Estudo sobre distorção em soldagem

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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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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Ill

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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.

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

•

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

•

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)

:

•

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.

8

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

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

w

Uc•H

•H

o•H•POQ)

U•HQI

XQ)

Si-p

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01

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W•HQ(Ufdo

II

rH (N ro ^ in <o r^ 00• •

O o O o o o o o o

Q)

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(D Qj U -P -H r- in CN CTi rH in o "^ KO

l3^iH > fd C! CM CTi CN cr> 00 00 .H 00 o '^

(d rH W fH (L) 1 r- rH rH O r-- 00 f^ CTi 00

M fd c cii e o • • • • • • • • •

0) X: fd W nH iH m "^ in in vo VX) KD «X)

> ^ -H --ri^ +J 'd 1 > |CM

t^ o•=:r •

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cy> <^ "vT r^ 00 0^4

(N •

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o• •

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w

^ in «x) «J3 r^

'=1' rH '^ U3 r-\ ^Q) o •^ o CN 00 <o 00 ro

g OJ • rH CN o r^ CM KD

fd o • • • • • •

(!)

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r- 00 o <T\ rH r- r-{

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m ^ in vo r^ t^ 00

CM o o rH G\ CN <y\ cr>

iH (N (N r~- vo r^ OJ in 00 r-iH •

O •

o•

rH•

o•

00• •

CT\

•

rvj

•

CN ^ in VD U3 r^ r^ 00

<X) O in 00 r-\ rH •>* kO '^

H (N "* in r^ in cyi o <D CM

CT> •

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ro•

00•

CN• • •

O• •

rH n •^ in <X) KD r>- 00 00

<N 00 ro rH 00 CTi r^ <o oo

o o rH a\ o\ -nT r^ 00 MD oiH •

o •

en• • •

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o• •

rH•

in•

rH ro •^ in >x> r^ r- 00 00

Q^-WUlWCU•riC

(U UUT3 WC

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)

•H C+J •HO t^(U

5h 1n •HP CO

w 1+J

h^ >H Cm (1)

< cu gEH 4:^ OJ

•p

c•H

•HU) fd+J MC pQ) CO

BQ) -p

c(d td

rH +J

a CO

03 C•HQ u<D

TJ

S

^U H CM ro '=:r in KD r- 00 (j\

II• • • • • • • • m

Pi

4->

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e —Q) C

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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ro•

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t «

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•

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T3r^ CM 00 KD 00

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•

rH• • •

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in•

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( <y\ 00 00 vr> CM r^ rH •

a\ n 00 a\ in •^ 00 r-» CN(N •

• • • • •

in• •

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iH a\ ^ 00 00 in (T>

rH CM (3^ CM o^ 00 00 in ^ r^CN •

• • •

C7^

• •

00•

00•

rH•

CM •^ in in MD r^ r- 00

in KD ^ 00 rH r- t^ "^

<o H CM CM CM "^ •<* "^ CM >X>

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• • •

CN•

CTv

• • • •

.H 00 •^ in «X) KD r-« 00 CO

<u --CO CO

CO 0)

•H ^(DUO'd CO c (

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

m(X)

<E^

en

4->

(U

w

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U(T5

(U

C!

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PC•H

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LD inVX) *X> in00 • 00iH •

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ro ^(^ in .H O^(N • in

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in•

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in•

CN ro^ -^ ro 00CN • r^ r-i <:r

CO • • •

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LD • CN CN VOQ) rH • • •

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00 <-\ cr> vx)

CN ro r^ r^ inrH • rH cr> r^.H • • • •

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CN in VD 00

in in r^ rHrH rH r^ •^

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CN in vxi 00

in CN roro r- CN r~- roiH • \o r-iH t • • •

CN in vo 00

in in ^ "ÛD 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

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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j Tlii- :'.

'.

;"-r -'::': ;:::: ;:

: : ': :::: n^^:::f ::n :•:'. '::'.':

:

;

'

:

r: : r

'rll

:.::r:: :

:;:•«;;:'''' :::; '.:.': iri: 1 xk .'d-

' '-^ 2

- .:t-;.I - . .

;::; ::;;-':':

::;: ^ •f-iii ^.: 1

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"J I* ^

i : i J_U."

\

:

:

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ry v- I\--) . :

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\

: : 4,

0- 1

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];.

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-

-:-

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.:,.. -• -'' .....--- •— —

1

1.

1

y t>

!i

T1

--—r—i-. --- -: —

J^' r"1

. --..«

- -.'.- „.

; LX1 ir

-

i

i

1

1

(

-:

t

.._.

-.-^^:=^.^ •:- —T'

.... .-..• -T-

-- -•-

!,:''^

:>

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—

j

1

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t

1.

1

--- ...

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i

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i

- :-: •'• ;;: . ;.. ... - - .„il

i "•

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-J

•>

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% 1

o

r 1

>->•J r >^ 17>

A1 "J

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1- . r-•

No:t 1* -^ -3

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


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


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


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

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

•:• :':: -H- ^r: •Hi-

r-- "•:; 1

:

'*r

;;;:'"h..;.; "" '- -" i.ui

.

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""

1

i

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-—r-- f:r

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rirr -•ir--:-

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i-^"ft '-.'}.

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—• -i-r- -i-ir

1k .

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\ . ; :: :;:;i;;^ :::: ::; ~~- -':.' •fir "•:- — • -"

-rr- r.t* --:-

i .:,.:.

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--;— .... :-:-

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•- ~.-. rr: —r- f::--—-

-rrf--- / '— — ~ '-' -.y: :f- -T-

::::'l

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•— — •-- :-

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~*'T* ;^~"" "^"

f"") ..

^

1.—.—

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-T.•" •--.:.;..: :-

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_ l'

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1

1

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I

V*""^

"/v" r% '

-)

f /^ r>» «»\ 4» ••' (.1

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,, rj .-4 «-J -2 rs Z >.^ I- ^ ^' ' ") ^^ - ;

^ 3 ^ M ^

'•J I J »* '4 «'

LEHIGH SPECIMEN

(No Sawcut)


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


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«

LEHIGH SPECIMEN

(With Saw Cut)


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.

61

W

<

-P

o

fd

•P•H15

C

e•HU(U

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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"


Force Applied = 5 X 10-" lb/in

a = 10 psiJ o

Young's Modulus E = 10 x 10 psi

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

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

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27 28 37 30

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39 22 21 30 311

4C 31 30 39 40

41 40 3 9 49 50

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43 61 60 68 69

44 60 59 67 68

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76 64 63 71 72

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FLEMEM PROPERTIESf

1 TO 316 TYPE •CSTG' THICKNESS .2

37 TO ICl TYPE 'PSR' THICKNESS .2

CCNSTANTSt

.

E 300C0. ALL

PCISSCN . 3 ALL •

JOINT RELEASES

1 TO 8 FORCE Y

47 55 58 66 74 82 9C 98 FORCE X

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JCINT LOADS

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LOAUIKG »TWC' • 1^ = 0.2*

JCIAJT LOADS

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9 FORCE Y -20.

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.

LOAD INC 'THREE' ^^ = 0.3'

JOINT LCAUS

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18 FORCE Y -10.

LCAOING 'FOUR' 'R=O.A'

JOINT LOADS

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STIFFNESS ANALYSIS

LISr CISPLAClMENIS STRESSES ALL

13

LCAUI MG - ONE K = C . 1.

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JOINT / \)\ SPLACLiMENT

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56 C.C17C744 -0.0027654.J}J C.,01 3 43-6i3 -^D^ 006^1-62-4-.

59 0.0140073 -0.0034514-fin 0. 01 r40 2- -o.oo6Sb m 7

61 C.0057607 -0.010^637.Ji2 QâQ-li^-li*2 .--0^131 5a 62^'L.

6 3 -C.CC2102 8 -0.0 18 1929.-6--4-_J. =C.-a0j6i:i^_g^ ^..0.^-02QLL4-9..

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bh O.Clll.'i')! -O.OOAOWl zh.J^S) Q.J:JIJ/a3£C ^..aC6dii.3^J

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7S C.L12-UC3 -0.00i;>l67.^_6 C-^lJ_2i)iiC -ij^(iaZZ-'l2Ji

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91 C.C112139 0.0002513112 . a.iijj:3i)B2 i}âac4xa2-93 C. 0086121 C. 0005602_Q_4 0.0056152 C.0CC640995 C.C012466 C.00C6271.Ji) ^JUj:-£L5_lî:4 iDâCQ5^'^_97 -C.012B944 0.0004011.33 0.^X1093326. i)âaL7Lî2-9_

100 0.0092382 0.0030051101 .0077172 0.004544 1

I •

102 C.C05C456 0.0061566.1C3. (^ôaiîbz jo*ac-Vx53Ji-

104 -0.C050369 0.0085373-xot .

,-(xji-i^M5az u*aaaa-L3J-

LOADING - SEVLN R

RESULTANT JOINT DISPLACEMENTS - SUPPORTS

JOINT'

/ DISPLACEMENT '

X OISP. Y OISP. Z nisp

1 G.O -0.0768688J2 CUD -0 . 7^-14 1 463

5

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7

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C.O -0.05978 21(IJX --j:_,i} 5 4/i2ii5_

0.0 -C.0 5C1130r.o -0.04 66 5 ill

C.02540C2 0.0aÔP] >{D35 D-.D.

0.G196082 0.0aâijjji35 S).ô.

74 0.0154283 0.0JL2 r . 01 41 951 ^0.

47S^S

5866

90 0.0131449 0.0.-^-8 L^(li.Ur3J^ D^D.

RFSMlTANf jniM T SPI A f.h M E r^ T S - FUIF- JlilNTS

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x ni.sp^. Y_DISP. Z--D4,SP«-

r. .ooj.'î n^ :£L. 7AO'^^410

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1

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24

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7

2823

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36

3839

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46

49.50-

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61-62-

63

65

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68.Ji9_.

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72

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79 -O.COCIOIO -0.00944 19

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81 -G.G16'i9'10 -0.0i;??952.33 aavL3-5.Zj.4Z -D . C C5-9J_?

84 C.01?n33 -C. 0011047.3.5 c.(.:09_ijjai -D^£aL9ZJj>86 C.00603C1 -0.003^989

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91 C.01267C1 0. 0002755.32 C^ 1 1 Ulk £).* aQ.Q.4_4il3

93 C. 0097220 0.00C6077J±^ 0.0Ci^33CrS 0.0006857-95 C.0013945 0.0C06554^_6 -Caia5_3^J.5 £!âQCS-6il5-

97 -0.0145783 0.00C3854_^L9 aaiia2-559 DâaL9-63i-100 0.0 1044 17 0.0033845101 0.CC872C9 G. 00 51 166102 0.0056993 0.0069294

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104 -G.C056972 0.0096009.I£5 ^C-.iU_4X.3£a_- X)*Ca9ii.Uiil_-

LOACING - EIGHT R=0.8

RESULTANT JCINT DISPLACEMENTS - SUPPORTS

JOINT / DISPLACEMENT

X IJISP. Y DISP. Z DISP

1 CO -0.0816633-2 C-^D -0.0792.5023 CO -0.0759463.4 Ô^ .^0*aZ0ii-5^4D-

5 0.0 -0.0646051.6 Ô^) -Dâ5.9iX72-2-

7 C.O -0.0545346-3 C*£ -n.0 5C8 22047 0.028-^854 0.0

.-5-5 uaL2-42Il-5-4 Qui}.

58 0.C2167C7 0.0.3jb auiLaH3i:3 sx^q.

74 C. 0169330 . 0.0-32 r . 01 '^5654 0.09 0.0144105 0.0-9_a aa)-L2J-fl-43 -C-Q-

RFSl.ll.TA NT jr iNT D I Sg| A O f" ME MT S - FREE JOINTS

X-JirSi^.^ JY^DISP-. L-£U^P*

9 0.001^2/? -n.0H0H557

10 C.002A66U -L.C^'^00^7 27.-IJ. 0-.G0-71ii70 -0-0 7.9 3905I? 0,CQ2?5b^ -O.C73fD959

.Ji3 = G.^0-DCi)3.C9 _î3-.a6a5.4-21

14 -C.001^767 -0.0627940_L5 -r. .(/Q] ^^-02 -G.0S7/.SHH—16 -U.G(:2 30 56 -0.0 5-?4'59:^

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18 0.C1C77C1 -C.076b671._l-9 L.-aOJa4iJ9 îD., 74 6-53-1

2 CO 14? 5 72 -0.072810^3_2J r .ri 0^976^H -G.C66912422 -C.CGC1811 -0.0621822.23 ûai023J.J7- îD-i:5-ZiC/+5

24 -C.0032121 -0.0535986._2-5 -CO 43:9J J -CQSai6A:i.426 -0.0090650 -0.0473883

_2-Z C .CI77371 -Q. 0672871 —— _2 8 0. CIO 7 133 -0.06 5186 5

.JL9 aai2X3î3ii -X)-0ixa3IU-5-

30 C. 0047820 -0.0547995._3j :_â .-00X12-092 ô-asi-6iio2-32 -C .0027585 -0.0493286

-33 -I .00 4 1331 -0. 047415834 -C. 0061001 -0.0455S45.25 ^C^Xli2J5-8Z -0-Q^3-4-L9j9-

36 C. 0236985 -0.0509101.32 (^^JXL2Ji:iBl ^0-048-77-3-4-

38 0.0250820 -0.0377371-19 ( .CO 5324 1 -0.03 7336140 0.00C7911 -0,0384618.^J. r=.C.Jja2JJi29. ^0-a3a4J7-1.2-

42 -0.0043543 -0.0399194.A3 ê-..aoj3055- .ô-os^s/^j-j-B-

44 -0.0155904 -0.0384098-A.5 .026] 252 -0.022 83 Û46 C. 0130166 -0.0247029

.AJ^ (^J^-CAShZ ^0-aa9-23-33--49 C. 0082490 -0.0203232

.-50 C.C03J^3Z -0-0Z5i>-4Jx3--

51 -0.C0C79C1 -O.C2955r^-52. I -C .0 38909 -O.Q32i:i42653 -G.0C799C6 -0.0331861-54 ^jL-XU_7.^33S. -0-012-7-9-1-6-

56 0.0213593 -0.0036277.51 (^.IU-63955- .^0-aa3-L3ilO-59 C. 0172888 -0.0044316J^ n . C 1 ? 6 1 3 q -0. 00 8 49] 7

61 C. 006^528 -0.0141193.-6-2 OÔOiZJJO r--0--CI9Ld2i^.

63 -C.00266C4 -0.0228724.-hJt :rC.^0£LiL?-1^3 ^JD-.025a9-3-4-.

65 -0.C186072 -0.0255132_6J C . C 1 6 3 5 8 6 -0.00 2 64Q 3

6 8 CO 13 73 58 -0.00 514 92.-6l9 L. 90-030 î3-OCaza63_.70 C0C374C2 -0.0130267

.-XI • =C.CJliAJ33 ---0-£i6fx523_.

72 -C.C0PC375 -0.0191021-13 -C0UL6-a3Z =^0.019^932-6-75 G.C158702 -0.0015396-7-6 C^C-LliL9-92 .-_0-,^DZB_a.i-9_.

n • C0103524 -0.0049923._7Ji 0^005-5004 --0..00 77.2-T0-

79 -0. 0001577 -0.010472680 -C0075774 -0.0 12^005

'.T -c .(; l!i l*^ ] 1 -u. ui. '>'•> n>v

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8 4 0.01 350 6H -0.00 173 35

.ul5 L^jXaUJO. -Ô..OC2LîiL.

B6 C.C06569'-3 -0.00 3''t627

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.B3 -u.j:iXLC2C -oâ(nap.ziL .

n 0.0138879 0.0002925.JJ^ C.ail2-833a jQâO.(X^lA3-.

9 3 C.C1C6462 0.0C06379_Ui r .006 9??^ 0. OOP 70 R 4

28

9 5 G. 00 151 16 0.0006 58 7

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9 7 -C. CI 59903 G.C0C3465.-9-9 (^jr^^3A2k X)^CaZLAix2-

lOC C.Cl 14494 0.00 369 79

101 C .0 09 5 6C3 C . C Q 5 5 8 H

8

102 C.C062445 0.0075656.aj:3 a^XLLUJîtiiZ D^caoziui?-104 -C. 0062526 0.0104727JLT^ - -C.X_L5^-61C iD^CLOîiyi)-

LOADING - NINE R=C.9

KESLLTANT JOINT D I SPL ACEiMENTS - SUPPORTS

J CI NT / DISPLACEMENT

X DISP. Y OISP. Z DISP.

1 CO -C. 0850353-2

' C—

C

-0.08264943 0.0 -C. 0793785-^ O-C ^0âZ4J-La45 CO -0.0680318-6 CJD ^0.*aa2Ai:Ai3

7 CO -0.0577312_a C-L -0.05384994 7 C03 10150 CO.5l5 ^C-.J?-2iaij56. -oâ.

58 C.C232868 0.0-66_: L..^C-2C13Z9. -C-a.

74 0.0180622 0.0_a-2 C . C 1 6 ^ 8 8 ] 0.090 CO 153533 0.0.^3 aîlL3iiZ3J: XUJD.

RES U L T A N T J n I N f DIS PL ACE MENTS - FR EE J (1 1 N T S

niNI j:==- --T_-_--^ --==- - "_-_aj 5i?L A-CE-HEnj-=----_-^---- =-—_-_/_

X._IH-SP-. Y_OLSJL. zâu_sp^

r. 0037056 -0. 08 4 238^1-101

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0.0024349 -0.0771531. _ a âaui933 t-jd^dzz Q-0ii8-

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16 -0.0 0?^. I') 3 -O.C^J66 1Crf pg.J.J t^C...0-C4 95l3 4 „-0.Q 5292.7a ---^

10 C. 01100.33 -0.08C0133._l-9 C.^DnjJ*£!.25 ^D^QIQl-UxB2C 0.01A5717 -0.0^63407

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22 C.0CCC462 -0.06566;]C.23 -CaKIZ2JJ6 -iD^D6Gri2il5

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36 0.0244242 -0.05530013J (LJlX3223t -i:â^xZ'L913.-.

38 C.02634C9 -0.04335763S COO 56 5 85 =01^40772740 C.00C8980 -0.0414421.^1 -C-Jia2315Z -JJ^Q4 2Z1ZJ-.42 -C.0045449 -0.0425252.^3 -aaiojjxia -D*a4.2.L9jj3^.

44 -C.C164535 -0.04082343i3 C. 02 7784 2 -0.027526146 C. 0134052 -0.0201706

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63 -C.00285C8 -0.0245219.64 -0 . QOiiJ-lZl .^£)*a^6iiX0S_65 -G. 0197060 -0.0272200-6J L . r- 1 7SL±Jlfl -n.0O?H57^68 0.0145584 -0.0055768.3S CL.âa-9_4J3j4 r--£L,_CQ95-a9A-.

70 G. 0038843 -0.0140553.-7-1 L-a^QJll3SJiA r--C..i:'lZa8-9_4_.

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77 C.G1C9639 -0.0053913.-J-B il^CJyhJBJ5 _rD..CQ813iL4....

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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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1?7 1 .6 Y..0

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130 l.B 0.6

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Ti? 1.8 1 . '^

133 l.P. l.B

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

lAC 2.2 CCS

141 2.2 0.?(

142 2.2 0.6

143 2.2 l.C

144 2.2 l.'^

14^. 2.2 l.B

14 6 2.4 C.C S

147 2. A 0.4

148 2.4 O.B

1^9 2. A 1.2

150 2.4 1.6

151 2.4 2.C

15 2 2.6 0.0 S

15^ 2.6 C . 2

«I^wwî^B^Mi»îejîi_wjjmjw ^ -E.1

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t

157 2.6 l.n

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162 2.8 1.6

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16 A 3.0 0.0 S

165 3.0 C.2

166 2.0 C.6

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169 3.0 1.8

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•" ~a~F^rFN T INCIOFNCES

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62 61 70

Ta 62 "ic 77

vf~70 ~lh 77 (

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----.__? 17

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23 AA 4 9 57 r

2A A9 62 571

1

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^

26 62 77 71

2~7~71

„_P6

2e 77 85 86

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30 76 75 85 1

31 75 84 85

32 A 3 17

•

—

33 A 17 23

3A 17 30 23

35 23 30 36

36 30 44 36

37 36 44 50

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3S 50 57 63

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41 63 71 78

42 71 86 7R

43 78 86 92

48 4 18

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A9 2^ IR.^.6.

50 IB 23 31

51 2? 3 6 31

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56 58 6 3 72

57 63 78 72

58 72 78 R7

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60 87 92 10?

61 92 101 10?

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64 91 90 100

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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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106 106 105 114

107 105 113 114

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109 8 19 25

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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

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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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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

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142 26 33 39

143 33 4 7 3 9

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146 53 60 6 6

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14^5 74 ttQ Bl

15C P.I 89 ^5

43

1-^

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153 1C4 118 lie

154 110 118 124

155 118 123 124

156 118 117 123

157 117 122 123

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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

18C 67 66 81

181 67 81 96t{

10? 81 95 964A

TfI S6 95 110

IBA 9 6 1 10 125

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

197 ^0 67 54

1S£ 54 67 82

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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209 135 141 142

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

i21^ <;7 1 1 1 126

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^

ÂR lî'^ I'fR 15546

?Ji(^^ lê 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

27<^ 153 158 159

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

283 161 16C ]<S7

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78 5 160 159 166

286 15S 165 166

28 7 \^<; leF 16 5

17? 158 16A 165

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 -cciêefc4 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

-0.C1C2O54-0^0093475

11 o.cQ_,0

-C. 0088387-ajL.0 0«3_74 3

1314

CC0.0

-C0Cei29g-0.0078257

15

75cc0.0037623

-0.0076729CÔ

64 0.0033174 0.0.iC : Ci^XJii5.2_ (I^JI.

ôq

1C50.00287230.0027397

CO0.0

w 1131 19

0.00259310_^002509L

ccaô

128i34

0.0023517CX022B97

0.00^Q_

1401 /nf

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

?^^ -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.OOfiCRÂ

%^C -0.0006377 -0.0077892-^1 .T_C_._0^3aa2.^ =Qâ0_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_ -Qâ_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 -XâaA.CJL3if81 -0.0004387 -0.0C44C83^^ -0.0010615 -0.004731983 -0.0C19478 -0.004778^-^-5 ^0.^11)2 970.1 -D.â-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_^^_02iîz'i zQ^'iQjineiyjICO 0.0077166 -0.0002'^69 51—̂ ^^J- Q-._C_C2.C59L _-0âC0a_^.i)2^'^^ 0.001343A -0.0015100—1-^^^ C.C0069]? -0.002 I??9iO'* 0.0001/-I79 -0.CQ26CT6

~ ~

_IC6—• :0_._00?40'19_ -Q^Q_0_C_4_0Ri}

^*^^ 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 zQâ'^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_â3J1^^ 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^^ -£îa_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 Dâia5_9_6J^^"^ 0.0016514 0.0004093^-^ 0.0^094R2 C.CQ C7^<;21^1 -0.0000412 0.0009532

-|f-^^T.a_ô_ojj4Qa i3^caL(L5_6i

1^3 -0.0022087 0.0010752^-^^ Q-._o_ojj76a Dâa.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

o

Ct

]2

_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

-0.^^191726-0.0201 2C5

IC11

0.0c.c

-0.0 184778-C,_0174939

12

J 3

.0

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

16

X iDlSP._ Y _L.L5Pj 2 CilP.

0.0022795 -0.031204717

la19

2L2122

-0.0001187-C,_e0r)_4i]7 7

-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

o

et

^': ). 1^:50090 -0.0187906-j-6 c^moj:^£8. -Qm.q.ibjsjj 53"^7 -0.0009056 -C.C1782453fl -3_.lOO_U5aL rG...Cl6_83C_83*^ -0.001123? -0.0160329^^ -C.C012^t73 -0.015^^ 39^»1 -0.0015 845 -0.0 14490 7

.^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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127 -0.0167630 -0.0145021.129 0.._ai_39171 -DiL0am7_8i)_130131

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Lr/\OIKG - FIGHT X=0 .80

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.

JCTNT DISPL /^CPMENT

_X_D1SP._ _Y _C_I3_P_, J_C1S_P_

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112 -C.C138322 -0.0211376.II4_ 0_,OJJ29L2. -a^Q_(UAA32.115 0.0144934 -0.0038031.iL^ (L,D1LZ±QSL -a^(mJD231.117 0.0060289 -0.01C2160_LLa 0.OC22 73 7 -0.QI 31851120 C. 0157704 -C. 00*20124.iZl D_^DI2^5.L4_ =aâQAZ29I!.122 0.0086712 -0.0069871.121 D^0D^9il/iL -Q.^(LCS33Ji2.124 0.0010481 -0.0117214-L2_5 -0.002380^ -0. 0134087126 ( -0.0095897 -0.0154177.L2_7_ -JLim52ZL4_ -O.^Q13S32Q.129 0.0152600 -C.0C11962.L3-(L D_>DI^9 5.72_4_ ra.iIQA39i?2.131 0.0021566 -0.0079276132 -0 .00 523 4 R -0.0099987133 -0.0132856 -0.0109919-125 Q_^0Jî,554 -D^1(1L^?JA.136 0.0065545 -0.0034901.121 ^^Q_._QO_0 9_61I -iDâCL517_43.138 -0.0087570 -O.0C645P71 39 -0.01 68797 -0.0069863141 0.01^0824 -C.00C1427

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143 0.0032649 -0.0017331;j44 ^^Q_^G_C^424 7 = J:^0C-2_5_L4_C.

145 -0.0123539 -0.0029828-1A2 0.012C949 (l^JiaiI56L7_

J.148 0.0069553 C.00C7626.J49 - Q..J)PI)J 20 J. (I6i6_P^_

150 -0.0081030 C.00C564C.i5i_^ - (I._0_L5 65 14 QâQ_0_15/U_1^3 0.0132433 0,0008246154 0.0095762 C.CC22213

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.

165 0.0122690 C.C0221fiCItt Q_._OOBaiO''t. ai0iI6_0_12i.167 0.C030079 0.0087350168 -0 .00^1^21 C . 1 r 1 9 c ^

) 69 -0.0116281 0.0106335.170 .-^0_,_OJ_521i4. £i.C_LC6i3iii.

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 DÔ.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^îZid =(:â6.fLP.x5J3.

ZC -0.00072?4 -0.06^^t?:i7

.21 d.-O^Jjîa^L rQâa.lJî/1.22 0.0050^2? -0. 07^92-^22? C. 001 6 79a -0.0 rAR46 1

2 ''^ - C . ^^ 7 I 7 - C . C 6 9 6 5 3 6

.2 5 ^C_.j:)J}i2i2LL =Qâ6_5i)3i.9.

2^ -O.OC16200 -0.061195A21 -_(L.J:X'2255.T rQâi633J.f.28 -0.0036A79 -0.05370B629 0.QK972 P - 0.079211

73

30 0.00A5612 -0.07^676711 (L,j:ir2)21U±. r:Qâ^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

-C.CC9720eQ_^J7913(I__.

-C. C495C65-Q^Q_lC166i

44 0.0066437 -0.0653920.4 5 0î)X:5£6i5. -£â6_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 rÔ^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 â..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

f

J

hi Q_Ol036'M / r{2rO.^C130 2

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 Ô.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.

104106

C.0CC49230.0176457

-C.C2C478C-0.0032201

1C7JOB1C9110

0.0 126 82

3

^3_._0nj5S8L__.0.0033350

-C.0C03319

-0.0074120___-Q^0LL7_4_8^

-0.0156276-0.0185933

111112

-0 .0068947-0.0146756

-0.0215281-0.0225887

114115

0.01855820.0154005

-C.0012C42-0.004J098

116117

0.0108048.0063129

-0.0076C59_ -0.ail0322

118120

0.0023419C. 0167982

-0.0141988-0.0C21694

12112^

0.0134248C.0C9I47C

-G.CC45660-0.0075469

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

._L'1q_ ____JL.i}r6919iQ_ -aJlC2J333_.137 -0.0010578 -0.0058181

._L18_ _-_Q_,i)i:52i2.4_ -Q»_(ICii5J9l_.

139 -0.0179729 -0.0075351141 0.P149964 -0.0001613142 0.0105553 -C. 0008599.1^1 (1.^S>3^^2lZ -^_âaL9_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

.JJLC -(I^0_C_8j54iC Q^(1QS13J33.151 -0.0166868 0.0003018

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. -Q.._0_lJ'-5.^+6 5 0^003947_8159 0.0121275 0.0029051J 60 Q_._Q0^92 9 6 OÔa5.?JJ_5161 -0.0003527 0.0066842 'j

^f

162 -0.0C84386 0.0073842

1 63

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0.0075109Q*.0.QZ33i>^ .J5C.OC63028QAQ.Q5Jl6i.i.0.01081410« 01127?70.0 113022

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*

\1

1

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76

APPENDIX 2

Elliptic slit , 'LSI' type „


1- Plate division , the numlDers are node names ,page 77

2- Input data, pp 78 - 97 .

3 -Output data , pp 98 - 121

7B_

H ^ p p r M r> T y -, I rI I

T P 'T r '"^ L T T i

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2- Input data, pp 1 24 - 131 .

3- Output data, pp 1 32 - 139 .

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RESULTANT JOINT DISPLACEMENTS FREE JOINTS

JOINT -DISPLACEMENT /


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0.1441357"0.09^52^7"0.1802178~ari81H09 8'

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4 9

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0.2180954"0.14681590.1311033"0. 1113^504^

0.0267071

0.0541130

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LOADING - FOUR R=C.83


JOINT DISPLACEMENT-

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LLI^UING - FlVt R=1.0

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JUINI / DISPLACI^MENT /

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1JOINT / tTTQDiArP MP M T — // UlorLALt"uNi

X OISP. Y DISP. Z OISP.

a C. 0288491 -0.7896380"9

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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 •

1

-j- 1;

141

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n

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1

1

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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

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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

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96 3.50 0.50

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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

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

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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"

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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

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>

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^''^

.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

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 '•

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-


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-


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 ''

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


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


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'-^

'

-0 . J 2_5_4 9 P _ ,_

-0.0034636-O.OO5O0I6

-0.0 18 15 2^-0.0168801-r. 01 65 25 1

-0.0 163125-0.015Q419-0.0156880

Jj5-0_

B

47484950

0.00656200,00485220.00716820.0035032

-0.0174525-C.0171O65-0.0149 079-C.Ol A620Ow 51

52-0.0001187-0.0019432

-0.0142730-0.014046

5354

-0.00^2 518-0.0060692

-0.013 77]

4

-0,0134 7745556

0.00766650,0056342

-0,0121742-0,012245 3

6263

0.00791550.0059362

-0.0096045-0.00 9 69 4 9

6465

0.00399890.0016314

-C. 0098 115-0.0102026

6667

-0.001C567-0.0020300

-0.0110 3^0-C.Ol 1201^

6869

-0.0034060-0.004 7 84 5

-0.0113295-0.0 1 12441

7071

-0.00681870.0068791

-0.011144'"-0.0091287

7273

0.00629650.0049978

-0.0080998-0.0082849

7475

0,00599060.00413 52

-0.0065743-0.0068878

7677

0.0G29C010.0060391

-0.C0744 5 7

-0.0044*3207879

0.00492170,0060445

-0.0049416-0.0017741

•8182

0.00571450.0'^43 78 5

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-O.OCb7 37-3..

-0.00') 3931

-0.000.3658-0.0 0J-52iL-

217

4 66.1L7A 76

LL8A 6 7

1 19

,00^1507. ll^Q(lLa.73S_-.

0.00127610..Q 0/17.0 3ii_.

0.0 04430.000 196 1

0.00 1162 3

:0^DJ59ia.-0.0024864

0.00390090. 0044026

468 -0.0000572 -0.00494S2_L2_C_ -I}j.0(l(l±T3A .-a..0D555-Z5..

469 -0.0004300 -O.CCfcC488A2J -.0 ,0.0.^(1332 -J1.,DD 635 6_8.

.

470 -0.0004775 -C. 0061622±22 -0> 00 O647R -0.0060562471A724 74

.415_

47 7

4 7P

0.0025867 -0.0003477_0_» iD.O 2 4 5.5_5_ - a^.C.0i:jX3 9

0.0022555 -0.0010023._D_.^DLI4.4_4^ -CL.i)iIL8^5I0.0012084 -0.0C2753C0. 00 7 096 -0.0037138

479 0,0004739 -0.0042509_4 8a D^QQaZT2jl -Q..Di)„4_3331.

481 0.0000159 -0.0067327.4 8 2_ -D^.IâO.ZZl(L -a^0JI6J5J23.^£3 -0.0005727 -0.0059805JlM -0. 0009] ] 3 -0.0059021123 0.0025266 -0.O0C6568

.-^£6 J)^JDiDZL3_3_C -a.Djli3i)5fi.125 0.0016540 -0.0021439

.-4:a'L D^QCLLUJm -0__.IlO.Z9_8ia.

126 0.0007700 -0.0039691488 0.0O OA727 -O.OO^Q-^??

-0.00 58346-C^XC5_6I51.-0.005A632

127 -0.0002037-4 8.9. -0.000 7 22 3

128 -0.0012521-AS.C ^^.o£!Ziaa.a -Q..iKir3i}Z4..

491 0.0026012 -C.CCC6096-^t-23 Q .0 2 446 2 -0.0 009 260494 0.0020346 -0.0016043

-4:9.5_ JD_.IIQ 15.02.4 -Q^îlQ23JlS.496 0.0011820 -0.0031159-4S.1 J?.,D00a33J> -C^JlCJBlhQ..498 0.0002468 -0.0045647499 : -0.0008522 -0.004781 4

500 -0.0014874 -C. 0047515-5-ai .DÔaZTdfLl -CLjlQJ^2.1ka.130 0,0026772 -0.000553930.2- .O^D0.Z5.1^ ^^C.JIGJIP.423.

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

0.0027518 -0.0004843. D.. Q 2<iL6Jl - (L..ilCXJ^ 1 a

.

0.00 244 19 -O.0C1G10 8

^L2_ JD..QQ224^3J_.513 0.0018319

,001 442-1-0. CC188C9-0 .00245"^^

a^

'.J1 ')

/3L6.

517

o . iJ 'J C 9 ^^ 2 6

_0.Q00 7.^'«B.

-0.000866 1

5.1-8 -G*Qaii325i5 1')

1 i l

.002Q0A4ii*i}il2i32JL5_.

- 0. 00?H976-0.CCJ13 5?-0.00=5 216 5

-C^.0C32Z6.5_.

-0.0CC19P2^0.0DC3-SLZ3_

218

5 20 0.0026fi8«L3>L^ JD^Q0.252i3_5 21 0.00 23236LkQ £l^QaZ1.2rL0_

522 0.00172311-^n .00 1 ?S74

-0.C0C61 16

_-jL.i3i)-0.349-6..

-O.OOUUA-.Q..^0..ai3fi9,2

-0.0019^6 1

-0.002^79752 i

.A42_524

_L43_

0.00 068 80-i].Q0.0Z56i>_-0.0008703

0(VL5_LD_6_-H

-0.00 260 39-r-Jl,i)i)2 64.5.L.

-0.C026C2 5

_-0_^iDD2 5 aL5-.

V

'nAf,

'-nyg^iftTir "inT "aatsm

1

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

c

r Xo•J >

._„^-^JJ

ftPPpNinTY e-»H-cl._TTi

Yv~D'r" p [_ A M f^ <^ T R'f^ ^ *=

I I M T T ^. M T I IT M r T c P c

jniMT rOOPHTMATrc,1

^1 . o

1 , C Q

u

6

-7^-

n

1

1 . c

-1

1

1 ^ 1 c: . O c; , O

T-T1 q 1 c.n o n , p.

on ^ .^^-^- -•^r^7n

'K'^T\

"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

r59~~"6orc~

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.

C

35.0

^ 1>'8~ ~65'.""o"" 50.0

"6"9" ~68.T"

70 69.0 33.0

---"69.~(D"'

~7'2~ ~7 0. C" 0.0 S

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

• —

-

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

,

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

"r20'l'2 57c 8"0.0

121 12'5.C ICO.O j^f'^.

r2Yl30.0 5.0

1 ? A 13 5.0 ro".o

r2Tl3 5 7o~To~.o

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

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 '

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^ !

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

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

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- '^

'

?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>-^

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


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


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


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_

7818377749

)-yO

'+5

C. 1 7 1 9 6

.00 7A09 2

-0.0^711 1 4

-0.C661C5A .24]A 6 0.00 6 2237

.Q.._0_Ql.^iJ5a

6 .01904060.0163652

-0.06 35254r0j,06 2.5LVL-0.06 35 B7

4

•0.06358465C51

0.01298 5 3

.0096548-0.0633 6 02-0.0621/25

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

- C.0CC29010.0000625

] 1 /

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_

C203AC9J_3_p_

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 '-

-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

li 1

-^91^2

93

-0.000354''f

.0 1645510.?lOO_7_L625_

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

----PiQi^_8.3_5_7_

-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

-_Q-.-P2J7iil^_0.0276352

--Q_.il?^j6 50_0.0132089

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_

.0184630• ^0_._0J2J182_

-0.0C182C6-Q_._0J3 73 3 6

-0.0115527—r0^02.12_C64-0.0281572~

--P^0117_0iU_-0. 004967C^Q_,_0i.35_249_-0.0019759

--0i00 2.9_2_4__5

-0 .0057924"-rP^0LL39j;_6_-0.0C1C268rQ_!j00j39^2.-0.0041720

_r0j.00_013_6_7_6

C.000C582~P^C0(11_C_9J_-0.0000316

-ZlC^_0CC376^-0.0013112

_ r J, 16_3_4_1

C. 0004332— 0^0 0L5_5_7i0.0043193Q_5j)0_l_L499______

0.0032004-Pi.00 6.0_2_?i,

C.01C7197- Q iO 0_2_L0 6J_0.0041362—Ci_CCr7i09______C.01C8254

-5i0 L37_2J

J

0.0163499-QiO_l_7_7_lJ 1

-0.136162 3~

^:0_5_1_3_5_3605_-0. 1339058^0_^13 159^8_4_-0. 1286461r0^ 1 2 (i4^2_99_-0. 1236860-0.116 7514

J_l

1C2

0.0291098_0!,.0_3JJ7.3,2_

.03305540.0352572

0.0_C.0_

.o'

0.0

— -..PJA

106111

.03560200.0354419

0.00.0

115123

0.03512840.0344795

CO0.0

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

•0. 1283832•0.1346282

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

-0.13C5931-0.1296018

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

-0.1215C59-0. 12C2646

3C31

0.01249180.0096667

-0.1186211-0. 1155466

3233

0.0026604-0.0052449

-0.1109441-0.1071942

3435

-0 .013 179 5

-0.024O740-0. 1047851-0. 1009749

3t37

0.0179192.0157872

-C. 1178100-0. 1 164249

3839

0.01826540.C162CC9

-0.1142333-0. 1 127592

4041

0.01432790.0088652

-0. 1105272-0. 1039273

4243

0.01300620.0161714

-0. 1111283-0. 1095176

4445

0.01.01

3618717672

C. 100. 10

4 3 918C 3 6 C 5

4647

0.01048050.0036489

-0.0958327-0.0924369

4849

0.01737710.01 56972

-0.1082509-0.1065831

k ' 5051

0.01385140?LQ.l_3j4-5i'I

-C. 1039626-0.0977237

52

335455

0.01.00

•0.000.01

267308.44 3?2160821596

•0.09_0_.09

O.OBO.OH

46515C13_0,5_

595 7590946

S6

3J.58

336C

0.00617B5-0t_0J_66 00 7,

.01''-.9Q''«9

.0._0JJ.2 7.2 7

0.0126A5qo.orn335

-0.0R52776-OilOfLQiL^A-0. lC^<4^7a

rP_, 10_2.LL5_5_-0. 099 312 2

-O.Cg9?5PC

c'A6

6 2

63-0.00806430.0101211

-0.0776229-C. C794264

6465

0.00404990.0145055

-0 .0 7694 8 2

-0.08 10 37 2

1 666 7

0.01599380.0086868

-C.066924C-0.0671262

6869

-0.00200580.0202447

-0. C686204-0.0506123

7C71

0.01262260.00 565 3 3

-0.0535321-C. 0579228

'

737A

0.02903100.0282540

-0.0067860-0.0137320

7576

0.02750620.0242565

-0.021C296-0.0391339

7778

-0.0224526-0.0391175

-0.0632498-0.0613659

7980

0.C1934550.0102026

-0.0402696-C.C476814

8283

0.03080200.0303854

-0 .0061926-0.0124672

8485

.02811660.0248348

-0.0 2 5428 1

-0.0314141- 86

87.0135962

0.0002818-0.0415140-0.0503886

8889

-0.00696130.0203347

-0.0533895-0.0323613

9C91

0.00918880.0325887

-0.0402440-C. 0053650

i

9293

0.03085390.025679C

-0 .0161696-0.0253263

9596

0.0 3 3099 8

0.0219987-0.CC9CC19-0.0241940

9798

.0.0345051.0297736

-C. CC37450-0.0137948

99100

0.0131827-0.0044 806

-0.0253163-0.0335885 .

IClIC3

-0.04111640.0340218

-0.0377666-0.00593C5

1041C5

.02322320.0351659

-0.0161448-0.0023586

1071C8

0.03407630.0304966

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1- Plate division , numbers are node names , page 249

2- Input data, pp 250 - 26? .


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$ BEFORE SAVhCLT

ELEMENT INCICENCES

17

9 18

17 18

17 -i;^ A3

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18 A3 45

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11 ^5 4 7 5 3

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31 125 276 127

32 125 127 135

23 135 127 136

34 127 126 136

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36 i2e<

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37 137 139 145

38 139 144 145

3 2 16

40 3 16 46

41 18 45 46

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46 8C 90 1 C 6

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46 1C6 1C5 125

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51 138 145 146 ^

52 146 145 162

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61 202 23C 271

62 271 22C 239

62 229 22C 253

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65 25^ 2 53 2 6C

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67 202 271 221

66 231 271 239

69 221 229 249

71 249 254 257

72 254 260 257

72 257 26C 263

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82 244 24 9 255

8 3 255 249 257

8A 255 257 263

6 6 222 221 244

67 244 221 249

88 202 221 232 -

8S 2C2 232 233

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91 222 244 25 6

92 244 255 256

9 2 256 255 262

94 256 262 264

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116 251 256 264

117 251 264 265

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123 ci 67 115

124 91 115 116

125 116 115 140

126 140 163 164

127 164 163 164

128 164 203 204^^

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131 252 251 265

260

132 IC

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148 69 81 83

149 8 3 81 92

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151 El SI S2

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153 C2 <;6 98

154 C6 '^A 116

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156 96 116 117

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158 117 126 13C

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165 14 7 164 165

166 165 16 4 174

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168 176 174 185

169 174 16^ 165

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171 185 188 190

172 19C 186 2C5

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175 174 164 164

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177 205c"

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251 84 83 93

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257 1C2 ICl 123

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262 99 12C 121

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270 121 120 131

271 12C 13C 131

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273 134 133 143

274 122 132 143

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27 7 143 151 152

278 132 1^2 151

279 132 131 142

28C 142 15C 151

281 131 13C 142

282 130(

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285 142 141 149

286 142 149 15C

287 153 171 180

288 153 152 171

289 152 17C 171

290 152 151<

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291 151 16^ 17C

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293 150 168 169

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296 171 179 18C

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301 169 \i^ 111

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305 179 176 167

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3C7 167 192 193

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31C 178 166 192

311 186 191 192

312 177 176 186

313 176 165 186

314 186 185 190

315 166 19C 191

318 194 211 219

319 194 193 211

32C 1,9 3 21C 211

321 193 192 21C

322 192 2C9 210

323 192 191 209

324 19 1 2C6 2C9

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32 6 211 21C 216

329 210 217 216

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366 14^ 165 168

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RESULTANT JCINT C ISPL ACEM ENTS - SUPPORTS

JCIM / DISPLACEMENT

X nisp. Y cfsF. Z CISP.

1 0.0 -C. 12362202 ûSl -Q,121Â2i^3 C.C -C.1I«^8 7 36

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RESILT/!NT JCINT DISPLACEMENTS - FREE JCINTS

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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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276 C.C333514 -0.0 4 67880-L23 TlL.n£Ji3A12 -0.052iAg_413C -C.CC95Ca4 -0.0534436

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STIFFNESS ANALYSIS

LIST CISPLACEMENTS STRESSES ALL

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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 -DÎ9_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.

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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-

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APPEEDIX 8

H-slit alujiiinura ,'OSIG' type .


1- Plate division , numbers are element names and node names ,

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2- Input data, pp 304 - 321 .


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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^

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


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. oâ

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_ÔDJJ350.0.00171610.0013750

-0.0733365-Qâi3.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_OÂ_.

-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 -Dâ6_«A3J7_A9 0.0034790 -0.061765C50 0.0032827 -0.067175951 0.002871A -0.0668C65

.32 CLÔA^ltl rJQ.i.066_7_3_3X

53 -0.0005P93 -0.0665969_3J* ^Q_.JK13951^ =0â6 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 -Dâ6A58_70__.65 0.0045575 -0.0673177.ht ^CL^-r)i^9aLi -D.â6_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 â>JK)3JaQ9. rQ.â642r).43_73

,0.0037941 -0.0636907

-JA 0.0027790 -0 .06^ 110275 0.0013413 -0.0627652.16. -CLJ:)3D213iQ. -Qâ6.25AZB_77 -0.0025117 -0.0622293-la â..JIDj4_a64.6. = JDâ6.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 â.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.ÛJyJ3^125 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 -f3âa25.Qi,_6_

137 0.03?0017 -0.0049034.138 i.^n3^3n{ti =i?.âai3_a7_6_139 ( 0.0?91854 -0.00968031 ^0 0.0?71?58 -0.01?0938141 0.0?4690? -0.0144947

-IkZ fl,JD.219âi -0âL7_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 aâ237362 -_0_.i314ia7JL.152 0.0207542 -0.0165378

-L53 -0.Q0192O8 -0.02344771*^4 0.0054706 -0.0192248

--133 Q^(1131309 Î>.^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

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6-;

SI005

nkag,

^ study

on ofT:;--9e

""'tilt''" '°'^^82

1^

thesN476

A Study on shrinkage distortion of butt

m

'lit

Estudo sobre distorção em soldagem

Documents

Transcript of Estudo sobre distorção em soldagem