
136 94 ICS 104
137 104 109 118
138 1C9 117 118
13S IC c> 20
140 10 20 26
141 20 33 26
142 26 33 39
143 33 4 7 3 9
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146 53 60 6 6
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153 1C4 118 lie
154 110 118 124
155 118 123 124
156 118 117 123
157 117 122 123
15R 117 116 122
159 1 16 121 122
160 116 115 121
161 115 120 121
162 115 11^ 120
163 114 119 120
16^^ 1 lA 1 13 119
• ~"165 124 12^ 131
166 123 130 131
167 123 122 13C
16 8 122 121 13C»
169 121 129 130
17C 121 120 129
171 120 119 129
172 119 128 12^
174 11 IC 26
175 11 26 40
176 26 39 40
177 40 39 53
>^iC 53 67
179 53 66 67
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181 67 81 96t{
10? 81 95 964A
TfI S6 95 110
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185 110 124 125
^18f- 125 12A 131
p187 125 131 137
188 131 136 137
18S 131 13C 136
190 130 135 136
191 130 129 135
192 129 134 135
193 129 128 134
194 12 1 I 27
195 11 40 27
196 27 4C 54
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1199 67 96 82
200 8 2 96 HI
201 96 125 111
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203 125 137' 132>
2 04 132 13 7 144
205 137 143 144
2C6 137 13 6 143
207 136 142 143
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210 135 134 141
211 134 140 141
212 13 1? 27
213 13 ,27 41
2 14 2 7 54 41I i
2 15 ^1 5 ^« 6 R45
21^ 5^1 f2 6fl
217 63 F2 97
21 e.c? 8? 111
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220 111 132 126
221 126 132 138
22? 132 lA'^ 138
2 23 138 14 A 150
22^ 1A4 149 150
225 1A4 143 149
22f 1^3 14 F 149
227 143 142 148
228 142 147 148
22C 14 2 lAl 147
230 141 146 147
231 141 140 146
» 232 14 13 28
23 3 13 41 28
234 ?B 41 55
235 41 68 55/
236 55 68 83
237 68 <;7 83
238 83 97 112
239 97 126 112
2A0 112 126 133
24 1 126 138 133
242 133 133 145
2^3 138 15C 145
244 145 150 157
245 150 156 157
246 15C 149 156
247 1^9 155 156 U^
^AR l^i'^ I'fR 15546
?Ji(^^ l^e 154 155
?5C lAP l'<7 15A
?5J 147 153 154
12b? "I47 146 153 -
w —?5 3 U6 1^2 153
254 15 14 28
25 e 15 28 42
25f 2e 55 A2
257 4 2 55 69
2 58 55 83 69
25C 6S 83 98
2ec se 83 112
261 98 112 127
262 112 133 127^
263 127 133 139
264 133 145 1 39
1265 139 145 151
266 145 157 151
267 151 157 163
268 157 162 163
\
269 157 156 162
27C 1^6 161 162
27 1 156 155 161
272 155 160 161
2 73 155 15^ 160
274 1^4 159 160
275 154 153 159
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277 153 152 158
2 7B 163 169 170
279 163 162 169
2 8C 162 168 169 Hk
? P 1 \( 2 K I 1. (-. P
?R? K.l 17^7 16n
47
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286 15S 165 166
28 7 \^<; leF 16 5
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FLf:MENT PPGPERTTFS
1 Tf 287 TYPE 'CSTG' THICKNFSS 0.20
CONSTANTS
E 30000. ALL
PCISSCN .3 ALL
JOINT RELEASES
1 TC 15 FORCE Y
75 8A <5C 99 105 113 119 128 134 140 146 152 158 164 FORCE X
LOADING 'ONE' •X=C.1C'
jri M LCACS
1 16 FCRCE Y -10.
LCACING 'TWO' •X=0.20'
JOINT LCAOS
1 2 1 FORCE «Y -10
16 FCRCF Y -20.
LOAOING 'THREE* 'X=0.30'
J CI NT LCACS
1 29 FCRCE Y -10.
16 21 FORCE Y -20.
LCACING 'FCUR* »X=0.40'
JOINT LOADS
1 2A FORCE Y -10.
16 21 29 FORCE Y -20.
LOADING 'FIVE' 'X=0.50'
jnj NT LEADS t» /
1 ^i!^ ropcr Y -1 c.A8
16 ?l 2 9 3^t FHRCF Y -20.
LGAQINr, 'SIX' • X = 0.60'
JOINT LO/^ns
1 ^R FCRCF Y -10.
16 2 1 2^ 3^ ^3 FPRCE Y -20,
l.CACING 'SFVEN' »X=0.70'
JGI M Lr/>,ns
I ?:6 FORCF Y - 10.
16 21 29 34 43 48 FORCE Y -20.
LOAOING 'E IGHT« • X=O.BO«
jniNT LCAOS
1 ^1 FCPCP Y -10.
\e 2 1 2 9 34 4 3 4 8 5 6 FORCE Y -20,
LOACING • N INF* •X=0.90
•
JOINT LOADS
1 70 FORCE Y -10.
16 21 29 34 43 48 56 61 FORCE Y -20.
STI FFNFSS ANAL YSIS
LIST DISPLACEMENTS STRESSES ALL
A9
LHAOING - CNR X=0. 10
RESDITANT jniNT IS PL /^C FM ENTS - SUPPORTS
JCINT / DI SPLACEMENT
> dTsp. Y DISF. Z CISP.
1 0.0 -0.0 18 52^02 CO -0.01666123 CC -cci^eefc4 0_._9 =0 4.Q_L33543.
5 CC -0.01227916 0_.0 -Q^(ILL4_7J6.7 n.n -0.01074849 CC -C0C97C98B
10
0.0CO
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11 o.cQ_,0
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1314
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15
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64 0.0033174 0.0.iC : Ci^XJii5.2_ (I^JI.
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1C50.00287230.0027397
CO0.0
w 1131 19
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cca^o
128i34
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0.002162C, 0021160
0.0CO
152 0.0020 134 0.0.Li^ 0_^^JD191^6^ Q^^.164 . cod 185 12 CO
RESLLTANT JC I M I SPI. ACEM EN T S - FREE JCI MS
JOINT / CISPL /SCFMENT
XOISP. YofSP.~
7CISP.
16 C.CC12313 -C016247611 -0.._0J)02.3 6^ rQjL0 1_3_7_12_8.
18
19-C.00C2592-0 .000 2 3 30
-0.0118305-COLC4 82 5
2C21
-0.0001980C0C23128
-0.0095192-C0128208 /
~ 23.0005709
-0.0003089-0.0 1277A6-0.0117733
2 4
23-6.0004105-0 .0003942
-0.01057^5-CCC^614C
2 6 -0.0003 54''. -0.008881727 -CCCC3685 -C.CC£C599
m
?^^ -0 .0005''»'33 -C.00760<^62f ^0_._002 7Q7 7 rQ.JlL0_ai.33 503'^ C.0C0;^573 -C.0lCa33A31 -Q_.l)0.Q4Ql?. rQj.lL0_?J.B3"?? -C.0C0S2'+3 -0.009A6^033 -O.CC051^ 8 -COOaS C A A3^' 0.0029916 -0.009?6Aa3^ C_._0_CJ5^7L -Ct_Q.C9_27iiC.3^ o.oooo?7A -o..oc<:?].5c37 ^-lO_._0_COA84 9 z0j^00_8_963 7.
38 -C.000601.7 -C.CC8A94533 -0.00 CSa I? -O.OOfiCR^A
%^C -0.0006377 -0.0077892-^1 .T_C_._0^3aa2.^ =Q^a0_73i)i>9^? -O.C01A450 -0.0C6969Cit! 0_.303iIB.L -Q,_007iK)2 3.^-^ 0.OCO9117 -C,CC80G75
•0.000 15a'^ -0.0080513^6 -0.0005451 -0.0078908
-ifJ -_0_._PI)06 4 54 = C^0.C_73l?A9^^5 0.0033389 -0.0066778'
.J}5 Q_.3_019 93.0 = Q^0p_6J_C8 950 C.C004644 -0.006860^"
_3J -O.ooo?q37 -0.0069919^2 -C.000574C -C.CC69S84
_5.3 ^^0_.000 69 79_ r Q i0.C_6_9J3J5A -0.0009155 -0.0067190"
-5 5 .-r_C_._C0i44 5I -Q^0_C_63.J0i5<^ 0.0034469 -0.0054150-33 0.0012530 -0.005585858 0.0001838 -0.0058727-3i -_C_._C-C034 5.1 -Q^00_6J311_^0 -0.0005927 -0.0062A71-tl. 0_, 03351(11 =0^00_4J394_62 0.0021088 -0.0042466A3_ .00 78 6 5 -0.GC46314^^ C.000C456 -0.0050971^^- n_C_,i)^32Zl] -£._a.03A333_66 -0.0006000 -0.00562326 7 rJI,300SL64_ =Q.JKL5J-26i:_68 -0.0013237 -0.0057325^^ - 0.0024363 -0.00 5582 9
%
•
70 0.0034500 -C. 0025663-71 D_,331^59.<l r:Q^'U13J'333"^2 0.0005619 -0.0039651-^2. -_C333Q2 2_8_ -Q^a_CAA_5^ff^ -0.0003972 -0.0048740J-t .0029579 -0 .00 1 1066"'^ 0.0021931 -0.0017626.3S C^CDJJJQQ. -iD^Q a^lIPj6_7^ 0.0004382 -0.0034952
..8£ -1^CD33 6 8Z -X^aaA.CJL3if81 -0.0004387 -0.0C44C83^^ -0.0010615 -0.004731983 -0.0C19478 -0.004778^-^-5 ^0.^11)2 970.1 -D.^a-X1Ail8J26 0.0018473 -0.001595733 Q.._aD33331 r 0^ 0C2_A_pJ3.P? 0. 0003480 -0.0031A47""^ -^- 0001 20 1 -C.CC36283*^1 0.0024698 -0.0CC70A232 .0 ._0_0J33 2 5 ^ o_, 00 L5 /t3J"^3 0.0008323 -0.0022P573Ji C.._0_0_0^332 -3^0Ci86J33^55 -0.0002042 -0.0C3265196 -0.0006102 -0.0035424
r»
o
('•
11
-I'ii z0_^^_02i^iz'i zQ^'iQjineiyjICO 0.0077166 -0.0002'^69 51—̂ ^^J- Q-._C_C2.C59L _-0^aC0a_^.i)2^'^^ 0.001343A -0.0015100—1-^^^ C.C0069]? -0.002 I??9iO'* 0.0001/-I79 -0.CQ26CT6
~ ~
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^*^^ C.CC18C?? -C.0CC92O9-J- -2 3- lO_.^5Ji401 = Q*_aC_LA6J21^"^ C.CC05501 -0.0019706
-fl?"'^00 ^1^3 -C.0C??R6?
^11 -0.0009394 -0.002P254""
--11^ --0_,_C020 566 zQ^a'^3_0^06l'"^ 0.0025036 -0.000156"l
-:fff^^-.-O-DiliiZ. -C^00032^b
1^^ 0.0015465 -G.00C9527iP >00Q9500 -0.0013R76^^^ C.CCC3927 -C.CC1R040
'
120 CL.002287L = C ^Q C_C2_7J^^1 C. 0018658 -0.0005778
"'
J22 ^?^._0j0i2i5i -C^(LQ09_^a3J1^^ 0.0007722 -0.0013000
-H-7C«0 CO?oni -0.nm6C4 R
125 -0.0C03I57 -0.001647?"" ~
-fJ5---0-^-0^l'±Q.YL ra^0Jl?J_4Ri?
127 -0.0027045 -0.0C22467.I2-9_ 0_.0^22Ii2. = Q^0_Q_QJJ;4_51-'" 0.0014250 -0.0005933J^J C.CCG3512 -r.nnir797132 -0.0007557 -0.00 13640
~
m
fr-2. _-_0_._0Xig44 5_ -C^(ICL5JJ^135 0.001R4R7 -0.0001R73J^-^ a^-DDD^lQZ. = Q*_Q0_Q43JI1^"^ -0.00011B6 -C.00C7C91L15 -0.001?696 -0.000P5R7139 -0.0024551 -C.CCC9387Y±L J:>^DQ21^^ -£^ia_ar}_L3j6^^2 0.0014535 -0.0000849141 .C^Di:^49L^ -C^(LC_C25Ji.1^* -0.0006319 -0.0002999
—ij^ -C.^017iIL4 -0. 00036451^7
,0.00 17588 C.0CCC961
L48_ 0_.,(}JDia(I4^9_ a^QJlIU^3I^'•'^ -C.C00C333 C.C0C145R"L5-CL -zD_,DDllt9.L Q^Q_C_CJ3i:31-1 -C.CC22605 0.0001091
f^T— Q.O(:]W Cf. c.oon26n15'^ 0.0013933 0.0003428^:r a._c_oi:3i:22 X)..oac49i)71-^ -0.0005853 C.C0C5698151 -Q_._0_0J6994 D^aia5_9_6J^^"^ 0.0016514 0.0004093^-^ 0.0^094R2 C.CQ C7^<;21^1 -0.0000412 0.0009532
-|f-^^T.a_^o_ojj4Qa i3^caL(L5_6i
1^3 -0.0022087 0.0010752^-^^ Q-._o_ojj76a D^aa.Q32i>9}^-^ 0.0012805 0.0CC8889'
-fH Q-.00a4390 0.00129601^^ -C.C006034 0.0015169i69 -Q_i00J.6 778 O.00L5_8__2J_^''^ -0.0022046 0V0015865
LOADING - TWr X=0.20
t
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_Rf_SJJi-J P^ N L _.vLLI_NT._ L S (ILA CJ- MX- _N I S. _^ _SUJP.PQEL^i.
JCINT / niSPLAC^MFNT-
_)L-DJi^!_ Y_D_LSP_, l_DLSP_.
J C_,_0 = 0. ,.0.3^63 4 3-
2 CO -0. 0307897J 0,lO -0 .0279135^ 0.0 -0.02591445 0_.3 r:Q*_a2_35JtJJ.
ft 0.0 -0.02250537 C_._0 -Q,u.Q2UJ^32.9
8
0.00.0
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12
J 3
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0.0-0.016587C-QiL016JJ0
lA
15 0.0-C. 01551 12-0.0152098 ,,
It84
0.0075018C. 00^6090
0.0 ^
C.d9099
0.00620310^0C5 7L73
0.00.Q_
10 5
11 3
C.C0545060.0051573
O.C0.0
119 0.00-^9893 C.C.L^8. 0_.05^6 7 5.1 Q^Q_134 0.0045515 0.0L4jl CUDD429L1 _C^Q_146 0.0042060, 0.0
'tt' 152 0.0040020 CO^ 158 .0039271 0.0_144. ^ 0_, 003 6 8 01 0^0
RESULTANT J^T^^T C I S PL /^ CFM FNT S - FREF JOINTS
.IdUU A-_-_----=ri---_-_-_---rQLS.PL.AXJV£NI-"_-_"-r=--"_-_---Z.
1
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0.0022795 -0.031204717
la19
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-0.0004205:lC..J1O_03 74QC. CO 42 5 20
0.0012127
-0.0266604-.D.^QZiL971_-0.0206582-r^O L8_8JJ_1
-0.C26SC92-0.025R665
72 -C. 0004228 -C. 0232197.2 4 -cll0J)0J22I =Q^<izrjLej3.25 -0. 0007384 -0.0190020.2t -Q^^QOOMl^ -Qj>.1U3JQ2_27 -0.0007234 -0.01597292fi -C. 0010775 -0.015086329 0.0051342 -0.02221313_C Q.tOOO.510 3 j:0^0 2L8_2J_2_
31 -C.C0C6917 -C.02C311432 -a^0_0^974l ^Q^.aLa_7A53.22 -C. 0009854 -C. 017439334 0.0058749 -0.0187645
o
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.^Z ^-_C,J)_02B5 61 '1^(1)33221^^ 0.0063057 -0.C1596A4.^^ O_.j0517 515_ -Q^Q_L6i353^5 -C.0CG32C0 -O. 0161272iti? -0.00 1051 A -O.C157Cf <5
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LOAOING - NINE • X=G.90
rf^sultant jcim displacements - suppcrts
jITkt / dTsplacfmfnt
X OlSP. Y DISP. Z OISP.
1 0.0 -0.0841545.2 C_.X -Q^Q_a_2_4_3J.^.
3 0.0 -0.07^7706A Ljl^ -0.07752RP5 0.0 -0.0746*502.6 0,0 -<l<tOJJ233t.7 CO -0.0695996.3. O^D -Q.^(lh3Ji233-8 CO -0.067458010 Q^^ -0.063548111 0.0 -0.0609992.12 a,_C -Q^0_5JL3_i)_Ci3.
13 0.0 -0.0569868.14 CD -a^033J)31Q.15 CO -0.0540548-15 0. 3 270 5 0_J384 « 0.0258738 0.0
.9-fl D_.n2393.^1 Q^Q_.99 0.0217545 0.0-1C5_ Ci32(3515.] Q^CL.113 0.0193256 0.0_L19 0.01861 35 0.0123 0.C17350S 0.0
.124. ^Q_^Ji»j35^8 Q^Q_.1^0 0.01 58906 0.0.146 (IJ113J;5Q4 D^O.152 0.0147997 0.0_L1Q 0.014535^ CO164 0.0 136 3 20 0.0
RESULTANT JflM DISPLACEMENTS - FREE JOINTS
tJOINT / DISPLACEMENT
X diIp." Y~c7rsp. 7~dTsp.~
16 0.0037537 -0.083636517 0.0016388 -0.0791713
Ig C.OnO^v)!^ -0.07 39101._ia z.Q^.oM^^iZid =(:^a6.fLP.x5J3.
ZC -0.00072?4 -0.06^^t?:i7
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2 ''^ - C . ^^ 7 I 7 - C . C 6 9 6 5 3 6
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2^ -O.OC16200 -0.061195A21 -_(L.J:X'2255.T rQ^ai633J.f.28 -0.0036A79 -0.05370B629 0.QK972 P - 0.079211
73
30 0.00A5612 -0.07^676711 (L,j:ir2)21U±. r:Q^a^9_5Ji?2_32 -0.0012527 -0.C6A816231 zO_.JD2l'±1^^ =Q.._a.60_9_5i,9.
3 4 0.01453 82 -0.07 51963_5 C.009AA93 - C. 07 2163336 0.0029072 -0.0681629.21 ^C_,_OJ:j36Z9 2. = £*.Q.6_3_4_9i5.
38 -0.0022802 -0.0597195J_c ^T_0_»J3X2 9S0a -Q^Q_5_6_8_93I3.
4041
-0.0036745-0.0053095
-0. 05495^8-0.05 18048
424 3
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44 0.0066437 -0.0653920.4 5 0^i)X:5£6i5. -£^a6_CJ'_?J6.46 -0.0018942 -0.05732254 7 -0.0031283 -0.054744348 0.0211788 -0.0632734AS. D_,0J.Z44 6] -1 ^1.(12:2)3^)3.
50 0.0034708 -0.0565437.5.L r^O^Di^DaiLL -Q.^(1322AAJ:.52 -0.0028065 -0.051545453 -0.00 3933C -C. C49960154 -0.0059455 -0.04812275^ -JL,DI?9fiii.f_ -O.JdAJz33^fi.56 0.0238381 -0.054597152 0_,00 74 76_7_ =aiJ350_23i 8.
58 0.0010890 -0.048061659 -0.00 1 P681 -0.04710056C -0.0034523 -0.0463518-^1 0_,Il2i!nZLl -CJ1.^33J32.62 0.0132820 -0.0413265_tl Q_,2>2^Q.^'U. -a^(LkOA2AZ.64 -0.0001226 -0.0411661Jl3 -n. 0024541 -0.04 1682466 -C. 0038333 -0.0419017.67 -(L^233I^2 -2.^L'lZ032)3_68 -C.009C598 -0.0414922.3^ ^1JI1_6^504 9 -X^_04a3.Ll_4_7C 0.0261355 -0.0267869-U 0.00 8 3 094 -0.030235072 0.0026188 -0.0335922J2 -C_._C_0D6 83 9 rQAai5JL00i_74 -0.0028130 -C.C372345J 6 ^a..Jl2i8 8 5<i rQjL0LQ_6„5_92_77 0.0146449 -0.016374778 .0066253 -0.024 354 8
79 0.0019827 -0.02879803Q -L._Q0C5^4 7 r3j-.G1lJilJ2.81 -0.0032259 -0.033490582 - 0^0 OJ 41 24 jtO.. 0.14 9_5 8J?.
8 3 -0.013569 3 -C.C34 94 6785 0.0223667 -0.0042259
1
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PF 0.00l'>6 8? -0.02 534 059 ^0_._0.0_1321.C = 5..0^.PJ'JJi,
91 0.0179783 -C.CC5R76192 C. 01 C3605 -0.01 2930293 0.0049294 -0.C1P699894 ^O.OOJil6.2_ zO^QZ?J\J_^S95 -a.b6l8'241 -0.C25252496 ^HD_.00 4 5 50(I = Qi.02_6_9_2ft 7
97 -0.0104884 -0.0285654 A98 -0.01954 74 -0. 0285846ICC 0.0203286 -0.0018647ICL ^0_^_CJ49il2_ -Q^(I0_6_8_54^.102 0.0088730 -0.0122252iC3 0_. 004156 8. -0 i.0_l7033 6.
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C.0CC49230.0176457
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0.0 126 82
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111112
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114115
0.01855820.0154005
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116117
0.0108048.0063129
-0.0076C59_ -0.ail0322
118120
0.0023419C. 0167982
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123 0.005154C -C.01C3C69.L^4L 0_.001Q5_5.5 =Q*.Q_LZ_6J79.12'= -0.0025718 -0.0144C80126 -0 .0102001 -0.0165268127 -0.0194596 -0.0171260.1Z9. 0_.jDJ^2^1Q_ -C^G_(ULZ_8J3_
130 0.0101389 -0.004750611] 0_^_D_02Z42_8_ ::a^0JL83308.132 -0.0C56457 -0.0107605133 -0.0141324 -0.0 118100135 0.0134652 -0.0015402
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139 -0.0179729 -0.0075351141 0.P149964 -0.0001613142 0.0105553 -C. 0008599.1^1 (1.^S>3^^2lZ -^_^aaL9_13_3_.
144 -0.0C473C0 -0.0027625.145. -Q_J)JL3J631 -i)^0C_I2-^J_.147 0.0128752 0.0005790148 0.00 72 83 6 C.CC C7689149 -0.0003411 0.CCC6468
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76
APPENDIX 2
Elliptic slit , 'LSI' type „
The following presents :
1- Plate division , the numlDers are node names ,page 77
2- Input data, pp 78 - 97 .
3 -Output data , pp 98 - 121
77
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Straight slit , 'CSTG' type .
Ihe following presents :
1- Plate division , numbe'rs are element names and node names .jpage 123
2- Input data, pp 1 24 - 131 .
3- Output data, pp 1 32 - 139 .
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-0.3-5448/0-0.3319695
f 0.05093130.0808291
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0.0381340-D. "2 6 8" 519^5"
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26Z728
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0.0864744-"0.7Z39 7TJE
-0.2517479
A0. 06 8 B 4930.C919099
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-0.2221598
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3 233
34
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-0. 2083/61• -0.2098475
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-0.17 AC 7?
9
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-0.0533026-DTU8"05B~3"8~-0.1160896
133
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48
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0.0953897"C"
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.0113002
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LOADING - TWO R=0.50
RESULTANT JOINT DISPLACEMENTS - SUPPORTS
JOINT •DISPLACEMENT-
X DISP. Y DISP. Z DISP
1
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5
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0. 1506339-0.
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-0.1409023
1
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-0.1091548-0.12 32 304
4445
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-0.0619599-0.0364460
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5354
565860
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f
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0.0818811
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-0.0 /^96 42 f
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3
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-0.0081999T).'nu^rc"5Tr
0.02033086870
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0.04C42380.02194960.0499531
/ I
7 2
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X).07Z^^5r0.0738095
LUADING HRbb K=0.66
r:esijut7\"nt^ JO r(srr^m"SP t?vcETiENrs~ -"sriTPTimTT; ^\
c«
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X DTSPT
DISPLACEMffNT
Y'"DryF.'135
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0.0
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6
4 6
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0. 1379033Cr.TTr4^^D"0.U54167
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0.0"070
0.0
RESULTANT JOINT DISPLACEMENTS FREE JOINTS
JOINT -DISPLACEMENT /
X DISP. Y DISP. Z DISP.
8T^—
10"TT"
12
1415'
16
0.02 62 73 7
"0^."in."D7030""
0.0310741(J.C2205810.0080292^a".Tr5D7T54^'
0.0506972""0"."D74^2U8"
0.C694405
-0.6642418-D7&5T^"9^t"3-
-0.643429 8
-0.6334 95 2
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-0.6337957-D76"T41S(D"7,-3"
-0.6124594"T7~18
~T^~20
22~7T-24
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26~Z7""28
30"31-
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44"45^
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0.0703990
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0.0253100"•0.0203677~Cr."03r8mT5""
0.1420336"Or."n.73B3r0.1626024
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0.1367357"C".T8in"6D2"0.1453409
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-0.3726331"^"D733"5ar95""-0.3018763
•
0.09 5^0 520.0301653•a."tr4irr3"67"
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0.1441357"0.09^52^7"0.1802178~ari81H09 8'
0.1661752
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-0.2296998^'7rB9T973X-0.1465153-T)716 52556""
-0.0829486-070487640"-0.0398991
4 9
50^152
54"TIB"
58"6T)-
61
64
0.14570310.1517542a.l?57B72C.07594?5"0.T0942470.09R0603
-0.0681182-0.0246995-0~.053r6'B2'
-0.10437420^17 26 443^
0.2180954"0.14681590.1311033"0. 1113^504^
0.0267071
0.0541130
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-0.0256419-r);aTiT"5T-0.1037567-"D";T297^0-0.0228660
136
66"hi68
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71
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-0.1557252"G-;"D"51"61^'B8'
-C.C822977C.15427B5
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0.0265541"0706^^271:2"
0.09171420.093161CrT7~
LOADING - FOUR R=C.83
RESULTANT JOINT DISPLACEMENTS - SUPPORTS
JOINT DISPLACEMENT-
X DISP. Y OISP. Z DISP
1
3^"
5^5~
7
47
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0.0"Cr.TT"
0.0
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0.2171218"C^.^^153570.1730714
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55'
5 7
0.007a0.0
0.162/6/60.1537723ir.1"37)lD64'"
TTTO"0.0TI70r"
5T6369
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1 1
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0.0959952
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137
?-4 ~ ~ -a.0T9?5 58"2 5 -0.0 6 64 8 79
- 0.544-848^-0.4817865
26 0.152932827 C.1321822
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2 8 0.17 54 35329 0.1469604^^ ^_^^.^g^^_31 0.1959851
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•"'- -"0.52Z6'667-0.5197136
32 C. 1583 1653 3 0.2142721
35 0.1090916
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3 7a4~75T)
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37 -0.05508263 8 -U. 159851739 0.2291063
^tT) Or."r6"67737~
41 0.2333664
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43 0.1097060-D.Tff&4^Z155~-0.2091449
44 0.214942545 0.2170081
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51 0.1442784 -0.06540705 2 0.08^^1^53 0.0088350
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54 -0".~n."4XI"997
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41
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67 -C. 0124751
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72 -0.1815396
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0.1080755
LLI^UING - FlVt R=1.0
FrEST;iT7\T^T jan^rDi^PEa-CTR-TrNTs -^
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JUINI / DISPLACI^MENT /
X DISP. 7 [JTSV. "TDISP.
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ikRESULTANT JOINT DISPLACEMENTS - FREE JOINTS
1JOINT / tTTQDiArP MP M T — // UlorLALt"uNi
X OISP. Y DISP. Z OISP.
a C. 0288491 -0.7896380"9
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120.02633490. 01 2011
6
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140.05595380.0576967
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1516
0.08252820.0795026 -0.7399804
1 r
18
20
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7T22
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26
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1
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2930
3233
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C. 15510040.1073206
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140
APPENDIX 4
Straight slit with circled ends , 'CSTG' type .
The following presents |:
1- Plate division , the numbers are node names,page 1 41 ,
2- Input data, pp 1 42 - 1 52 .
3- Output data, pp 1 53 - 163 •
i
1
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141
o
n
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1
1
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c T M r M n- c
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20 1.00 1.50
2^ I. CO ?.5
30 1.00 3.50
31 1,00 4.00
3? 1.25 0.03125
33 1.25 0.25
39 1.50 0.03125
^0 1.50 0.50
41 1.50 1.00
42 1.50 2.00
43 1.50 2.50
44 1.50 3.00
45 1.50 3.50
46 1.50 4.00
47 1.75 0.03125
48 1.75 0.25
49 2.00 0.03125
1 5C 2.00 0.50
51 2.00 1.50
52 2.00 2.50
53 2.00 3.50 /
54 2.00 4.00
55 2.25 0.03125
56 2.25 0.25
62 2.503\ 0.03125
63 2.50 0.25
64 2.50 0.50
65 2.50 1.00
66 2.50 2.00
i.67 2.50 2.50
6fl 2.50 3.00
69 2.50 3.50
70 2. SO 4.0'':^
14471 2.552 0.15
7 2 2.640 0.2 24
73 2.62 5 0.375
74 2.75 0,25
75 2.75 0.50
76 2.75 0.75
77 2.87 0.22
78 2.875 0.375
79 2.97 0.12
80 3.00 0.00 S
81 3.00 0.25
82 3.00 0.50
83
1
3.00 1.00
84 3.00 1.50
85
i
3.00 2.50
86t
3.00 3.50
87 3.00 4.00
^ 88
1
3.25 0.25
89 3.25 0.75
95 3.50 O.CO S
/
96 3.50 0.50
97 3.50 1.00
981
3.50 2.00
99 3.50 2.50
1001
3.50 3.00
101 3.50 3.50
102 3.50 4.00
103 4.00 0.50
105 4.00 1.50
106 4.00 2.50
107 4.00 3.50
108 4.00 4.00 u-'
114 4. so 0.00 S145
115 4.50 0.50
117 4.50 1.00
118 4.50 2.00
119 4.50 ?.50
120 4.50 3.00
121 4.50 3.50
122 4.50 4.00
123 5.00 0.50
125 5.00 1.50
126 5.00 2.50
127 5.00 3.50
128 5.00 4.00
129 5.50 0.00 S
130 5.50 C.50
132 5.50 1.00
133 5.50 2.00
1- 5.50 3.00
135 5.50 4.00
136 6.00 0.00 S
137 6. CO 0.50
139 6.00 l.CO
140 6.00 1.50
141 6.00 2.50
142 6. no 3.50
143 6.00 4.00
ELEMENT INCIDENCES
1 2 1 10
2 1 9 10
3 10 9 16
^ 4 10 16 2 5
5 16 24 25
2 5 24 26 xUS
f
2 5 2^ 3 3
8 76 32 33-146
cj 3^ 3? 39
10 33 39 40
11 3^ 4,7 4 8
12"'
40~~4f
'
49
13 4R 49 56
14 4 9 55 56
15 56 55 62
16 56 62 63
17 63 62 71
18 63 71 72
19 63 72 73
20 73 72 74
21 73 74 75
22 75 74 78
23 74 77 78
24 78 77 81
25 77 79 81
26 81 79 88
27 79 80 88*
28 80 95 88
29 3 2 10
1
30 3 10 17
31 10 25 17
32 17 25 27
33 25 33 2 7
34 27 33 40
I
35 33 48 40
)
36 40 48 50
37 48 56 50
,
38 50 56 64
39 56 63 64 \\} ^-o
40 64 6 3 7 3
Al 64 73 75147
^.^."7 5~ "78 82
43 78 01 82
^44 82 81 88
45 82 8 8 Q6
46 88 95 96
47 4 3 18
48 3 17 18
49 18 17 27
50 18 27 41
51 27 40 41
52 41 40 50
53 41 50 65
54 50 64 65
55 65 64 76
56 64 75 76
57 76 75 82
58 65 76 83
59 76 82 83
60 83 82 89/
61 82 96 89
62 83 89 97
63 89 96 97
64 97 96 103
65 96 95 103
66 97 103 117
67 95 114 103
68 103 115 117
69 103 114 115,
70 117 115 123
71 115 114 123
72 117 123 132V^
-)
73 11.4 129 123i48
74 123 130 132
~75 T23 12^ 130
Y7~Y3o"i?"9 "iYt
P 78"~n2^l3 7'^T3'9
79"l20 136 137
SO 5 4'~l'a
81 5 r8"~Y8'
82 Tp 41 28
_________.^^
e^~Wi 65 "5I__ _^ __ __
86r~'6 5 83 84
87~~"84 83 97
8? 84 97 105
89^" 97" 117 "105
1" 90""lo"5~117T25
gl 117 132 125"
I
_^
92 125 132 140
93"T3 2~ 139 1^0
94 6 5 19
95 5 2 8 19
96~T9 2 8 ~T9I
97 6 19 20
98 "20 r9 29
q<f~Y9 28'~"42
Too 29 42 43
101 43~"42 ~T2
f02""'28 5r"Y2"
Y^ ^^ ^ 5^
104'~^'52~~5r~"66
- ro5"""52^ ~66 ~~67 '^(^~'
106 6 7 66 RS I4y
107 51 84 66
foT"66 ~"b^ 85
109 85 84 98
110 8 5 98 99
111 99 98 106
112 84 105 98
113 98 105 106
llA 106 105 118
115 106 lie 119
116 119 118 126
117 10 5 125 118
118 110 125 126
1191
126 125 133
120 126 133 141
1211
125 140 133
122 133 140 141
^ 1231
7 6 21
124>
7 21 22
125 6 20 21
1261
22 21 30(
127 21 20 29
1281
21 29 30
1291
30 29 44
130)
30 44 45
131 29 43 44
1321
45 44 53
133 44 43 52
134 44 52 53
135i
53 52 68
1361
53 68 69
137}
52 67 68
138 69 68 86 .HA
f
13^^ 68 H^) 86
lAO 68 67 R5150
1^1 86 85 100
1A2 86 100 101
1^3 8 5 99 100
lAA 101 100 107
1^5 100 99 106
1^6 100 106 107
M7 107 106 120
1-^8 107 120 121
1A9 106 119 120
150 121 120 127
151 120 119 126
152 120 126 127
153 127 126 134
15A 127 134 142
155 126 141 134
k 1561
13A 141 142
1571
8 7 23
1581 _
7 22 23
159 23 22 30< /
1601
23 30 31
161
1
31 30 46
162
1
30 45 46
163i
A6 45 53
16A>
46 53 54
165 54 53 70
166 53 69 70
1671
70 69 86
168
1
70 86 871
^ 169 87 86 102
170 86 101 102
171 102 101 107 iv'J>
r
If/ 11,/. 1 ui ruvr
fTlTTba 107 122
"i'tT lo 7"r2^r ~l??'
vn
175 12? 121 127
176 122 127 12R
177 128 127 135
178 127 142 135
119 135 142 143
FLEMENT PROPERTIES
1 TO 179 TYPE 'CSTG* THICKNESS 0.10
CONSTANTS
E 30000. ALL
POISSON 0.3 ALL
i_.
JOINT RELEASES
1 TO 8 FORCE Y
80 95 114 129 136 FORCE X
$ PLATE LENGTH L0 = 12, WIDTH V^I0 =8WFLD WIDTH W=l/16, WELD LENGFH CENTER
$ "tITc ENTER L^5".57~ENf~cl^CLE~DrAl^rEl'lR^0r^^^ WIDTH WS = l/37,
$ SIGMA0=8*10**4
LOADING 'ONE' *X=0.20'
JOINT LOADS
1 FORCE Y -1.00
9 FORCE Y -2.00
16 FORCE Y -1.00
LOADING 'TWO* •X^0.40*
JOINT LOADS
1 FORCE Y -1.00
9 FORCE Y -2.00
16 FORCE Y -2.00
24 FORCE Y -2.00
26 FOOCE Y -l.no
LOADING 'THREE* •X=0.60'
JOINT LOADS TT
1 FORCF Y -1.00"
"9" F OR C F ~f "-Y. "00
r6^nRCF Y -2.00
152
24 FORCE Y -2.00
~?b "for C E "y - 2". 00
T2' FOR C E "y -2.00
39 FORCE Y -1.00
UTacTi N G"
""f dUR~»" "•X = .~8b~«"
~J cTn T~ L DADS
1 FORCE Y -I. 00
'9""foI C
E^Y ~^2~.~0
y
1
16" "Fln'RCE"-y _^_^__
3
24 FORCE Y -2.00
4
26 FORCE Y -2.00
^ , , .
32 FORCE Y -2.00
6
39 FORCE Y -2.00
7
A7 FORCE Y -2. CO
t49 FORCF Y -1.00
9_
LOADING 'FIVE* •X=1.00»
D
JOINT LOADS
1
1 FORCE Y -1.00
2
9 FORCE Y -2.00
3
16 FORCE Y -2.00
4
24 FORCE Y -2.00
5
26 FORCE Y -2.00
S
32 FORCE Y -2.00
?
39 FORCE Y -2.00
J
47 FORCE Y -2.00
3_
49 FORCE Y -2.00
^_
55 FORCE Y -2.001
1
62 FORCE Y -1. 00
I
DUMP TIME
TIME BEGINl^''^
o
o
.153
i_.
LOADING - ONF X=0.20
^RESULTANT JOINT D
I
SPL ACFMENTS - SUPPORTS
JOINT DI SPLACEMENT-
"~"Y~n~i"^p7X DI SP. 7 DI SP.
1
2
0.00.0
•0.00COG
86755P2036
3
4
5
6
n n
0.00.0
-O.on-0.0^' 78239
_7_07 5 8
'662545R597
-O.GO(
O.'^O
2223
•0.0003790•0,0005535
-0.00 5 364 8
-0.00 5 2 56 7
?
7
8
0.00.0
-0.0055113-0.905 3223
3
8095
0.00178140.0016919
0.00.0
4
114129
0.001493 3
0. 00136620.00.0
;
136 0.0012693 0.0
6
7
RESULTANT . JOINT DISPLACEMENTS - FREE JOINTS
JOINT /- MTcni ArcMcMT /111
U I S 1-' L A L r n t ' N I—
9
X DISP. Y DI SP. z nis^.
9
10
0.00055570,0002291
-0.008 306 7
-O.OC79604
1
1617
0.00092740,0002137
-C. 007469 1
-0.0070741
?
1819
-0.0000497-0,0001827
-0.0C665C3-0.0059258
3_ _
2021
-0.0002045-0.0002Q43
-0.0056892-0.0055396
2425
0.00127440.0007213
-0.00 6 506 9
-0.00 6 54012627
0.00147180.0005111
-C.0C58308-0. 00 5 P 841
7
2829
-0.0001555-0.0O0 38 5?
-0.0056535-0.OO5269P
8
3031
-0. 0007266-0,0010494
-'^.OC 5^276-0.0048734
9
3233
0,001636?0,0011581
-0.005213 1
-0.00 524 3 5
k
3940
0.00176940.000 869 1
-0.0046604-0,00473?? '
1
1
4142
0.0002667' -0.0003452
-0.0047954-:0_.00A6 974
43
^^4546
0. 000499?•Oj.000 7 59A•0,00101340,00 14 60 3
-0.0046137_-l0^i):04 5 64 3
-0,0044638-0.0043^37
o
o
4_
7_
9_
2_
5_
t
47_4PL49505152
0.00189680.O0 1/.'!»32_
0.00200 39Oj.00lO7.2_p_
-0.0000 32 5
-0.0005760
-n. 0041231 .c4^-r_.^' 04.1 5 5 5
"-0.00 3 573-C.00 3638C-0.0037827-O.OO ^8 597
5354
-0.0012293-0.0017601
-0.0038132-0.00 3 7 361
i
5556
0.00209060.0016213
-0.003O127-0.0030352
6263
0.002 147 7
0.0016731-0.0024 177-0.0C24355
64656667
O.OOll 7640.0005213
-0.C024674-0.0026032
-0.0002759-0.0005766
-0.0029503-0,00 30299
6869
-0.0009712-O.rO] 3669
-0.0030872-0.0030747
7071
-0.00195280. 0018989
-C. 0030519-0.00 2 300?
7273
0.0017 5200.001A234
-0,00 2046 7
-0.002O89<^7475
0.00166560.0011952
-0.0016660-COO 17458
7677
0.00086610.0016692
-0.0018929-0.0011422
7819
0.00138860,00 1664 2
-0.00125 5 8
-0.00045268182
0.00158330.0012462
-0.0C06717-0.00 10315
8384
0.000 702 7
0.000 2 28 6
-0.00 15237-0,0018316
8586
-0.0005444-0.001A379
-0.0022<^36-0.0023653
8788
-0.00202810.^^01622 5
-0.0023572-0.0003506
8996
0.00106660.001 3840
-0.0C0 9416-0.00040 8 2
9798
0.0008943-0.0000371
-0.0008238-C. 0014264
99100
-0.0004874-0.0009653
-0.0015585-0.0016495
101102
-0.0014567-0.0020403
-0.0016039-G. 0016910
103105
0.001^2610.0005453
-C.C002010-O.O0C6334
106107
-0.0004208-0.0014433
-0.0009711-O.OCl 0809
108115
-0.O020C340.0013505
-0.0010996-0.0000758
117118
0.C0104330.0001464
-0.00014B4-C. 000 38 26
119120
-0.0003 74 5
-0.0008872-0.0004566-0. 000 50 7
5
121122
-0.0014202-0.0019571
-0.0005336-0.0005439cVG00039 9
0. CO 00 A?
2
123125
0.00133250.0006534
126127
-0.0003311-0.0Oi^9S7
-O. COO OOP 5
-0.00 00317 '
128130132133
~134135
-0.00192260.0012543
-0.00003950.0001340
0.001020 8
0.0001835-0.b00~8'^77-0.0019035
0.0002553C. 00040620.000457 3 //A0,0004661 '•
o
o
1
137139140141_
143
0,00121870.00094760.000 6054g.,000_3 2BL0.001380 3
0 .00 18^^95
0.000? 6410..pOC49] 3
0.0006691_0_._0.CCR9 72,
0. 000 966?r.000^7u?
133_.
LOADING - TWO X = '^.40
RFSULTANT JCINT DISPLACEMENTS - SUPPOfUS
JOINT niSPLACffMENT-
X DISP. Y DISP. Z DISP.
1
2
345
6
0.0
0.0
0.00.0
0.01510120.014^94 7
0.0143387•'I.-OJJ 3 4150.0127092•0.0 1136^^9
7
8
80
?5114129
O.n
J3_.00.0034995_p.003 3 2^2_2
0,00293030.00?6ai?
0.0107165•C_._0_10_354 3
0.0^^00.0
136 0.0024917 0.0
i RESULTANT JOINT DISPLACEMENTS - FREE JOINTS
JOINT DISPLACEMENT-
X DISP. Y DISP. Z DISP.
910
0.00078390.0005129
-0.0148300-0, 01452^3
16 0.0014993 -C. 01431042 17 J}j 0^0 5640 -0_._OJJ^J^.Ii
1819
0.0000711•0.0003118
-0.0128202-0.01148 91
2021
0.00038250,0005600
-0.01104'=>0-0.01077] 1
2223
•0.00072910.0010632
-0.0104360-0.01023'^6
2425
0.002 17740.0013916
-0.0134 84 8
-C .0 1318 502627
0.0026910O.OOlOORl
-0.0121029-0.0117723
2 8 -0.000 2 29 8
8 29 J-.9_.Q[^P727_7_30 -0.00 1400 3
, 3L -C.OOZQZQb^32 0.0031542
k ^33 0.0021583
-0.O11G299_-CL._0J0_26 8 2
-0.0C98G04_-Q_.J?0_9^50I8-0.C106014-0.0106 5Q0
3940
0.0034535 -0.0094169C. 0016498 -0,0095367
41
4344
0.0005042-.O.C006_41A-0.0009539-0.00 1462?
•0.0095675•_0_._C09_2010.
•0.00002720.0089240 vsT
o
o
m
45 -0.0019590 -0.008725C 155
146 -0.0028369 ^^C . 8 5 8 7 g_
47 0.0037241 -0.0082844
J48 0.0027935 ^-0.00 8347 7
49 0.0O3Q''U6 -0.007157150 0.0020568 -O.C072B3851 -0,0000811 -0.':'0 7S138
45253
-0.0011113-0.00 2 3843
-0. CO 7 5 84
3
-0.0074 752
% 545 5
-0.00341050.0041166
-n. 007^214-6". 006018 3
f.
56 0.0031678 -p. 00606??62 0.00^2304 -C.OJ4R188
J63 0.0032760 -^.004855764 0.0022P34 -0,0049191
g65 0.0009915 -0,005186066 -0.0005574 -0,00^8390
467 -0.001126? -0.00597676B -G. 0018920 -0,006077069 -0.0026595 -0,006046470 -0.0037968 -0,0059993
1
71 0.0037299 -0,0045 83872 0.0034359 -C. 004 07 7
C
773 0.0027794 -0.00^l64''i
74 0.00 32651 -0,00 33173
375 0.0O2326R -0,003476976 0.0016752 -C.00376^-^7
477 0.0032742 -0.002273478 0.0027145 -0.0024996
1 79 0.0032660 -0.0CC9O0 5
81 0.003104^ -0.00133646
82 0.0024320 -0.002052183 0.0013565 -0. CO 30 3 86
784 0.00C4252 -0.003634485 -0.0010721 -0.004476086 -0.0028054 -0. 0OA664987 -0.0039553 -0.0046452
988 0.0031834 -0.0006 96 9
89 0.0020792 -0.0018719_
96 0.0027104 -0,000810^97 0.0017405 -0.0016368
1
98 -0.0000853 -0.002830?99 -0.000O611 -0.C03C884
2_ 100 -0.0018924 -0.0n32659101 -0.0028484 -0.0033307
3102 -0.0039891 -O.OC 3 34 3
103 0.0027961 -0. 0003^994
105 0.001G593 -0.n0126C8106 -0.0008308 -0,^019313
510 7 -0.00?8267 -0.0021470108 -0.0039237 -0.0021818
6115 0.0026A83 -C.000is?g117 0.0020424 -0.000?991
7118 0.00 02 820 -C.00C76 8 9
119 -0.0007384 -C. 00091678
120 -0.0017415 -0.00 10183121 -0,0027841 -0.0010696
9122 -0,0038376 -0.0010894123 0.002613-' 0.000C752 ,
m 125 0.0012785 0.000 07?'!.w126 -0.0006520 -C. 0000339
1127i28'
-0.0027377-O".' 00 3 7 71
9"'-0.0000819-r.O0CO971
?__ 130 0.0024611 0.000 2 59 9
132 0.0020C1
7
C.O00A948 ^{;,,
133 0,0003583 0.0007841 ''
o
n
134135137
140141
n. 00 1.66390.0037^520.002 39210.0018 591O.Of^ 11871•0.0006444
O.onrcyjq0,00^08961C . 5 1 5 7
0,00130 400.00 17462
157
142143
-0.002 7086-0.0037275
0,0018 7900,00 18867
1,
Ida d fn g"-~ ~Tm e f
"
X = 0.~67d~~''
8
RESULTANT JOINT niSPLACEMFNTS - SUPPORTS
JOINT / — dt^^piatpmpmT // UlorL'Al^cici^'
X DISP. Y DISP. Z CISP.
1
1
2
0.0O.n
-0,0202986-0,0199403
5
3
4
0.00.0
-0,C19631A-0.^186197
3
5
6
0.00.0
-0.0179 5 56-0.0162740
4
7
8
0,00.0
-0.0153912-0,014880 8
<.
8095
0,00509550.00 4 8 344
0,00.0
6
114129
0,00426050.0038987
0,00.0
7
136 0,0036246 0.0
•
9
RESULTANT JOINT DISPLACEMENTS - FREE JOINTS
JOINT / — — r^TCnl APCMCMT /
'0
/ — IJ i S t-^ L A L t " b IN 1
M
X nisp. Y DI SP, Z DISP.
'2_
9
100,00094060,0006927
-0.020C500-0.0197869
'3
1617
0,00182480.0008964
-0,0195005-0.0188825
'4
1819
0,0002702-C.OO0 37 18
-0,0181233-0,01643«6
!5
2021
-0.0005213-0.0007814
-0,01 58433-'^.015468'i
!6
2223
-0,0010316-0.0015021
-0.015002 7
-o.riA7ioi
U
242 5
0,00 26 7940.00197^1
-C.0lP8^15-^,0185984
>»
2627
0.00348440.0016314
-0,0178546-0,017 1300
ll
2829
-0.0001944-0.0010039
-0,015 885 6
-0,01 '+7 8 30
M3031
-0.0019862-0.0028629
-0,0141194-0.0 I 36 93 5 /
ii
3233
0,0042462' 0,0030331
-CO 16 53 2 5
-0,016 2 53439
4142
0.00483820_.0n2 330A0,0007714•0.0008574
-0.01-^6572.-C0_14403l-0.0141486-0.013332 3
t»
(I
43 -0.00 13 34 5 -0.013064 7
14445
-0.0020685 _
-0.00 2 7 89 3
-0 . 1 29050 J-58 .-_
- . 'U 2 6 1 ^' 8
14647
-p , 00 /+0 3 36 ._.
""0.C0 5 3P90-0.0 124 133-0.f>l>6354
1 48 0.003924? -0.012 7?b?49 0.0'^57469 -0.01082^>4
450 0. 002 8 60
P
-0.010971151 -0.0001246 -^0.011085 4
% _ 52 -0.0015 752 -0.01103 31
53 -0.C0 34C92 -'^.0108492
^54 -0.0048714 -0.0 10621555 0.0060181 -0,00902 8 9
156 0.0045528 -0.00 908 8462 0.006 1854 -0.007192^
863 0.004 7331 -0.00 725 3 6
64 0.0032493 -0.0C734Q94
65 O.f^O 13685 -0.00 7734566 -0.^^008290 -coot 579967 -0.00 16211 -0.008745868 -0.0027189 -0.0G8869O
1
69 -0.0038187 -C. 008813470 -0.0054471 -0.0087403
) 71 0.0054235 -0.00^840972 0,004980^ -0.0060813
373 0.0039990 -0.006214974 0,0^47324 -0.0049455
475 0.0033316 -0.005185476 0.0023739 -0.0056214
s77 0.0047531 -C. CO 333 7078 0.0039162 -0.0037249
679 0.0047465 -0.001340681 0.CC4 503 5 -C. 0019895
782 0.0034994 -0.0030551
#83 0.0019190 -0.004516384 0.000 568 1 -0. 005384485 -0.0015 601 -0.0065760
986 -O,0OaO43? -0.006828187 -0.0056973 -0.0067920
'0 _88 0.0046255 -0.001035589 0.0029P86 -0.0027834
I 96 0.00 3 92 64 -O.0012O4297 0.0024965 -0.0024303
'198 -0.0001514 -0.004 186 5
99 -0.0014043 -0.0045553'3 100 -0.0027448 -0.0048086
101 -0.0041179 -0.0048967'4 102 -0,00 5 7665 -0.004^^111
103 0.0040597 -O.00C595 3
s 105 0.0015152 -0.0018743106 -0.C012179 -0.0028644
'6 107 -0.^040962 -0.00 3176 1
108 -0.00 5 6 86 2 -0.00322257 115 0.0038459 -0.0002315
117 0.002 9 58 3 -0,non4525'I 118 0.0003979 -0.0011583
119 -0.0010 80 7 -0. CO 13731•1 120
121-0.0025323-OVod40 40^
-0.0015295" -0.00l'603 5
'
^ 122 -0. or 5 5710 -0.00 16 314 '
f 123 0.0037977 0. OOP 102 4
II125126
0.0018 500 0.0000819-0. 000^531 -0.0000876
12 127128
-0. CO 3 9 76
6
-0.6054 800-0.0001625-6. 00 01840 ~ .^f-
P., 000 37 1 1130 0,0035772 .
f»
<l
13213 3_
134135137139140141142143
0.0029067o.nnn51690.00241940..00542e6_6.00 3 4 7900.(027C17
C) . C .T 7 C 5 8
_0.p0 1 1 1 1.1
0.0r.l24'»4
.0_v0„01?62 4
0.00074390.00 138 OR
159
0.rni7236 O.OO1075R- . Q 3 7_9 ^_'S0 2. 5 6 LV ;} 3 9 36 1 • C . C 2 6 9 3 5
-0.005417R 0.CC2704?
LOADING - FOUR X=G.SC
RESULTANT JOINT DISPLACEMENTS - SUPPORTS
JOINT DISPl ACE.MENT-
X DISP. Y DISP. 2 DISP.
>
1
?
0.00.0
-0.0244147-0.02 4090 4
3
3
40.00.0
-C.0238209-0.0228374
4_
5
6
0.00.0
-0.0221868-0.0203627
;
7
8
0.00.0
-0.0193178-0.01 P6908
6
8095
0.'^0651000.0061686
CO
114129
0.00542800.0049665
0.0 •
0.0
•136 0.0046197 0,0
9 -
RESULTANT JOINT DISPLACEMENTS - FREE jrU'TS
JOINT / riTcnt APuMcNiT _/
'1
U I b r L AL ri b !N I
2
X DISP. Y DISP. Z DISP.-
'3
9
10
0.001061R0.000818R
-0.02416^4-0.0239363
'4
1617
0.00206670.001 1415
-0.0237171-0.0230710
5
IB19
0.0004397-0.0003785
-0.0223618-0.0205501
'6
2021
-0.00^6178-C.Q009 501
-CO 198 5 14
-0.C194126
'7
2223
-0.0012730-0.0018 50 5
-C. 0188451-0.01848 5 3
'8
2425
0.00303630,002 362 7
-0.0230232-0.0227746
26272829
0.0039702
-O'. 00 00 8 20-C. or 1203
-0.02 208 78
_-Q_»JP_2J.39?4-0.019^638-0.0 18 5974
3031
-0.0024569-0.0035374
-0.017779C-0.0172494
3233_
3940
0.0048848 -0.0208813J?.'^037392 -Q.. 0^062880.0o'5 74 7 -0.019 33 800,0030024 -0.0187010
r»
•
•
7
41
4243444546
0.^011 18 1
-0.n<^09803'-0.001622'-^
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RESULTANT JOINT DISPLACEMENTS - SUPPORTS
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479 0,0004739 -0.0042509_4 8a D^QQaZT2jl -Q..Di)„4_3331.
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132 0.0023535 -0.0011431-^C3 .0 019340 -O.on 776?133 0.0015255 -0.0024401-5C4_ ; i).*DD 1L4_5J] -0.^)03^922.134 0.0006436 -0.0036033.5.Q5_ -jO..OOQ2^92 .-Q_.Di)33i)5d.135 -0.0015512 -0.0039293-^01 .0028400 -0. 00073^15C8.510.
511
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219
APPEl^DIX 6
H-slit ,'CSTG' type .
Th6 following presents :
1- Plate division , dark numbers are element names , light num-
bers are node names , page 220 ,
2- Input data , pp 221 - 234 ,
3- output data, pp 235 - 247 .
220
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"55 57.0 30.0 22'!)
81 75.0 0.0 S
82 75. C 5.0
83 75. C 10.0
.,
bt~ 5770 37.0
57"~60.0
58 6G.C 5.0
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60~' 60.0 15 .0
61 60. C 25.0
~b2 ~60.0 60.0
63~""6r."o~
64 61. C 4C.C
"65""'6l75~
6^6""""6570" 30.0
67 65.
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"6"9" ~68.T"
70 69.0 33.0
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73 70 .0 5.0
74 ~70.T 10.0
~~7"5~ "7070"
76 70.0 25.0
Ti '~7C'.T
78''~7oTo" 100.0
79 73.0 30.0
8~0"'~737c' 37. C
84 75. C 20.0
85 75.0 25.0
86 75.0 35.0
87 75. C 50.0 L^ ^
88 7'?.0 60.0;;;;*_
22k-
89""7B.O 30.0
90"~78. C
91 80.0 5.0
V g"2~"aorc
g^B 80. C 25.0
^4 85. C 0.0 S
"95 85 .0 10.0
91: 8570k
97 90.0 5.0
s"fi""9c7c
9"9"'^cTc AO.O
100 90. G 60 .0
l'cl~~90."'c 100.0
l'o"2 95. C
103 95.0 10.0
lb7» 9 5.0
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l'C5~ 100.0 5.0
106 105. C O.C S
r07~ 10 5.0 10.0\
^^fcTloi".^ t
1C9 1C5.C AO.O
llo 110.0
lU~ir5'. C" 0.0 s
112 115.0 10.0
ri3~Ti5rd 30 ."b
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llA 12c. C 5.0
115 125.0 0.0 S
116 125. C 10.0
Tr7~l'257o 20.0
118 125.0 40.0
TiVl^'570 6'd7c
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1 ? A 13 5.0 ro".o
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r2'6'l40.0 5.0
127 1A5.C 0.0 S
\2Q~~\^^^Q 10.0
iz^'T^b'^^ 20.0
130 l''i5.C AO.C
r3Tl'5070 0.0 S
r3'2"l"5c7c 5.0
133 150.0 10.0
1^4~150.~0 20.0
rsVlsC.T 30.^
136 150. C AO.C
l'37^l50.C 60.0
H 1~3TI50.~0 100.0
ELEMENT INCinENCES
\ 2 T ^
"Z 2 9" loI ^ •
3 10 9 lA
- ---------
5 14~
13 15
6 lA 15 16
7 T6~~r5~Tl
-^—-------^^
9 21 20 22
fo 2l~"22'"'23~
-----22'~'27
12 22 26 27
'V^ '
LA 27 28 29
i
15 29 ZS 37
T6"~2 8 3b~~3T
T7~"l7~"36~""3"8'
Ts 37 Fb 39~
T^~~39"T8"~'^T
"20 A 2 4 3~ " ITs
^'^6
21 43 42 48
~2"2" "43" "4^8"" 49
2 3 49 48 58
24 48 57 58
25 49 58 50
26 50 5g 59
27 50 59 60
28 5C 60 51
29 51 60 61
30 52 51 61
31 52 61 55
32 55 61 65
33 55 65 63
34 63 65 66
35 63 66 67
36 67 66 70
37 7C 66 69
38 70 69 T^
39 79 69 76
40 79 76 85
41 76 84 85
42 76 75 84
4 3 7 84
44 75 74 83
4 5 74 73 83
46 73 82 83
47 73 81 82 1 '
r
c
4B 73 72 81 dc.(
W ~3 ~211
50"" ---
"lo"
51 U 10 14
•5~2~"
"li~ ~r^~'
5T''1^" "iT" 17
54 17 16 21
5~5~""T7"
----24
5'6~'"2!" 23
57 24 23 27
"5"8"'"24" 27
^^"""27" 29 30
60 30 29 3 7
---~30" 37 40
'bf"37"'T^7"
6 3 40 39 ^3
"6^ ~4d"'~4T"
» l)^ "43 " 4c 50
66 40 50 44
bT 44 To 51
6"8" 44 'Tf. £. ^
69 45 51 52
---45 ~'^2
---46 ~'b2 55
72 46 55 53
-- "53"""55" 56
74 56 5 5 63
75 56 63 64
76 64 63 67
80 80 70 86
__.64 ~61 'Ti
/
78 67 70 71
-- ~^1~ ~7C" 80^ !
r
81 70 79 86i'«rt.-><UWM,>^t<'i« «j.'
.22i8
82 86 ?S 89
83 79 85 89
£4 89 85 93
1C8 54 64 68
s's"~93~'~8 5"'
84
86 84 92 93
87 84 83 92
88 83 91 92
89 83 82 91
9C 82 81 91
91 81 94 91
92 3 11 4
93 4 11 18
94 18 11 17
95 18 17 24
96 18 24 31
97 24 30 31
98 31 30 40
99 31 40 44
100 31 44 45
101 31 45 41
102 41 45 46
103 41 4£ 47
104 47 46 53•
105 47 53 56
106 47 56 64
107 54 47 64
109 68 64 71
110 68 71 87
11
1
67 71 90
112 71 80 90
113 90 £C 86 ^- i
€
(
IL4 06 OS 90
115 '90 89 96
ri6~~ PS 93 96
117 9 6 9 3 9 8
_^______------
iCg 98 92 95
120 92 91 95
l"2~l 95 91 ^7
123 5 '^ 12
r2'4~~T2 4 18
------j;-g---2"5
126 25 la 31
.^-- 25~Tl 4T
129 6 12 19
1^30 19 12 25
132 32 2 5 41
1Y3 22 Al A 7
_^- ^- ^^ -^
135 32 54 33
136 33 54 62
137"54 68 62
138 62 6 3 88
_ --^ ^^ ^_^ ^^
14"0~~88 87 100
141 87 99 100
-__ ^- ^- ^^
-_^ -- ^- ^^
I — -'-
144 96 104 99
'
r4^5~"96~~98 104
r46~" 9a~ 95 103
L
147 95 97 10^ _._,, 230
"~lAH ^7 94 10?
150 7 19 33
152 7 33 34
153 34 33 62
r5'434~ 62 1^
l'55 7 7 ^6 2 ^B
« 156 77 88 100
15 floo 9 9~ I'd 9
r5'a~~99~rd4 109
159 1C4 98 ICa
l'60~"S8~fc"3~ id's
161 103 107 108
162 1C3 97 105
163 1C3 1C5 107
1
164 97 102 105
165 105 102 106
166 105 UC 107
"
167 105 106 110/
168 77 100 101
170 100 119 120
171 100 109 119
172 1C9 118 119
173 109 104 113
174 1C9 113 118
17 5 118 113 125
--^-l04 'loa" 113 1
177 ica 117 113
---Tl3''iiT 125
179 125 117 129
c
(
100 lOH 107 112 231
T8Ti.oo'~rr2 11^
"l8Yll2"'ir^ 117
1B3 117 116 12A
"foT' 1 T 7~r?"^" "l"? 9
TaT 1 2 4~ IJe"1 2 9
186 107 110 112
TeTl^Tc'uT 112
T8'8~"U2~rr4 116
189 116 114 122
~r90"~U6~r22 1^2 4
~r9"r"i"2 4~r2"2 126
192 124 126 128
l'93'l2 8T2"6 1^2
~l'94~l06~lll 110
195 110 111 114
T9"6~ir4'Tll 115
1.97114 115 122
198 115 125 122
T9V122 123 126
'~200~l26 r23 127
201 126 127 132
""2"02~r2 7"T3"l l"3 2
"203" 7 34 8
204 8 34 35
"205" 35 34 7 7
"ToV~3 5 77 78
207 78 77 101
"Tc'8"Toi~r26" 121
'~2"d'9'~l2l~r20 138^
210 120 137 138
"TlT ~1^0~Tl9~13 7
"Tl2 ~U9 130 137 'i- '^
'
r
?A^ 11^ 118 130
YlV 130136 137
232
215 ~\W 12^'"iTo
216 130 125 135__ __
>"^'i'^" l3C"Ti"^
w _218 Y?5'T2'9' 135
219 135 129 134
2To~"I29'~r3"3" 134
'I29'~ 12^87
222 128 132 13 3
"element properties
1 TO 222 TYPE 'CSTG' THICKNESS 5.0
CONSTANTS
E 21100. ALL
PGISSQN .3 ALL
JOINT RELEASES
1 TC 8 FCRCE Y
-^----------^--------------^^^^ ^3j^ FORCE X
LOADING 'ONE' •R=0.33»
JOINT LCACS
1 FCRCE Y -125.0 p ^0 Shi (<:/
' /A^. v^
9 FCRCE Y -250.0 tfi
____ll"rCRCE Y -250.0
____15~FGRc'e~ Y -250.0
20 FCRCE Y -125.0
LOADING »TWO» 'R^O.SC*
JOINT LCADS
1 FORCE Y -125.0
9 FORCE Y -250.0
13 FCRCE Y -250.0
15 FCRCE Y -250.
20 FORCE Y -250.07,
9P PPRTF Y -7^0.0 l>-^
c
26 FCRCr Y -12 '3.0 d:):)
loadIng"' THREE' •R=0.66'
JCIM LC/^CS
1 FCRCE Y -125.0
(f "fcrce'T -250.0
iT f'crce Y -2 50.0
15 FORCE Y -250.0
2"0~ "fcrce" Y -250 .0
22" FCRCE Y
26 FORCE Y -250.0
fe" "f'c'rce' Y
36" "f'crce" Y -125.0
LCACING ' FOUR' 'R=0.83»
j Ol N T "l c'a'd's
1 FORCf Y -125.0
9 FCRCE Y -250.0
lY'pORCf Y -250.
C
lY"force Y -250.0
20 FCRCF Y -250.0
22 FORCE Y -250.0
26 ""fcrce Y -250.01 .
^
28 FCRCE Y -250.0
~3i'forcTe" Y -250.0
38 FORCE Y -250.0
42 FCRCE Y -125.0
loaoIng 'FIVE* •R=l.O*
JOINT LGACS
1 FCRCE Y -125.0
9 FORCE Y -250.
C
13 FCRCE Y -250.0
15 FORCE Y -250.0
20 FCRCE Y -250.0
22 FORCE Y -250.
C
l->
2 6 FCRCF Y -250 .0. -..v 234
28 FORCE Y -250.0
36 FORCE Y -250.0•
^3 8 FCPCE Y -250.0
• 42 FORCE Y -250.
48 FCRCE Y -250.0
57 FCRCE Y -125.0
STIFFNESS ANALYSIS
LIST DISPLACEMENTS STRESSES ALL
I
*
•
'
-
1
-
-
'
-
-
*
"
-
—
-
-
-
-
235
LOADING - {JfME R=0.33
•RESULTANT JOINT DISPLACEMENTS - SUPPORTS
J CI NT / —OTCDIAPCMPMT // — U 1 br L AU t i'^C 1\ ! /
X DI SP. Y DI SP. Z CISP.
1
2
0.0 .
0.0-0.0603966-0.05 94 46 8
3
4o.c0.0
-0.057641C-0 .0548952
5
60.00.0
-0.0516929-0.C4S5426
7
8
0.00.0
-0.0469618-C.C436242
72fil
0. 00 94 R 460.0101828
.0
0.0941C2
.01113030.0115812
0.00.0
1C6111
0.01170690.0116545
0.00.0
115123
0.01154490.0 113149
0.00.0
^
127131
0.01099090.0107025
0.00.0
1
RESULTANT JOINT DISPLACEMENTS - FREE JOINTS
JCINT / DTCDIArPMPMT _ _ /t UljrLAoti^tlvl _z
<
y DI SP. Y DISP. Z CISP.
9 0.0023112 -0.059962910
11C.C016332.0009699
-0.0567361-0.0572288
1213
-0.00008380.0044323
-0.0513172-0.0582331
1415
.00314060.0063772
-0.0572006-0.0559198
1617
0.00446640.0026746
-0.0546722-0.0532781
1€19
0.0006865-0.0005046
-0.05C9567-0.04689CP
2021
0.000.00
8116454060
0.05C.C5
1299 3
C83932223
0.00898550.0066415
-0.0464355-0.0466015
\
2425
0.00414970.0003240
-0.0465S03'-0. 0450660
2627
0.00961740.0075242
-0.0431111-0.0431986
262c
0.01006690.008207^
-0.0400898-0.0402264
3C31
0.000,00
6042031658
0. 04•0,04
C3277C 4 7 3 R
#
3233
-0.00021 84 -0.0401 168-0.0027739 "'""~' -C.C39C707 236
3435
-0.0055505-0. 0099883
-0 .0 381519-0.0367230
3637
0.010 33 39
0.00868 80-0.03753«;7-0.0 3 7569 6
3{:
390.0104803.0089846
-C.C351463-0.0351999
•4041
0.00718750.0024572
-0.0352C2C-0.0354174
4243
0.01050160.0091414
-C.033C911-0 .0330806
4445
0.00620010.00 3996 3
-0.0328712-0.0328676
4647
0.00325610.0006737
-0.0317370-C.C315697
484c
.01045910.0091840
-0.0311325-0.0311276
5051
0.00749490.0054567
-0.0310435-0.0306526
5253
0.00449150.0025357
-0.0303998-0.0300203
54 -0.00151530.0039542
-0.0297884-0.0290964
5657
0.00176920.0104216
-0.0286136-0 .0294274
5659
0.00917200.CO75719
-0.0294332-0.0295C53
6C6 1
0.0063717C.0048258
-0.0294807-C. 0282849
6263
-0 .00356600.0033037
-0.027C376-0.0262223
k_
64'
650.00104370.0048932
-0.0258942• -0.0263705
r 6667
0.00525760.0027956
-0.0222105-0.0224143
686S
-0.0012269.0.0065509
-0.0235052-0.0L69915
7071
.00408410.0016543
-0.C179832-0.0195592
7374
0.00945730.0091Q41
-C. 0022953-0.0046428
,-'
7576
0.00893950.0078533
-0.0071055-0.0131868
7778
-0.0087023-0.0149055
-0.0219912-0.0214369
798C
0.00625100.0031921
-0.0136131-0.016118?
828 3
0.01005730.0099141
-0.0C2C96C-0.0042192
8485
C. 0091488.0080574
-C. C0e6C29-0.01C6319
868 7
0.0043178-0.0003827
-0.0140657-0.0171868
88
9091
-0.00307210.0065699
-0.0183481-0.0109842
0.00278710.01066 36
-0.0136821-0.00 18188
9293
C. 0100757.0083443
•0.0054835•0.0086012
9596
0.01084010.0071071
9798
0.01132120.0097233
-0.0030569•-P_i 8 2 13_2_
0.0012734•C. 0046977
99ICO
0.0040727-0.00216^0
-0.0C86126-0.0 1 14540
101103
-0.01536830.0111636
-C. 0129586-0.00 20L9 6 237
1C4105
0.00749700.011554 9
-0.0054907-C.00C8C25
1C7ICR
0.01 118540.0099645
-0.0011830-0.0023424
A_109lie
0.00503610.0115795
-0.0045927-0.0004125
W" 112113
0.C1128050.0078895
-0.0005533-0.0016493
1160.01150300.0110585
-C.0CC14020.0000482
1 17118
0.00987890.0051061
0.0000936C.00CC87C
119120
-0.0010550-0.0077965
-0.0000050-0.0003537
121122
-0 .01475160.0113350
-0.00C52000.0001914
124125
0.01094170.0077315
C.00C66920.0018635
126128
0.01106260.0105146
0.00048830.0013495
129130
0.00936030.0050552
0.00254380.0045468
132133
0.01063600.0103142
0.O0Ca8390.0017358
134135
0.00911910.C072547
0.00324210.0045619
136137
0.0049158-0.0011751
0.00579440.0069375
138 -0.0144677 0.0075483
• LOADING - TWO R=0.50':
RESULT/iNT JOINT DISPLACEMENTS - SUPPORTS
JCINT / DI SPLACEMENT ._, /
X DISP. Y OISP. Z CISP.
1 0.0 -0.08316492
3
.0
0.0-0.0822756-0.0805988
4
5
CO0.0
-C. 0778629-0.0744478
67
0.00.0
-0.0719271-0. 0686974
8
720.00.0141539
-0.06399650.0
8194
0.01519690.0166110
C.
0.01C2i-06111
123127
0.0172828-Q.!lO_1_7j:^70I
0.0173931.0._Q_1_7_2J2 2
0.01689310.0164152
0.0o_.o,
0.0
0.00.0
131 0.C159887 0.0
RESULTANT JOINT DISPLACEMENTS - FREE JOINTS \V
~Saiir\ / __ ^r:^^-i-)i si'LACI N'hNf-- -—^-z^^^— 7 _
X DISP. Y DISP. 7 CIS P.
9
IC.0026702
0.0021107-0.0828182-0.08 1702 1
11
12
0.00 152010.0001 746
-C. C8C3047-G .0741282
X
131^1
0.0G5210C0.004 1355
-0.0814128-C.C8C465g
r 15 .00764520.0C602C4
-0.0795391-0.C783687
1718
0.00417950.CC18C77
-0.0769282-0.0740731
192C
-0 .00036830.0098885
-0.068566-0.0765035
2122
0.00770740.0119113
-0.0754935-0.0728663
232 4
C.0C915990.0061781
-0.0716518-0.C7C2744
2526
0.0C098540.0136484
-0.0663891-C.067C885
2726
0.01022180.01 44762
-0 .0666077-0.0612 57 8
293C
C. 01152970.0084478
-0.06144C5-0.06 14481
3 1
32C. 0045327
-0 .0000490-C.C6C9726-0.0593643
333A
-0.0038929-C.0C8G128
-0.0576265-0.0562619
35 -0.0145131 -0.05415883i ^Q_._0_l_50015 r0^0 5.IC_7_0J_
37 0.0123732 -0.057125C.3g ^0_._0_1^25 2 8 rQi0 5_3_ZPi)3_
39 0.0128910 -C.C53289040 0.010 1308 -0.053261 1
^1 0.0035608 -0.05311544 2 ^Q_._0JL_5 26 8 3 rJC:i_a49.9_9_4J.
43 '0.0131560 -0.049971344 0.0C87977 -0.04961574546
0.00573880.00472 5 7
-0.04C391C-0.0476073
4748
0.00100490.0151870
-0. 04 7 162 1
-0.04697194950
0.01323370.0106758
-0.0469659-0.0468278
5152
0.00778270.0064619
-0.0461590-C. 0456^50
\
5354
0.0036942-0.0021 590
-0.0449719-0.0^43429
5556
0.00576720.0025790
-0.0436593-0.0428196
5758
0.0151223.01 32157
-0.044386C-C.0 4 4396C
5^ 0.0108043 -0.0445181_6.0 ^CC_9_C^5 r 0j.Q4 4^7_9_7_
61 0.0069433 -0.0425497_6_2 -
._0P_5_1J 3 2 ri^_. 4 2P_2_0_
63 0.0048285 -C.0393C0764 0.0015138 -0 .038716565 0.007157C -0.0395796_66 5.QP_7_76J0 -0_?.Q13?A^_^_67 0.0040906 -0. 0335466_6_8 -0»0Sn73 3'2 -0_.0 3^_CJi^Ji_6c 0.0097395 -0.02540997C 0.006C296 -0.0268926 V'
r
o
(#
71
J 3
74
0.0024260.Q_._0J_4ll32
0.0137195
•0.02922940.00.3420J_0.0C69366
239
7 5 0.0133384 -0.01C61687677
0.01 17062-0.0127 5 42
-0.0197095-0.0327C6 6
787 9
-0.0218 53 3
0.00 92 9 64-0.0318510-0.0203373
Nk
8082
0.00472490.015009 3
-0.024C835-0.0031307
f 8384
0.0147954.0136483
-0. C063022-0.0128499
8586
0.0120114C. 0064121
-0.0158802-0.0210090
878 8
-0.0005357-0.0044726
-0.0256489-0.0273472
8990
0.0097819.0041435
-0.0164C11-0.020^282
9192
0.01591460.0150350
-0.0027159-0.0C81877
9395
0.01244240.0161770
-0.0128411-0.0045638
9697
0.01059650.0168952
-0.0122912-0.0019010
9899
0.0145079.0060881
-0.007C125-0.0128634
ICC101
-0.0031419-0.0226133
-0.0171022-0.C193237
103104
0.01665910.0111904
-0.0030150-0.0082022
1C51C7
0.01724330.0166931
-0.0011987-0.0017695
1C81C9
.01487390.0075433
-0.0035042-0.0068785
1
110112
0.01728080.0168364
-C.0CC6185-0.0008331
113114
0.01179320.0171687
•0.0024858C^0Q.C_2_L3_9_
116117
•0.01650910.0147550
0.00006010.0001160
118 0.0076665-0.00146 5 3
0.0QCC827-0.0000760
120121
-0.0114495-0.0217504
-C. 0CC6102-O.00C8524
122 .
1240.01692140.0163 383
0.0002784C.CCC9820
125126
0.01156870.0165199
0.0027318C. 0007191
V
128129
.01570740.0139921
0.0C199200.0037531
130132
0.00760120.01 58898
C. CC67C090.0013067
133134
0.01541160.0136359
0.00256580.0C47F99
135136
0.0108671C. 0073963
0.00673580. CC85512
137138
-0 .00 16327-0.0213374
G. 01022510.0111138
•
^
f LOADING - THREE R=0.66
RESULTANT JCINT I S PL ACFMFNTS - SUPPORTS '. - • '
f
i *
<
vj a I N T 1)1 SPLACtMENT-
~"Y"Drs"p'.
240
X DISP. Z CISP.
1
2
0.00.0
•0.1C3C8250. 1022472
3 C .0
0.0•C. 10 067*35
0 .09807205
6
CO.0
0.0947222•0.092142C
7
8
0.00.0
-0.088751 1
0. C8298161281
0.0188581.0.0202427
0.00.0
94102
0.02211 700.0230047
0.00.0
106111
0.02325060.0231492
0.00.0
115123
0.02293870.0224949
0.00.0
127131
0.02186970.0213101
0.00.0
RESULTANT JOINT DISPLACEMENTS - FREE JOINTS
J CI NT / nTCOIATCMCN'T // Ul^rLAutl^tlNI /
X DISP. Y DI SP. 7 CISP.
9 0.0029493 -0. 1027629
*
IC11
0.0024341.0018854
-0. 1017208-0. 1CC4184
}
1213
0.0004487.0057705
-0.0944473-0.1C146C2
1415
0.00479100.0C84707
-0.1005676-0. C997226
1617
.00702050.0053045
-0.0986357-0.0972638
1819
0.00285700.0000346
-0.0944844-0 .0884972
2021
0.01097370.0091034
-0.0970650-0.096C956
2223
0.01329600.01 096 64
-0.0940173-0. 0928414
2425
0.00820770.0 02 02 54
-0.0913330-0.0862467
2627
0.01536570.0125951
-C. C9CC692-0 .0890107
2829
0.0172144.0 139590
-0.0856597 '
-0.08443343C31
0.01053420.0061620
-0.0830733-C. C8C9619
323 3
0.0004418•0.0047225
0.070.07
7674251926
3435
0.0101439•0 .0185568
0.07•0.07
341530684 1
r
3637
0.0186957.0148182
0.070.07
908928539 I
3839
0.01911720.0157569
4C41
0.01215570.0048007
4243
0.01917420.01 62017
•0.07•_p.0 7
-0.0 7
•A* 7-
0.060.06
2 4 617254772 3079C^7_7_2_
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-0.C661C5A .24]A 6 0.00 6 2237
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6 .01904060.0163652
-0.06 35254r0j,06 2.5LVL-0.06 35 B7
4
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0.01298 5 3
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X
i
5253
0. 008 26 36.0048980
-0.0612895-0.C599C93
W^' 545 5
-0.00258340.0075594
-0.0585479-0. 0583681
5657
0.00344690.0189343
-0 .0569537-0.0601176
5859
0.01634650.0131 772
-0,0601341-0.0603136
6
6 1
0.01 10305.008 84 81
-0.0602210-0.0571992
6263
-0.0065358.0063384
-0.0530123-0.0524392
6A65
0.00205110.0093469
-0.0514541-0.0529612
6 6
670.0102455Q.0C53842
-0.0443453-C.0446598
6869
-0.00219300.0129518
-0.0463983-0.0337880
7071
0.00797880.0032407
-0.C35757C-0.0388199
7374
0.0188042.0182811
-0.0045525-C. C092C99
757 6
0.01777490.0 1559 59
-0,0140982-0. 0261866
7778
-0.0164591-0.0282611
-0.0431458-0.0419675
}
798
0.012368C0.0062914
-0.0270016-0. C31979C
R283
0.01999330.0197100
-0.0041555-0^0 8365_2
8485
0.01818290.0159995
-0.017C567-0.0210778
8 6
87C.0C8541C
-0.0005682-0,0278820-0.C3399C8
ee89
-0.0 566 7 8
0.0130251-0,0361774-0,0217569
9091
,00555560.0211943
-0 .0270922-0.0036035
9293
0.02002560.0165737
-0.0108625-0.0170316
9596
0.02154060.0141235
-0.0060532- 0_. 1 6 2_ 9_7_7
9798
0.02.01
2492293221
-0.0025207-C. 0092967
99ICO
0,0081731-0,0039338
-0,0170615-0,0226715
1011C3
-0,02937770,022J:76.8
-0,0255754tOj, 3 9_9 7 2_ _ .
104
107108
0,010,02
4927729509
0,020,01
2222298137
1C9110
0,010,02
0128730002
112113
0,020,01
2413057516
-C,ClCeBC8.rPj0gi5_8_9'A-0,0023504
-0,0091495-_0_sQ0g82 3 6_
-0.0011145-0,00 332 8 8
1141 16
0,02.02
2853519B40
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u
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Jii}119
i20121122
0. CI 96676.0._0_l_P_32 2g0.0016R'500 .JLnW9B0•0.02832500.0225301
0\00Cj^oq
c . c
0.00•b . bo0.00
011930_0_Ap_l_
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1244103585
242
124125
0.0 2 1762 5
0.0154 71 I
0.00127700.0035471
\
126128
0.02200490.020935Q
0.00C93900.0026074
r 129130
0.01867250.0102521
C. CC490970.0087534
132133
0.02117970.0205486
0.0017134CCC33641
134135
0.01B20620.01 45564
0.00627600.0088189
136137
0.0099854-0.0018889
0.01118830.0133561
138 -0.0277954 0.014497C
LOADING - FOUR R=0.83
RESULTANT JOINT 01 SPLACEMFNTS - SUPPORTS
JOINT / niCDIAPPfi/PNiT // U 1 o f^ L A L t r t i\ 1 /
•-
X DI SP. Y DISP. I CISP.
1 0.0 -0.12061262
3
0.00.0
-0. I 198094-0.1L83131
>
45
0.0CO
-0.1 158616-0.1126758
6
7
0.00.0
-0. 1102142-C^L0L0C8^
8
72•0.0
0.0237562-C.1CC491C0.0
8194
0.02548020.0278095
CO.0
1C2106
0.0289069.029 20 6 3
0.00.0
HI115
0.02907780.0288176
0.0CO
123127
0.02827180.0275043
0.00.0
131 0.0268158 CO
RESULT/5NT JOINT DISPLACEMENTS - FREE JCIKTS
J CI NT DISPLACEMENT
X DISP. Y DISP. Z DISP
9
100.0031106.0026188
-C. 1203024-0.1 10-50 2 2
» 11
1213
15
16
0.0021093g5_0p_0_6_9^2.'0. 0060840"0._0p_5_1582.
0.0089196.0075646
-0.1180554
-0. 11S047C-0_.1L8LB_2A-0. 1173815-0. 11^3381
'C '-
r#
-oUFcy-xy—>"j^vro^,7 y^mLi. -f^JLJt
J.
3
0,JK)36 715 r.C. lL?.3Aaj ^i^i19 . 0.0006068 -0.1065?72?C 0.0U5?31 -0.114887921 0.0098237
0.013905 6
-0. 1129284-0. 1120843
Z'i
7 4
0.01184640.009412 4
-C. 11C9487-0 .1094378
?526
0.00 32 34 8
0.016C189-0.1042223-C. 1086547
>2726
0.01365190.C179351
-0.1075939-0. 1049272
293C
.01520450.0121053
-0. 1037C24-0.1021562
313 2
C.0C80044.0013171
-0.0992308-0.0948226
3334
-0.0051 948-0.01 18782
-0.0916559-0. 0895265
353 6
-0.02204500.0194553
-0.0862257-0.1003339
373f^
0.01642700.0205508
-0. 09^1260-0.09 542 59
39AC
0.01717670.. 0137035
-C. 0940452-0.0922835
4142
0.00648320.0212454
-C.C87497C-0.0869650
4344
0.01737180.0120812
-0.0882561-C. 0856237
454 6
0.0092547C.008C09R
-0.0831172-0.0795674
4748
0.00228620.0210519
-0. C77563C-0.0825827
4950
0.0177281.0139268
-0.0825203-0.08 199C8
5152
0.01124270.0101297
-0.0792153-0.0775126
^ 5354
0.0063689-0.0026348
-0.07"49122. -0.0723736
555 6
0.00955330.0045579
-C.073437P-0 .0710635
5758
0.02092330.0 176943
-0.0782443-0.0762643
5S «
600.01412500.0120172
-0.0783524-C. 0778806
6162
0.0107557-0.0075371
-0 .0726472-0.0654454
6364
0.00799920.0 02 8 206
-0.0657695-0.0641607
6566
0.01165600.0128801
-C.C667241-0.0555369
6768
0.0068258-0.0023218
-0.0556248-0.0575572
697C
0.0163565C.010C837
-0.0421689-0.0446179
71 0.0042407 -0.0463627_7_3 0._0_2_3^8 97 -0,.g0 56_6_9_0
74 0.0230386 -0.C114700J_5 0.0_2_2_4P9 7 -0 . 1 7 5.6JJi
76 0.0196879 -0.03264577 7 -0 .019714 5 - 0.C53294778 -0.0340217 -0.0517729_7^ C^015_6_2ii2 r0j.03 36 2_9_9
80 0.0080361 -0.0398283_8_2__:! g._02 5i69 7 rOj.Q0 51.7_3_6
83 0.0248186 -0.0104151 -X^-i'
64'
0.0229160 -0,0212386
r.
a
li 1
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93
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0.0? 67^638"Q..JL2_52 095_0.0?CF9?5(I.J127 9 2 3 _0.0178399
. 0.!_C282.71C__
0.0243233
•— --0-iC341C23_-0.04221R9
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-0.027C6A6Z0j^0136839_-C. 0044843
---0^0L3.5J_6J_-0.0211831r 0_. 7_5_2_88_
-0.0202573-__^0_j_003l33 8
244
._0_l_p 4 6 5 3-0 .0044284ZQ_._0__3 5^ 5 4 90.02787570_5ILI38_5713
0.0288362•P_27_? 2 9
i
0.0249364PJJ92 9i
0.02889360^^i_8l64_L.0.0199086
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3O^t-00X59_3J___-0.0177277—-1.-0_3_4J6 69__0.0283131^Q-._0_2J3 6450.01958070.02J66800.0263487
-0_._0_2J^4 740.0131432.Q.._0_2 66 543^0.0258733 "
-C.2_C21^7_il_
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0.0032004-Pi.00 6.0_2_?i,
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106111
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0.00.0
115123
0.03512840.0344795
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127131
0.03357050.0327542
0.00.0
RESULT /iNT JCFNT 01 SPLACEMENTS - FREE JOINTS
JCINT /- ntCDI ATPMPMT -fIJ 1 brL AC tn riN 1
X DI SP. Y DISP. Z DI SP.
9 0.0031175 -0. 13 585351011
0.00264340.0021837
-0. 1346599-0. 1336191
1213
C.00C92340.0060821
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1415
0.0C519720.0C88908
-0. 1337583-0. 133C135
1617
0.00760430.0061927
-0.1319761-0.1306701
1819
.00423710.0013188
-0. 1281 IC2-0.1228356
2021
C.0114071.0098359
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2123
0.01362360.0117681
-0.1279249-0. Vlkll'b^
24 0.0097680 -0.12 5217425 0. 0C44985 -0.12037742627
0.010.01
5459634323
0. 12
:Q^L2.
48085
2829
0.0169824'O .0147843
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3C31
0.01249180.0096667
-0.1186211-0. 1155466
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APPEivDIX 7A
Lehigh speciLnen,no cut ,'CSTG type .
The following presents :
1- Plate division , numbers are node names , page 249
2- Input data, pp 250 - 26? .
3- Output data, pp 268 - 271 .
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LOADING - CNE UMFCPN
RESULTANT JCINT C ISPL ACEM ENTS - SUPPORTS
JCIM / DISPLACEMENT
X nisp. Y cfsF. Z CISP.
1 0.0 -C. 12362202 ^uSl -Q,121^A2i^3 C.C -C.1I«^8 7 36
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RESILT/!NT JCINT DISPLACEMENTS - FREE JCINTS
JCINT / DISPLACEf^ENT
X DISP. Y CISF. 2 CISF.
9 0.0034677 -0.1224273J_C -C.CCC S46 C =l£ . 1 C L9_£^JL
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360 28 37 272
361 272 37 38
362 55 27C 60
363 6C 62
364 6C 61 62
36 5 6C 7C 61i
366 60 69 70
367 6C 270 69
368 7C 69 273
369 69 82 273
370 71 70 273
371 12 11 2 73
372 72 273 83
373 273 82 83
374 98 97 107
37 5 98 107 109
376 107 108 109
311 1C8 107 118 l^
i I
< I
n
(•
378 107 117 118 294
379 107 97 117
3eC 119 lie 126
381 12C 119 126
m --^120 126 130
383 lie 117 126
38^ 126 111 129
385 126 129 130
386 1^9 154 156
387 149 148 154
388 154 148 165
389 154 155 156
390 154 166 155
391 154 165 166
392 167 166 274
393 168 167 274
394 168 274 176
^ 395 166 165 274
396 165 175 274
397 274 175 176
398 19C ;95 197
399 195 196 197
400 195 206 196
4C1 190 189 195
4C2 195 169 2C5
403 195 205 206
4C4 2C6 2C5 275
405 275 2C5 214
406 2C7 206 275
4C7 2C8 2C7 275
408 2ce 275 215
409 275 214 215
^IC 252 265 266v*.
ri
, ^
r%
€
A
411 i3 82 92
412 92 97 98-295-
ELENEN7 PRCPERTIES
I TO 412 TYPE 'CSTG' THICKNESS l.CC
cc^ST/i^TS
E 211CC.C ALL
PCISSCN C.30 ALL
JOINT RELEASES
1 TO 8 FORCE Y
139 144 161 181 230 253 259 FORCE X
$ FORCE APPLIED = ICCOO KG/CM , SIGMAO = 10000 KG/Cf**2
LCACING 'CNE* 'UNIFORM*
JOINT LCACS
1 FORCE Y -50.
C
17 44 52 78 89 FORCE Y -100.0
IC3 FCRCE Y -63.
C
114 FCRCE Y -13.
C
STIFFNESS ANALYSIS
LIST CISPLACEMENTS STRESSES ALL
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296
LOAC ING - CNF LMFCRN
RESIJLT/^NT JOIM C I SPL /^C EMENTS - SUPPORTS
JCINT / DI SPLACEMENT
X DISP. Y DISP. 2 CISP.
1 0.0 -0.20193192 c^j:; tSl.jzqi23ij^3 0.0 -0.2C1C719A ^0_a3 -D^I9_3„0_9 8A5 C.C -C.1884C45Jb (I*i3 -D^lflCeXL1 C.G -0.1842340.B {L._G rj:. ie426C_L13S 0.0A90257 0.0lAk C^(L41,C5S1 Q^Ql.
161 0.0A51241 C.Clei ^0_.J13J333] o^a.23C C. 0301626 C.C2 53 O.Q2i?2622 Q_^Q_25C 0.026625E 0.0
RESULTANT JCINT C IS PL ACEMENTS - FREE JCINTS
JOINT / DI SPLACEMENT— ^
X DISP. Y CISP. 2 CISP.
9 0.0056779 -C.2CC5199_il ! -O.CC57fliL3 -O.La5,64.3_7-
11 -C.CC678C5 -C. 1617326.12 ^iL._Q0i)4iaZ -Q^18^22S_8_13 -C.CC13789 -0.184523S.14 G_J10D1942 -£l^ia.4_L5AJ.
15 0.0000976 -0.1842617-i£ r.rcrcp'si -c. i84?p.7617 0.0135303 -C. 1986922
.-1-6 C._CC£_4JJjC _-i)-.1917_4_6^7_.
19 -0.0102445 -0.1865546.21 =0..QiO_Ui)9 ^.0_.iada2 3 8_.
21 -C.CC46685 -C. 1662653-22 =:D-*JI0iI3_3:15 !HD_._La_i2Q55_
22 C.CC02283 -0.1843486-2-4 C.CQQi).5j6j! -.C^ie4.25_6i.25 0.0000261 -0.1842555
—2Jt -C..C.C225ii _-0_. I9Z17A1_27 -0.0067840 -C.1664C6S2_£ -0.0121258 ' -0 .186835029 -0.0153171 -C.ie66CCS
.^33^ r0.02J8_468 -0_.i8 57^86 6_
31 -0.C326195 -0.1849236.-J^2 rC.C509J02 -0_,18 3^*5C7
33 -0.0663199 -0.1832954"
yC{l^3^ -C.CeiC89E -0.1832266
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3^5 -0.0885186 -0.1632286 297A3 0._016.61.7J -0.19Al^3J. :
272 -C.C2C376A -0.182A622.3Jb -Q.Q13.025Q -0.1 80-^37137 -C.C17232? -0.1788149
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44 0.0263463 -€.1693974A3 Q_^Q1A3 4_1_4 ^Q_U^_£ ZtlA.46 0.0075937 -0.1873911
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.33 (L^(1232.()91 -.Q^llZJtJJ^.54 -C.C222485 -0.15694CC
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59 -0.088-^990 -C.1549C5C.i?£ -Q_^03J.Ql.tL -_0_.15.^Q3_9J_
7? 0.0355526 -0.1637000_eJLi -C.C362ei9 -C,I49266C62 -0.0401840 -0.1490817.±2 -C.._C3J_4J22 -i)^ I4.6_L£9_1_
64 -0.0659213 -0.1474774.13 -a*J18J._0D2Z -D^L4.I4-_GB3_66 -C.C6g499C -C.1474C86_6J -Q.QQ68541 -C.l-^lZSXfi,66 -C.C183C72 -0.1500628
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76 -0.1435071 -0.1451906.7fi (L^(1A£A£3Z -_C^15_15A5_6_.79 0.0288072 -0.1517318JJi C.C12C74C :ifl.I50Z443273 -0.0435228 -0.1416688
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.33 -(1^(1A3J333 _-0_.i316Z3.L.64 -C.C7C7858 -0.1332305
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. _aS a . 0.5 6iJ 9 8 -0_, 1 2 6 9 S 8_9_
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90 0.0324371 -C.1243416-SJ. -0.022379 5 ' -0. 118737C92 -C.037691C -0.1174523-9J r.Q..C.lLi6_63 -_C,J 1 5 ti5„4._94' -0.1193924 -0.1162687S3 jQ.C 4.72.24 6 -Si... .1 5 9 04 „96 -0.0314350 -C.lCC3CCeSI. -0.0397659 -0.0999684 1
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ICA 0.0'i05571 -0.C93C32f.JQ5 O.^Q3J*^h2^ -D.^a9L8J0_31C6 0.0133721 -C.C91268C
.J.0.7 -a..Q5JD2^34 -G,C12A13^1C6 -0.0523913 -0.0877309
_1£S =LC^Ci;j:_£Af±J ^^(L.JLLLL2±tlie -0.0833365 -0.0876521
.All -C_*aC12511 -JD^Q8_?3.5i3_
112 -0.1314296 -0.0873218.113. -0,JU'O3QQ3. -Qj^aOl^ZlB.115 -C.CC98973 -C.C849969116 ^^^ 2J3^35Jt =LiI.j:i3.5^3J9J_
111 -C.C4C36C9 -0.0862561.A13 ^-_(IJL5CJ_841 -X^..C8.(L2J3_S_
114 0.0621446 -0.084181C. _iJ.-9 -(i^C323211 r C_. C 84 6i CJ_
120 -0.0608084 -O.Ce42C861/1 -C.C81 8527 =J3_^a4Z53J_122 -0.1037564 -O.Ce44C6c
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illk C.j:5J_82Ca -X).<i.ClSA2I3_
125 0.0336380 -0.0721706126 -C.C5C465C -C.Cei_C52<L276 0.0475014 -0.066188712L -a^ja_8i]£Z -£!.*ai250_Z6.
129 -C.C4C3645 -0.0728810130 -a.J15D^14a -0^Q.13ZQ5^_131 -C.C71C685 -0.073353913.2 -Q.C 9275 Xa -0.0 7^43 16133 -0.1147968 -0.0734047_l3-4 ^0^X36^554 -C^C.13_4iIl_5_
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136 0.0429457 -C.C362C32L3-S < n.03 33AL4 - Q_..Q-323 3 1 8
137 0.0450272 -0.0159471140 -C_._C22j61.IZ -i:^Ci.4£-C5-6_
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APPEEDIX 8
H-slit alujiiinura ,'OSIG' type .
The following presents :
1- Plate division , numbers are element names and node names ,
pages302, 303 .
2- Input data, pp 304 - 321 .
3- Output data, pp 322 - 325 .
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116 5.54C 4.111
117 5.6 5.4
118 5.6 6.2
119 5.6 7.0
120 5.807 3.981 ^->
l?l 6.0 4.? 308---
6.C 5.C
1?3 6.0 5.4
l?A ^. C 6.2
m 125 6.193 3.981
126 6.4 5.0
127 6.460 4.111
128 6.556 3.831
12<5 6.6 4.5
130 6.8 5.4
131 ^.8 6.2
132 6.8 7.0
13 A 6.831 3.556
133 6.84P 3.848
143 6.981 3.193
135 7.0 0.0 S
i
136 7.0 0.4
1
137 7.0 C.8
138 7.0 1.2
139 7.0 1.6
140 7.0 2.0#
141 7.0 2.4 -
142 l.C 2.8
144 7.111 3.460
145 7.2 0.2
146 7.2 C.6 -
147 7.2 l.C
148 7.2 1.4
149 7.2 1.8
150 7.2 2.2
.151 7.2 2.6
152 7.2 3.C
153 7.2 5.0 r^~Y-
r
15 6 T.A C.C S309
IS^, 7.4 4.4 * ' V
1 *^5 7,5 3.6
A157 7.6 C?
w— 158 7.6 0.6
159 7.6 1.4
160 7.6 2.2
161 7.6 3.0
162 7.6 5.4
163 7.6 7.C
164 7.8 0.0 S
165 8.0 0.2
166 8.0 1.0
167 P.O 1.8
168 8.0 2.6
169 8.0 3.4
i7r 8.0 3.8
171 8.0 4.2
172 8.0 5.0
173 8.2 CCS174 8.4 0.2
175 8.4 0.6
176 8.4 1.4
177 8.4 2.2•
178 8.4 3.0
179 8.4 3.8
180 8.4 4.6
181 8.4 5.4
182 8.4 6.2
183 8.6 C.C S
184 8.8 0.2
185 9.0 0.0 S
186 9.2 0.2 •'\-' A
n
-_-*r_:.i=
187 n.2 0.6-^10
138 9.2 2.2
iRq 9.2 3.8
^ 190 9.2 5.4^ 191 9.2 7.0
192 <^.4 CO S
193 C.6 C.2
194 9.8 0.0 s
1C5 10.0 0.0 s
196 10.0 0.2
197 10.0 0.6
198 10.
C
1.4
199 10.0 3.0
200 10.0 4.6
201 10.0 6.?
202 10.0 7.0
# ELEMFNT TNCTDENCES
1 1 8 2
2 I 9 8
3 8 9 12
4 9 a3 12
5 12 13 19
6 13 20 19
7 19 20 23
8 23 20 24
Q 23 24 31
10 24 32 31
11 31 32 35
12 32 36 35
13 35 36 4?
1^ 3 6 ^3 4?
15 42 43 46
16 43 A7 46 ^ ^ '
(^
i f 'fO H 1 ^n311
IP A 7 57 56
19 56 57 65
?0 57 66 65
^ 21 65 66 80
2? 66 81 80
23 80 81 80
2A Fl 90 89
25 89 90 103
26 ^0 102 103
27 2 8 10
28 8 12 10
2«5 10 12 14
30 12 19 14
31 lA 19 21
32 19 23 21
# 33 21 23 25
3A 25 23 31
35 25 31 33
36 31 35 33
37 33 35 37t
38 37 35 42
39 37 A2 44
40 AA A2 46
Al AA A6 48
A? A8 A6 56
A3 A8 56 58
A4 58 56 65
A5 58 65 67
B A6 67 65 8C
Al 67 80 82
A 8 82 80 89
AP 82 89 91 ^ w
Q
50 ^1 ^9 103
1 01 103 10^
52 92 91 104
5 3 92 1C4 105
•""5A
~93'~"92^ 105
55 93 105 106
56 94 93 106
57 9^ 106 107
58 95 94 107
59 95 107 108
60 96 95 108
61 96 1C8 109
62 97 96 109
63 97 1C9 110
64 101 97 110
65 101 110 111
# 66 101 111 112
^^67 112 111 114
68 112 114 116
69 114 120 116
70 116 120 121f
71 121 120 125
72 121 125 127
73 125 128 127
74 127 128 133
75 128 134 133
76 133 134 144
11 134 143 144
78 144 143 152
79 143 142 15?
80 142 151 152
81 142 141 151
82 141 150 151 %\^
^
83 Kil ^''^C 1503^
aA 140 1A9 150
P5 140 139 149 %
86 139 1A8 149
i 87 139 138 148
88 138 147 148
89 138 137 147
90 137 146 147
91 137 136 146
92 136 145 146
93 136 135 14 5
94 135 156 145
95 1A5 156 157
96 157 156 164
97 157 164 165
98 165 164 173
99 165 173 174
1 100 174 173 183
101 174 183 1«^4
10 2 183 185 184
103 184 185 1869
ICA 186 185 1Q2
105 186 192 193
106 193 19? 194
107 193 194 196
108 194 195 196
1 09 82 91 9?
lie 83 62 92
111 8 3 9 2 9 3
11? 83 93 94
113 8A 83 94
114 8 4 94 95
115 84 95 96 %^
^
116 RS 8^f 96314
117 ns <^6 <^7
118 87 85 97
119 87 97 101
•""12C 87 ~IC\ 112
121 112 116 115
122 115 116 121
12^ 121 127 129
12^ 129 127 133
125 133 144 155
126 155 144 152
127 88 87 112
128 38 112 115
129 115 121 122
130 122 121 129
131 129 133 154
1132 133 155 154
r133 152 161 155
13^ 1^2 151 161
135 151 160 161
136 151 15C 160
137 150 149 160
138 149 159 160
139 149 148 159
1^0 1-^8 147 159
141 147 158 159
1^2 147 146 158
1A3 146 145 158
14A 145 157 158
145 68 67 82
r^146 6 8 97 83
147 69 68 83
148 70 69 83 1.r^
r
1^^ 70 8 3 8 A315
^iTc" ~7r 70 84
151 7? 71 8A
15? 72 8 A 85
B r5T 73 72 85
15A 73 85 87
155 74 73 87
156 74 07 88
157 75 74 8f^
158 76 75 88
159 77 76 88
16C 86 77 88
161 78 77 86
162 78 86 99
163 e6 98 99
16^ 86 88 98
»
165 98 88 115
166 98 115 113•
167 113 115 122
16 8 122 129 126
169 126 H29 153f
170 129 154 153
17 1 153 154 172
172 15^ 171 172
173 1S4 170 171
174 154 155 170
175 155 169 170
176 155 161 169
177 161 168 169
178 161 160 168
179 160 167 168
180 160 159 167
181 159 166 167
182 159 1«5R 166
209 59 58 67
...J
316
fsT 158 165 166
18A 15R 157 165
1P5 99 <;8 113
lH'
186 9q 113 117
187 113 122 117
188 117 122 123
189 123 122 126
190 123 126 130
191 126 153 130
192 130 153 162
193 153 172 162
194 162 172 181
195 172 180 181
196 172 171 180
197 171 179 180
1
198 171 170 179
199 170 169 179
200 169 178 179
201 169 168 178
202 168 177 178
203 168 167 177
204 167 176 177
205 167 166 176
206 166 175 176
207 166 165 175
208 165 174 175
210 59 67 68
211 59 68 69
212 60 59 69
213 60 69 70
21A 6C 70 71
215 61 60 71 311-
^?T6~~61 71 72
-------^2 -73
21B 62 61 73
^ ^2~lT~62"~73" 74
220 63 62 74
221 63 7A 75
2T2 63 75 76 ^
225 64 77 79
22^6^ 48 58 5^
727 49 4fi 59
.-—— — •"" "
228 49 59 60
~"22q 50 4S 60
231 51 50 61
232" 51 61 62
233 52 51 62
_—234 52 62 63
- —~" ~~ "^ """
235 53 '32 63
_ .— -——237 54 53 64
._
2"3B 54 64 7R —— -
239 2 10 11
^240 11 1^ 1^
241 14 21 22
242 22 21 25
243 25 33 34
2A4 34 33 37
245 37 44 45
2 46 45 44 48
T47~ 175 174 184
-t'•/
2^P 1P4 1P6 1B7318
?A^ 1P7 IPft in^
250 193 106 197
751 175 18A 187
•'" -^-~1'P7"T^3~ 197
?53 3 2 11
254 3 11 15
255 11 lA 15
256 15 14 22
257 15 22 26
258 22 25 26
259 26 25 34
26C 26 34 38
261 34 37 38
262 38 37 45
263 3 8 45 49
L26A 45 4P 49
r 265 4 3 15
266 4 15 16
267 5 4 16
268 5 16 17
269 6 5 17
270 6 17 18
271 7 6 18
272 15 26 27
273 16 15 27
27A 16 27 28
275 17 16 28
276 17 28 29
277 18 17 29
278 18 29 30
279 7 18 30A
2PC 27 26 38 -i|'Vj.|i,). It; """"V"
?Bl 27 ?R 39
Zn? 28 27 39
281 28 3<; 4C
319
7fi'^ 29 2B 40
285 29 AO 41
286 30 29 41
287 38 49 50
288 39 38 50
289 39 50 51
290 39 51 52
291 40 39 52
292 40 52 53
293 40 53 54
294 41 40 54
295 41 54 55
296 30 41 55
297 54 78 79
i 298 55 54 79
299 79 78 99
30C 79 99 100
301 55 79 100
302 99 117 118
3C3 100 99 118
304 100 118 119
305 117 123 124
306 118 117 124
307 119 118 124
308 119 124 132
3 09 124 123 130
310 124 130 131
311 124 131 132
312 131 13C 162
1-t313 131 162 163 '%
31A 13? Ill H>332(D
31^ 16? IPI in?
316 16? 16? 18?
317 18? IRl lf>0
318 18? 190 191
319 163 18? 191
3?C 176 175 187
3?1 176 187 18R
3?? 177 176 188
3?3 178 177 188
3?^ 178 188 189
3?5 179 178 189
3?6 leC 179 189
3?7 180 189 190
3?8 181 180 190
3?9 187 197 193
330 188 187 198
331 188 198 199
33? 189 188 199
333 189 199 ?00
334 190 189 ?00
335 190 ?00 ?01
336 191 190 ?01
337 191 ?01 ?0?
PLEM^NT PRCPFRTIFS
1 TO 337 TYPE 'CSTG* THICKNESS 0.50
CGNSTANTS
F 10000.0 ALL
PniSSGN' 0.30 />LL
JOINT RFLFASFS
1 TG 7 FORCE Y
135 156 164 173 183 185 19? 194 195 FORCE X
[•^ .T$ PLATE LENGTH=?0.0» WIDTH=14.0
$ WFin WinTH= 0.?5321
* WFIO |.FNGTH= 10,0
tH-PRANCH WIDTh=? .OfEND CIRCLFS ON H-BRANCH RAniUS=l.G»
i H-RRANCH OVFRALL HFIGHT^R.O
i FHRCF APPLIFC 5*10**3 LBS/IN , STGMAC=10**A PSI
LnAniNG 'ONE' 'UNIFORM'
jniNT i.nADS
1 FCRCF Y -1 .0
90 FORCE Y -1.5
102 FHRCF Y -0.5
Q 13 20 24 32 36 43 47 57 66 81 FORCE Y -2.0
STIFFNESS ANALYSIS
LIST DISPLACEMENTS STRESSES ALL
T^^
i
J- 322
LOADING - CNF UNIFORM
^RFSULT/?NT JOINT niSPLACFMFNTS - SUPPORTS
JCINT /- - nTCDIArPMCK'T — —
/
- U 1 c^r L AL t rnt <% I
X DISP. Y OISP. 7 DISP.
1
7
0.00.0
-0.073A3O4-0.0732901
3A
0.0Q..0
-0.0730776-0^,0729944
5
6
0.0o,ii
-0.0730816-0.Q.730L94
7
1350.00.033096A
-0.07204100.0
156 0.03374390.0342816
0.0. Oj.0
173iP3
0.03A71610.0350607
0.0. o^a
1851^?
0.035221?.0352738
0.0.0
1?A195
0.03506120.0 3 A 8052
0.00.0
^̂RFSm_T/iNT JOINT OISPLACeMFNTS - FR ^F JOINTS
.jritlL /?:- --i.-_-_"-r=----_--^-DJ SP LA_CJIM_FIS!I---—_--- -r---"_-^7_
XJU^P. Y_DL^P_, JLJD15P.
C.QC03B38 -0.073 3216
.!£.
11.12.
131 4
0.0008696_CL.OD0^9 91.O.OC06163_Q_^ODJJ350.0.00171610.0013750
-0.0733365-Q^ai3.L.80J_-0.0730569-5...ai31_42_B_-0 .0730734-0.0729197
15It17
J-8_
—
i«;
2J1
0.0010726!1^Q_Q_0_62J34.
0.0003482a*.0_Q_L916^.0.00186170.0025135
-0.07268^3_-J)_.r)7 25 7_8_2_.
-0.0727473-i)_.07226_O^A_.
-0.0727860-0.0726^^52
21_22_
23._?_A_
2526
0.0020233_Q._a017_853.0.0025314.Q.._Q03_23 6Q.0.00261060.0020396
-0.0724604-^^Q723^L59___.-0.07226AC-.D^Q7 2.0 620___.-0.0718601-0.^7 1^ 668
2 7
.JLP_
29
0.00177430,0005670
•0.0713860i0j,0714 75;>.
313^
•0.0021129:0..,aQ5J)3it5.
0.00313280.0038582
0.0715338lO-*.Q7C5 3J_0_
•0.07159450.0713521
-'^\-\j-
33 0.0031379 -0.0711063.3_4 Q_»J10J'JGQ3 -.O..Q7aS_07i).-
35 0.0036385 -0.0708076Ji, (l.*_Q0Jt33Bt -0.,.QlC'3/-t^_2L
37 0.0035580 -0.070?6?0_31 D_*_00.2J10.Z3. -Q.«_QJi2R338_.
39 0.0021158 -0.06^^035. A_C - Q., 00. 60 3 3 -D aO 6.95_873_
Al -0.00585?fi -0.0691174.A2 (L^J:>1*D^Q2 -5.. Q6.9-.9A0J._
A3 0.0047221 -C. 069683144 0.0038979 -0. 069351 9
323
45 0.0035501 -0.0690725j^h a^00„432.4Z -£I^Q6_9(I40J7_
A7 0.00A9A54 -0.0688052.j^f 0_.JiOA1136L -D^a6_«A3J7_A9 0.0034790 -0.061765C50 0.0032827 -0.067175951 0.002871A -0.0668C65
.32 CL^OA^ltl rJQ.i.066_7_3_3X
53 -0.0005P93 -0.0665969_3J* ^Q_.JK13951^ =0^a6 65_5j4_9_
55 -0.0119862 -0.065367156 0.0044948 -0.068143257 0.0050247 -0.0679629.51 Q_*00425i8 -Q^0 6_L5A03__.-59 0.0038847 -0.0671682,tQ. .Q_trL035 397 -D^Q6.t3_6_6J__.61 , 0.0035236 -0.0655600_62 0.0029068 -0 .065C96963 0.0010473 -0.0649150.i)4 z(L,J:1D23Q3Z -D^a6A58_70__.65 0.0045575 -0.0673177.ht ^CL^-r)i^9aLi -D.^a6_77_if^_6__.
67 0.0042944 -0.0667530JiB 0.OQ3941 -0.0'6633Ag69 0.0037252 -0.0658606.m (L.I1D3£^S.6. =D^QA^3J^jD_71 0.0036651 -0.0647818.J2 ^a>JK)3JaQ9. rQ.^a642r).43_73
,0.0037941 -0.0636907
-JA 0.0027790 -0 .06^ 110275 0.0013413 -0.0627652.16. -CLJ:)3D213iQ. -Q^a6.25AZB_77 -0.0025117 -0.0622293-la ^a..JIDj4_a64.6. = JD^a6.L933-9_79 -0.0100906 -0.0614059J&H 0. 0045244 -0.066616881 0.0048639 -0.0665914
.-B2 a._a.0423f}B _rO^D6 6_LQ.6JL.
83 0.0036249 -0.0651482.3A a.,Q03_637 6 j-J?_.ri6 2PC3_6_.
85 0.0041040 -0.0621180-SL6 -0.0026128 -0.059446887 0.0041632 -0.0605264
.JL9 a.^QXL4.4326 -^^Q6tQ373_88 0.0011789 -0.0568728
._9jC aJL0_43234 r5^Qt6.(I64_8_91 0.0041477 ,-0.0655852
_92 .0037<^09 -0.0 6 5148493 0.0035161 -0.064639134_.: ._0.Q3_45J
C
-_0_. 6 4 0_5_7_
95 0.0034727 -0.0631787_9_6 a«-0_Q3_83 70 -i)jO6Z0.56_9_97 0.0043033 -0.0606865^" -0.0027762 -0,0562197 i
I
ciq -0.00Sf>3Q^ -0.055R311 324.J.CjQ -Ci._aLH_q^.35 -D^03A?^\0J101 0.00A62?? -O.O*^flAA?0
..J_C2 0_^00A6 62^ -_Q.*Q6AR.L9.3
103 0.00-^3145 -0.065*S/^Rq1^4 .003896 "^. -0.0651TA4105 0.0035645 -0.06474?!
_a0_6 0._0J^J.3ft07 -0^Q6_4.2.4Jl11C7 0.003324? -0.06355R3
_J^J3 Ql*_0_034 65 -0ji.2.6?A8_6J)
1 CP 0.0038693 -0.0613408110 0,004330^ -0.059181 6
11? 0.0048R75 -0.0550030_lli ^Q_.003O4ia -5^G5.19.r^_3
113 -0.00?a3?6 -0.0521471__li5 a*_CL0JLJ_T7S -X^(:>5CJ_7ii)
114 0.0060730 -0.0509330116 . 0045610 -0*049156 6117 -0.0065178 -0.0467495
._1JJ -(I^0J392Za -i}^0 4_6^?_44i>
119 -C. 0220553 -C. 0454178._1_20 ^a.c00J2025 -0^ 0.4.4 P.3J)J'
121 0.0051331 -0.0417383122 - 0,0030292 -o. 0416148123 -0.0066800 -0.0412960.124 -Q..jnj^_5966 -D^.Q.^UJyJ3^125 0.0091869 -0.0380371.126 -Q_^0_02J5 0? -0^,13 5.2666_
127 0.0075689 -0.0332357128 0.011Q179 -0.0310369129 0.0027760 -0.0315591
.13(1 ^^1*00_6603Q -0^Q29.7_I7_7_131 -0.0152785 -0.0300276.132 ^T_CLJ124^J39 -D^1V1Q3S}3.134 0.0151415 -0.02^^421133 0.0115719 -0.0255821143 0.0186964 -0.0203147
.13fi .Q_.J13-?J8Q8 -f3^aa25.Qi,_6_
137 0.03?0017 -0.0049034.138 i.^n3^3n{ti =i?.^aai3_a7_6_139 ( 0.0?91854 -0.00968031 ^0 0.0?71?58 -0.01?0938141 0.0?4690? -0.0144947
-IkZ fl,JD.219^ai -0^aL7_U41_144 0.016?348 -0.0200919
—145- .0_^333f>6a -Q.^0.aLi39_5_146 0.0327708 -0.0033322
—1A2 0.031 81 2 8 -0.005511148 0.0303741 -0.0076824
.1-49 a..Q7„83Jll -J)_,Q0.1^^VL.150 0.0262707 -0.0120431
-131 a^a237362 -_0_.i314ia7JL.152 0.0207542 -0.0165378
-L53 -0.Q0192O8 -0.02344771*^4 0.0054706 -0.0192248
--133 Q^(1131309 ^I>.^QL5.63_51_157 0.0339205 -0.0008688
.A33 a^0334^5 9 ^JD^QQ.26-L49_1^9 0.0310711 -0.0060579I6Q 0,0269425 -0. 0094164161 0.0210298 -0.0125308
.JL63: rO*.CL0.599i)6 rJl* 01P6.2.5_9
163 -0.0251634 -0.0191380.-133 : ajLa3_45.001 ^_D_.i].QC6 63_5166 0.0^29406 -0.0032119 1.
167 0.0295894 -0,0056314^
rr'*^
If^P 0.0?''*6^^^t -0.0078^70_Ji;.9 O.^.OI7_^«35? -.O.OO.ILIB^ ^25170 0.01317R7 -0,0105A6A
„JJ1 aj)iLB^^55. -D.iLOLL'tOAD
17? -O.nnnq?!? -0.012R492_JJ4 0.03A7qQO -0.000A6Af!17S 0.n3A?A09 -0,0013R73
. iJ6__ 0^03JJ^7A -D*_aa26X5_9_177 0.0273689 -0.00A1S33
. JJJB 0^*mA2^B5. -D*aa5't2-4i>_179 0.0133*SS'5 -n. 0064867IPO 0.00419 6B -0.0 07516181 -0.0055496 -0.00825^2.152 -Q_^0J..53D4:Z -D..aa6795J__.184 0.0^51406 -0.0001661
._l_86 a*.03J5i501 5.^aQ.(10_7_6Jl_.
187 0.034R302 0.0002307_1A8 0.0277874 0^0,0 8 040189 0.0137354 0.0012130
.J_90 -O.._Oj03i5 74 0j^0(ILlA73_191 -0.0251415 C.0011087
.J._93 ^0.*03312R6. 0_,0 003403_1C6 0.0347258 0.0007609197 0.034127O 0.00202 04198 0.0316787 0.0039048.19 9 ^0._03124LA: 0^0077J.31_200 0.0045618 0.0101940.2.CL -CL._01305166 Q^0L(L^2_63_2C2 -0.0251017 0.01C9661
.,t>
Thesis . . 107S82N476 Nguyen-Tien-Ich
A study on shrinkage
distortion of butt
weld.
-iCi Jbrt •. . D I SPL A T
I I AUO ! 1 c ' »•
I4CCT69 S10069
\
6-;
SI005
nkag,
^ study
on ofT:;--9e
""'tilt''" '°'^^82
1^
thesN476
A Study on shrinkage distortion of butt
m
'lit