SAN FERNANDO VALLEY STATE COLLEGE
Northridge, California
CREEP BEHAVIOR OF A LATERALLY LOADED,
,,
SIMPLY SUPPORTED SQUARE PLATE
A thesis submitted in partial satisfaction of the
requirements for the degree of Master of
Science in Engineering,
by
Alois Joseph Baumgartner
January 19 68
The thesis of A lois Joseph Baumgartner is approved:
San Fernando Valley State College
Northridge ,California
January
ii
19 68
ACKNOWLEDGEMENTS
Grateful acknowledg ement is given to the The s i s Com­
mittee , e spe cially to its chairman and my principal advi s o r ,
Dr . L.
J.
Nypan,and to D r . V . W . Ander s on for their guid ­
anc e in my the s i s effort and the opportunity to attend their
lecture s which provided part of the fundamentals required
to suc c e s s fully conclude this proj ect.
The efforts of
Floyd Lee of the S. F. V. S . C. Machine Shop are appre­
ciated for constructing a fine te s t fixture.
Appreciation is extended to the Atomic s Inte rnational
Divis ion of North Ame rican Rockwell C orporation for
their prog r e s s ive "Learn- While- You -Earn" highe r edu ­
cation pro g r am , and to tho s e AI per s onnel whose co­
operative attitudes and efforts contributed to the form of
this the s i s .
Since r e thanks t o Roge r D. Elliott who prepared the
computer adaptation for the problem, and to Dorothy R .
Miller for typing this the sis .
iii
CONTE NTS
P age
ii i
Ac knowle dg e ment s
Ab str act . . . . . .
viii
I. Introduction
1
Appr oach
3
Theory . .
6
A. Defl e ction s and Moments .
6
B . Str e s se s and Str ains . . . .
10
C . E quiv alent Load Function
12
II.
III.
IV . Illustrative Example . . .
18
A. Definiti on of Problem
18
B. Application of Equivalent Lo ad Function .
18
l . C r eep L aw .
.
.
.
.
.
.
.
.
•
.
.
18
.
2 . Stre s s and Str ain Invar iants .
20
3 . Stre s s - Str ain Continuum v s Inc r ements
21
4.
Dete rmination of Cr e e p Strains . .
22
5 . D etermination of Reus s Con stant
23
6 . Integration of the E quival ent Load Function
26
31
C . Computer Adaptation
35
V. Expe rimental Effort .
A. E quipment
35
B . Specimen .
.
lV
..
35
CON TENTS
Page
C . Calibration and Preliminarie s
42
l . Mechani cal C alib ration .
43
2 . Thermal Calib ration . . .
43
3 . Tempe r ature Di str ibution Acro s s Spe cimen
51
D. Ac qui sition and Evaluation of D ata .
VI.
56
l . Development of Deflection Measuring Sy stem
56
2. P e rform ance of Gauging Sys tem
57
3 . R e cording of Data
58
Evaluation of Data
60
4.
Di s cu s si on and C onclu s ions
63
A. Obje ctive s an d Appr oach
63
B . Expe rimental Re sult s . .
64
C . Analytic al Re sults and Comput e r Not e s
66
l . Theor etical Setup o f the Problem
67
2 . Method s of Evaluation . . . . . .
67
3 . Effe ct o f M ate rial Pro pertie s .
70
D . Compar i s on o f Expe rimental With An alytic al Re sult s
71
E . R e sults of Analytic al St re s s , Str ain, and Load R elations .
79
Referenc e s
88
.
Bibliog raphy
90
Appendix - C omputer Data .
91
v
ABS T RAC T
C R EE P B EHAVIOR OF A LATERALLY LOADED,
SIMPLY SUPPORT ED SQUAR E PLATE
by
Alois J o s e ph Baumgartne r
Maste r of Science in Engine ering
January 19 6 8
Pres ented is the analytical development o f a mathematical model
de s cribing the late ral elasti c, as well as lateral primary and s ec ondary c r e e p defle ctions of a s im'p ly supported s quare plate due to a conc entrated load at its c enter .
A clos ed-loop c omputer program writ -
ten in FORTRAN-IVH languag e ha s been p r e pared and adapted for us e
with the IBM-3 60, Model-50 Computing System.
Application of the analytical exp r e s s ions and the computer pro­
g r am is illustrated with a p ractical example.
Evaluation of the s ample
problem is bas ed on the us e of Type 707 5-T 6 aluminum (0. 187 in.
thick) as the te s t materiaL
Two s pe cimens ( 1 4 x 14 in. ) are evaluated
analytically by s i mulating c entrally located loads of about 100 and 90 lb
and test tempe rature s of 5 50 and 57 5°F, r e s pe ctively.
An expe riment , paralleling the theo retical effort, was c onducted
and is de s c ribed.
A compari s on of the experimental with the analyti -
cal r e s ults show s minor to moderate deviations of the c re e p induc ed
viii
late ral plate deflections (-1 5 % maximum at plate c ente r s ).
Facto r s
pos s ibly contributing t o the dis c re panci e s are indicated and dis cus s ed.
Two primary c aus e s for deleterious effects upon the r e s ults of the
analys i s are inaccuracy ( s c atte r band) of the expe rimental constants to
b e s pecified for the c r e e p law postulated , and inade quacie s in decisions
applic able to the compute r adaptation of the the o r etical expr e s s ions.
The latter s ou r c e of potential misjudgments involve s the integ rating
technique u s ed when calculating the equivalent c oncentrated loads at
any grid point as w ell as latti c e size of the g rid system and number of
nodal points thr ough the plate thickne s s consid e r ed in the analysis .
Re sults of computational variatio::1.s complementing computer pro­
g ram, sug g e st that for good accuracy in the type of problem studied
he r e , an exact deflection e quation ( e . g . , Navier , Levy , or Ritz ) rathe r
than an approximate expre s s ion ( e. g . , Alg ebraic Carry - Ove r Method )
should be employed.
The s e r e s ults further show that a s mall g rid
point lattic e {4 to 8 time s the plate thickne s s ) and a larg e r number of
g ene ral nodal points ( 8 to 1 2 ) thr ough the plate thickne s s enhance
accuracy of the quantitative r e sults .
ix
I. INT RODU C T ION
The ability of engineering materials to r e sist the effects of long
term expo sure to high temperatur e s is of c rucial importance in many
engineering problem s .
A c la s sical tensile test attempts to determine ela s tic and othe r
proper tie s inhe rent in a c andidate de sign material, e . g . , elastic
modulus, Pois s on ' s ratio, yield strength, ductility, durability, etc .
Frequently, however, the r e s ults from this type of test are not indica ­
tive of a material' s ability to r e si s t high temperatures for extended
periods of time .
Systematic appr oach e s leading to a determination of the relative
m e rits of heat r e s i stant alloys have b e en develope d . P e rhaps, one of
the most important of the s e appr oache s is the tensile c r eep te st, akin
to the tensile strength te s t, which attempts to determine the effect of
temperatur e and time upon the m echanical behavior of a candidate
mate rial .
Ideally, the tensile c r eep t e s t provide s a c o r r elation b e ­
tween rate o f specimen extension ( c r eep rat e ) and time a t fixed tem ­
p erature and tensile str e s s .
Unfortunately, in practic e, data from tensile c r e ep t e s t s cannot
usually be applied dir ectly to a particular design problem . This lim ­
itation stems from the fact that only very few structural elements in
high temperatur e s e rvic e devic e s are s ubjected to a uniaxial str e s s
field o f constant level .
In
g eneral, the prevailing str e s s field is multi­
dir ectional and/ or of non- uniform distribution a c ro s s the structural
element being conside r e d .
1
2
A physical model , repres entative of this cas e , is a plate s ub­
j ected to concentrated o r distributed late ral loads.
Thi s typ e of
loading condition may be found in structural high temperature eng i ­
n e e ring in the areas o f exhaus t c ontrols fo r rocket engine s , support
points on j et engin e s, heat exchange r s in nuclear c o olant loops , and
others.
Often, the characte ristic of primary inte r e s t in this type
of structural ele ment is its lateral defle ction behavior over extended
period s of time.
A preliminary literature survey indicated that refe renc e s de al­
mg with the phenomenon of c r eep bending in laterally loaded plates
are rare.
It i s in view of this ob s e rvation that the effort entitled,
"Cr e e p B ehavio r of a Laterally Loaded , Simply Supported Squar e
Plate , " w a s cho s en a s my the sis subj ect.
II.
APPROACH
Numerous theoretical studie s were made prior to 1 95 2 in an effort to analyze the c reep phenomenon, and particularly, its effect on
str e s s e s and deflections ; howeve r, most of the s e studies were bas ed
on s e rious r e s t rictive a s s umptions .
One of the most prominent er -
r o r s was the failure to inc lude transient c r eep . A r eview, including
an extensive biblio graphy of the s ubject during thi s pe riod, has been
(1)
p r e s ented by Schwope and J ackson .
In 1 949, Shanley
( 2)
p r e s ented an inc r em ental method whe reby any
arbitrary c r eep function c ould be applied along with the effe cts of a
varying stre s s level . His work has b een a dopted, sinc e then, a s the
basis for analyzing the uniaxial creep of s everal types of structur es,
(5)
{ 3 ) beam - c olumns, (4)
including columns,
and rigid frames .
The problem p r e s ented h e r e will b e s olved by the equival ent load
te chniqu e .
The development and initial application of thi s m ethod to
the problem of c re ep bending of thin plates is attributed to Dr . T . Lin,
< 6 ?)
Univer sity of California, L o s Angeles . , The fundamental idea underlying Lin's equival ent load technique i s to r eplace the actual plate,
experiencing lateral c reep defl ections due to a c onstant load, with a
fictitious plate having c onfigurations, initial load, and boundary
(l) R ef. l:
(2 )
(3 )
(4)
(5 )
(6 )
Ref.
Ref.
R ef.
Ref.
Ref.
(7) Ref.
1 3 , Chap . l 6.
6 and 1 4.
1 5.
1 6.
6.
7.
. •
3
4
conditions identical to those of the actual plate except for the stipulation that the material of the fictitious plate should behave elastically
. ·
.
throughout.
Since•this condition results in a reduced magnitude of
lateral deflection for the fictitious plate,
a time-dependent distributed
load is determined and superpositioned on the fictitious plate in such
a way that identical magnitudes of lateral deflection behavior exist at
all times between the two plates.
Deflection characteristics of the
actual plate are then determined by applying an elastic solution to the
fictitious plate.
For a given thin plate,
the sequence of operations begins with
defining the initial load, boundary conditions,
and test temperature.
While the deflection behavior of both actual and fictitious plates is
identical at time,
t
=
O, the actual plate continues to increase its
lateral deflection following a smooth time-dependent function.
termination of the time-dependent equivalent load,
q ,
e
De-
that must be
distributed over the fictitious plate to result in identical deflection
behavior,
is accomplished by using well-known relations from the
theory of elasticity and plasticity as discussed in the subsequent sections.
The various analogies between fictitious and actual plates, includ-
ing their effect upon mechanical parameters,
The equivalent load technique,
are shown in Figure
applied subsequently,
cable to materials with arbitrary creep laws,
the often used n-th power correlation
{quite
is appli-
and is not based on
inapplicable for some
materials ) between creep strain rate and stress.
1.
5
r
FICTITIOUS P LATE
ACTUAL PLATE
XJt
= h/2
h/2
-I
I
(;;;'L ELASTIC}
qe
---
•
O
p
o
,� ----
z=
.,.
"
-
_
-z-0
-z=
e-,.
I
EQUIVALENT
DISTRIBU TE D
LOAD,q
e
e
-j - ·
__,!
--,
2
·£;: '. �
"e
AT T;ME t: 0
e=e•·,e''=O
::__
·
�
-h/2
�
AFTER TIME, t =0
e= e ' + e"
-
-e·
+
z=0�/3=0
e" = 0
Figur e 1 .
{3
: z -:-
h/2
h/4;1/2
PLASTIC
STRAIN, e"
f = prop. le)
e"
{3
z:
h
t' =prop. (e')
e" =
z
I
e"
z =
/21
·
:
ELASTIC
STRAIN,e'
{3 n
Strain Distribution - Actual vs Fictitious Plate
III.
A.
THEORY
DEF LECTIONS AND MOMENTS
Conside ring a s mall element of the actual plate, it will become
evident that its elas tic and plastic deformations are functions of the
c omponent stre s s e s , effective upon the element .
Ther efor e, the
analytical routine begins with a determination of the initial ( elastic )
stres s distribution in the plat e .
The primary analytical exp r e s sion
de s c ribing it s behavior is the deflection equation, wxy Clas sical
s olutions for the elastic deflection of thin plat e s have been p r e s ented
( 1}
( Z}
in the form of a double sine s e rie s, by Levy
by Navier
in the
( 3}
form of a s ingle sine s e ries, and by Ritz
bas ed on the principle of
·
minimum strain ener gy .
The g eneral form for each of the s e equations as applicable to
simply supported plate s is as follows:
Navier' s Equation ( g eneral distributed load)
0<>
w=
0<>
)D [ [
m=l
•
n=d
( l ) Ref. 2 , p. 1 0 9 , e q. l 3 0.
( 2 ) Ref. 2 , p.l l 6 , e q.l 3 9.
( 3 ) Ref. 3 , p. 2 - l 9 , e q. 2 - 3 6.
6
•
•
(l}
7
Levy's Equation (uniformly distributed load)
Ol
m=l, 3,5
Ol
tanhOl +2
20l y
m ·
m
m+
m
..::-L2v
h
cos
0!
cosh
L
cosh0l
L
2
2
m
m
y
y
•
)
w=
w
00
!�[ [
m= l
.
20l y
.
. m7Tx
m
s1nh
s1n r:::-L
X
y
Ritz's Equation (concentrated or distributed load)
00
•
•
•
•
.
•
•
( 2)
.
Lx L
.
I
!0 1 Y p sm(m7Txl LX) sm(n7Ty L )dxdy
0
n= l
L L (m
X y
2I
L
2
X
+
n
2I
2
L )2
y
y
. m7Tx . n7Ty
Sln--r-sm
L
X
y
(3)
From these equations, the sectional moments may be determined.·
MX
M
M
y
xy
=
=
� � oZ�
(- -)
-D oz
ax
+'"
.. . (4a)
ay
2
a 2w
w
+1-L
-n "
2
ai
ax
=D( l - v)
... (4b)
2
a w
•
oxoy
.
.
(4c)
Evaluation of these equations is practical with the aid of computers; however, the approach taken here will not directly involve
any of the preceding equations.
Instead, the algebraic carry-over
(l)
method
which was developed by Profs. J .
J . Tuma, S. E. French, and
K. S. Havner of the Oklahoma State University will be used.
( 1 ) Ref. 3.
8
In this m ethod, a general r ectangular plate is divided into (25 deflec -
tion and 24 s upport) g rid points a s illustrated in Figure 2 .
The mag-
nitudes giving clef lection, s e ctional moments, and other variable s
a r e obtained with influenc e c oefficients multiplied by s cala r s, determined from s ys tem and material constant s .
��
S UP P ORT G R I D P O I N TS
30
D EFLEC T I O N
G R I D POI N T S
I
6.x = Lx/6
6.y=L i6
y
( �;)
t =
Figur e 2 .
Simply Supported Rectangular Plate
The general fo rm of the equations of the algebraic carry - over
m ethod for deflection and moments due to a conc entrated load at a
distinct grid point, xy, are:
Deflection
wk£
x, y
=
(1
-
IJ
2)
3Eh 3
TJ k£.L . L . p
x,y
x
y
k.£
•••
(5)
9
Moments
k£
[Mx:ll x, y = [(-��-�) y +217�� - TJ(:! l) y) + (-11(�-�) +217�� -17� �y+l)) t2]Pk£(in.-lb) ;
�
�yJx,y
r,
k£
�(
)
•
)
(
.
•
(6 )
.
k£
k£
k£
k£
k£
k£
21
= -17
t p (i n. -lb) ;
+27]
+ -17
+27]xy - 17
x(y :.l)
(x+l) y �
xy -17 x(y+l)
j k£
Ll (x-l)y
1M k£ =
[ xy x, y
J
[-17 k£
J
k£
k£
k£
+
+
(x-l)(y-l) 17(x-l) y - 17(x+l)(y+l) 17(x+l) (y-l)
!
4
-
'
f.L p
.
.
•
(7)
k.€ (in. -lb ) ;
•
•
•
(8 )
The total deflection or magnitudes of moments a t a distinct g rid point,
xy, is given as:
Deflection
1
2
= w
w
w
+·
xy
xy + xy
k£=2 5
k£
25 = '
w
w
x, y
xy
L
k£=1
. . . (9 )
Moments
. . . (1Oa)
. . . (1Ob)
. . . (1O c)
·Similar expressions for the sectional shear forces are available
but are omitted here since the effect of shear forces upon the deflection
10
(1 2 )
of thin plates is neglig ible. '
D etermination of deflections and moments are made for both the all - elastic fictitious and the actual plates.
To avoid confusion, symbols for deflec ti.ons, moments, stresses, and
strains are identified with and without prime notations. Unless otherwise stated, primes are not used for variables associated with the
Single primes signify elastic components of actual
fictitious plate.
plate variables. Double primes identify plastic (creep) components
of actual plate variables.
B.
STRESSES AND STRAINS
The biaxial moment- stress- strain components in the fictitious
plate at any g rid point are from Hook's law.
f
=
f
=
X
y
f
xy
=
12 M
X
z
=
12 M
Yz
3
h
=
h
3
12 M
h
3
x
yz=
E
E
2
(1 - f..L)
-
E
(l
+
J.l.)
+
!J.e )
y
. .. ( l la)
(e +!J.e )
X
y
... ( l lb)
(e
X
2
( l - !J. )
. . .( l lc)
e
xy
Based on the relation of total stra,in e to thz a,ctual plastic strain
e"
g iven by Equation 12 and to the actual e;lastic strain e' Equations 11
may be rewritten as given by Equations 13 to express stresses in
the actual plate in terms of strains.
Total Strain
=
(F ictitious Plate)
e
(1 ) Ref. 2 , p. 81.
(2 ) Ref. 2 , p. 172 .
Elastic Strain
+
Plastic Strain
(Total Strain in Actual Plate)
=
e'
+
e"
.. (12 )
11
X
f
I=
=
=
[- 1 1]
1 - 1-L
E
�(e - e11 ) +iJ(ey- e )l
yJ
X
2 X
1
E
e +iJe
y
2 x
II
- 1-L
X
E
e + e )- (e11 +iJ e11)J
y
2 � x 1-L y
1 -1-L
... (13a )
Equations 13b and 13c are obtained by following the apnroach
inclicated for Equation 13a.
f
I
y
=
X
E
X�
G( e
e
+ uell)�
r y +iJ ) - (ell
y
2
r{1-/J )
II
E
fl -(e - e )
xy
xy- ( 1 + iJ) xy
...( 1 3b)
... ( 1 3 c)
The elastic strain components are expressed as
.
•
.
•
.
.
.
•
.
( 14a)
( 14b)
( 14c)
The displaaement fumctions "u" and "v" in Equations 14 are definedas
follows u
=
u - z (aw/ax); v
0
=
v
0
- z(aw/a y).
Illustrated by figure 3 is the physical interpretation of the function "u!'
B y combining Equations 13 and 14 Equations 1 5 result giving the
stresses in terms of deflection w and creep strain e".
f
I=
X
a
(av 0
a 2w
a w2 ) ( 11
[ uo
11 )�
z
z
- ex +iJe
+
ox - 2 1-L -a2
y
y
dx
oy 2
1 - 1-L
E
.
•
.
( 15a)
12
_..X
z
j.ooollf---
Figure
u = (u
-
z
aw)
ax
3 . General Slope -Displacement Relation in Plate
J
"
e
xy
C.
0
EQUIVALENT LOAD FUNCTION
...(15b)
...(15c)
At an y grid point in the actual p late, the sectional moments are:
X
M'
M
I
y
=
=
2
h
/
l-h/2 X
/2
th/2
£1
£1
y
•
·
z
z
·
·
dz
dz
.
•
•
•
•
•
(16a)
(16 b)
13
1.
xy
M
h /2 I
fx
f
y
-h / 2
=
•
z
•
dz
•
•
•
(16c)
T e se0tional m<»ments are e�-uressed in terms of p!J.ate deflection and
h
creep strains(Ref. Equation
Equations 16 and setting u
�
M
=
-
1
E
h/ 2
2
[o ;'
2
( 1 - J.L } h/2 Lox �
0
1 7 ) by substituting e<'uations 15 into
= v
0
=
0 for this ary.-,lication
1-L o 'il 2dz
z
+
2
oy
J
E
1 -
E
1
2
J.L
h/ 2
_
h/2
1
h/ 2
--..,2
1 - J.L -h/2
M
I
xy
=
j
+�
1 + J.L
/2
h
-h/ 2
::-:
Y
2
z dz +
(1
f
h/2
� , } -h / 2
,_
(< + l'•;)zdz
•
•
•
(17 a}
( e II+ J.Le II} zdz
y
X
•
•
•
•
•
•
e11 zdz
xy
(17b}
(l7c}
Equations 1 8 are obtained b y integrating the deflection terms in
Bquations
M
l
X
l
M
y
=
=
1 7 and substituting D
-D
(
)
02w
02w
--2 + J.L --2
oy
ox
(
M
X
)
"2
02
-D __.:!!. + J.L __3!.
2
2
ax
oy
M
y
-
-
=
Eh3 / ( 1 -J..I- 2}
h/ 2 '
(e + J.Le ' ) z d z
f
x
2
y
( 1 - J.L } -h/ 2
E
...
(1
M''X
E
_
J
h/ 2
(e11 + J.Le11} zdz
f
X
2
J.L } -h/ 2 y
M"
y
... ( 18a)
.
.
•
( 18b)
14
[,
2
Eh
E
I �
M
+ ___
+
xy- 12{1+J.L) ox oy {1+J.L)
h/2
e II zd zxy . - h/2
3
2
h/2
E
D 1- J.L) ox w +(1+
e zd z
o oy
J.L) - h/2 xy
(
1
M
xy
II
MJ
xy
... {18c )
The above equations represent the moment interrelations between the
lJ
all -elastic moments, M , in the fictitious plate and the elastic mo­
.
I
lJ
.
lJ
ments, M , and the creep "relaxation" moments, M
.
plate.
.
Briefly,
M
X
=
M
y
=
M
xy
=
II
.
.
, in the actual
M I+ M"
X
X
y
y
I
M + M"
M1 + M"
xy
xy
•
•
.
(1 9)
The general differential equation governing the deflection behavior
of a thin plate is
(1)
.
.
•
{20)
This equation is obtained b y inserting Equations 4 into the equilibrium
equation for an element of the platE: given by Equation 21
(2)
...(21)
Both of these equations clearly suggest that they are applicable for
cases where a distributed load, q, is involved. However, it has been
{l } Ref. 2 , p 82 , eq.l03.
{2} Ref. 2 , p 81 , eq.lOQ .
15
shown that these equations also apply to loading cases inwhich for the
general plate element, q
=
l)
o,< e.g., loading consists of one or several
concentrated loads. Equation 21 is rewritten as given by Equation 22 .
...(22)
Equations 23 are obtained by differentiating Equations 18.
-
D
0
2
(
2
2
o w
o w
+ J,L
--z -2
2ox ox
oy
)
-
(1
-
E
( )
2
J.L ) ox
2
1
-2
0
h/2 II
II
(e +J.Le)zdz ...{23a)
z x
�
1
2 . h/2
11
11.
{e +J.Le�zdz ... (23b)
-�2 2 -h/2 Y
o J.L ) oy
E
2
o M1
2
2
xy _ 0 w
_o
1
2
(
J.L
2
)
D
- oxoy oxoy oxoy
0
_
!
2
2 II
2E
0
e zdz
- l + oxoy
xy
(
J.L)
- h/2
... (23c)
Equation 2 4 is obtained by substituting Equations 23 into Equation 22.
[
4
4
4
4
4
ow
o w
o w
o w
o w
+D4+ J.L 2 z +--:r+J.L
2 z+(Z-ZJ.L )
2 2
ox
ox oy
oy
oy ox
.
ox oy
E
1 - J.l
2
o
OX
! X"
--:o:-2 Z
-
-
2
h/2
II
( e + J.Le )zdz
y
l
2
h/2 II
II
o
(e + J.Le )zdz
---..,2 2
x
Y
1 J.L oy -h/2
E
-
I
2
2
fV
II
o
2E
e
zdz = - (q + q )
l + J.L dX'd'YJh/
xy
o
e
2
... (24)
The expressions in the brackets (Equation 24) represent the differential equation of the plate (Equation 20), and is equal to zero for
the plate being considered (q = 0). Therefore, the equivalent general
( l ) Re£.2, pp.l 02 and l 80.
0
16
time-dependent load, q , that must be applied to an all-elastic plate,
e
if it is to have the same deflection characteristics as the actual plate,
becomes
q =
e
E
1-L
2
--
1
-
o
2
f
2
ox
h/ 2
-h/ 2
II
+
1
-
2
j
hl2 II
II
(e +�-te )zdz
x
22
y
1-L oy - h/2
E
II
(e + IJe )zdz +
y
x
o
2
o
2E
( l +1-£) oxoy
h/ 2 II
e zdz
xy
-h/2
J
----
... (25)
If the bi-harmonic operator given by Equation 26 is substituted in differential Equation 20, an abbreviated expression for the deflection
behavior of a thin plate is obtained (Ref. Equation 27).
'il4
...(26)
=
4
Dtl w=q
... (27)
A more general form for Equation 27 is given by Equation 28 obtained
by the principle of superposition.
. .. (28)
When all q's=0 except for q =q ' Equation 28 becomes
i
e
4
Dtl w
e
=q
e
... (29)
Equation 29 governs the deflection behavior of an initially unloaded plate under the time -dependent distributed load, q , thereby
e
providing an analytical model for the creep behavior of a laterally
loaded plate..
17
A review of the approach taken, leading to the expression for the
equivalent load, q , suggests that the same technique maybe employed
e
to find equivalent loads reflecting phenomena other than creep; e.g.,
thermally induced strains, residual stresses, load relaxation, etc.
The solutions presented thus far are quite general.
Analysis of an
actual problem, however, is contingent upon integrability of the expression for the equivalent load ( Ref. Equation 2 5) .
IV. ILLUSTRATIVE EXAMPLE
A.
DEFINITION OF PROB LEM
To illustrate the procedure of using the equivalent load function,
a numerical example is selected.
II
II
To determine the plastic strain
II
components, e , e , and e , the material behavior is assumed to be
x
y
xy
as described by a specific creep law. In view of the intended experi-
mental confirmation of the analytical results, an aluminum alloy type
7075 - T6 was selected for the material since it displays perceptible
creep behavior at relatively low temperatures ( r--.0
.5 0° F), and its creep
characteristics are well known. The nominal dimensions of the specimen plate are 1 4 by 1 4 by 0.187 in.
The other primary independent variables; i.e., the magnitude of
the concentrated load applied at the centroid of the plate and the test
temperature, were selected after preliminary computer runs were
completed.
In these preliminary runs, load and temperature were
treated as open parameters.
B.
APPLICATION OF EQUIVALEN T LOAD FUNC TION
1 . Creep Law
Evaluation of the equivalent load function, q , requires aknowl­
e
edge of the uniaxial creep behavior of the plate material. For numer-
ous structural engineering materials, an empirical equation, approximating the uniaxial stress-strain-time relationship, has been established in the following form,
=e
1+
e
tl
e
18
Bf
k
t
'
... ( 30)
19
in which the creep strain component is
e
II
(t)
=
A
•
f� k
eB • t
•
.
.
( 31)
It has been shown that the creep behavior of the test material (Al,
707 5 - T6) is closely related to a uniaxial creep la w described by Equations 30 and 31.
A, B,
The values for the temperature -dependent constants,
k, and E, which give the best fitting curve through experimen­
(l)
tal data points, have been determined by Mathauser and Brooks
and
are shown graphically in Figure 4.
6
E x I0- (psi) IOK
9
B
7
6
300
350
400
450
500
550
600
T E M P E R AT U R E (0 F )
Figure 4. Temperature - Dependent Material Constants
Evaluation of Equation 25 requires consideration of the biaxial
elastic -plastic stress - strain behavior of a general element in the plate.
( 1) Ref. 4
20
2 . Stress and Strain Invariants
Correlation of the material's plastic behavior with the equivalent
load expression necessitates invoking certain well-known stress and
strain invariants associated with the theories of plasticity and elasticity.
The second deviatoric stress invariant,
is given as
J
2
=
J , for polyaxial stress
2
(l )
..!_3
[<fx - fy)2
+
(f
- f )2
z
y
+
(f
z
- f )2] + 2 (f2 + f2 + f2 )
x
xy
yz
zx
... (32)
which, for a biaxial stress field, reduces to
... (33)
( )
The second creep strain invariant for polyaxial strain is 2
I 11
2
=
[ 11 11
2 ex ey
+
e 11 e 11
y z
+
e 11 e 11
z x
_
( exy112
+
e 112
yz
+
e 112)]
zx
... (34)
which, for a biaxial stress field, reduces to
I II
2
2
2 (eX"
=
+
e ,,2
y
+
e II e"
X
y
For a uniaxial state of stress, f = f ; f
X
becomes
y
+
= f
e "2 )
xy
X
= f
xy
... (35)
=
0, and Equation 35
•
•
•
(36)
In a uniaxial state of stress, the strain components are related to each
other as follows:
e
II
=
e
II
Z
; e
II
X
=
e = -p,e
Z
y
II
II
;
e" = 0
xy
Since it is assumed that volume remains unchanged during plastic flo w,
the effective Poisson' s ratio may be taken as, p, = 0.50, and
( l ) Ref.
( 2) Ref.
. .
7 , p. 2 20, e q . l 8 .
7 , p. 220 , e q l 9
21
e = e
z
II
e
II
X
= e
II
y
= e" = 0
= e
e
yz
zx
xy
= -e I 2
II
II
Then Equation 36 becomes
I
II
2
=
"
(3/2) e 2
e
II
=
J2/3·I 2
11
.
.
( 37)
•
Substitution of Equations 36 and 37 into the uniaxial creep strainstress relation, Equation 31 gives a similar expression in terms of
the second deviatoric stress invariant,
invariant, I
II
2
, is obtained; thus
J , and the second creep strain
2
.
.
•
(38)
3. Stress,..StrainContinuum vs Increments
The approach taken to determine , creep strains from the deviatoric stress components is discussed in paragraphs to follow.
This correlation is obtained by applying the Reuss incremental plas­
ticity theory.
(l)
Considering the equations to follow
it will be appar-
ent that application of the Reuss theory to the determination of creep
strains implies that the magnitudes for the deviatoric stress components, S 1, S 1, and S 1 , are considered constant.
x
xy
Y.
This suggests that
the theory with the inclusion of a specific value for the Reuss Constant,
K, is applicable for small time increments, .6.t, during which the mag-
nitude of the stresses may be considered constant.
are shown graphically by Figure
5.
These conditions
Subsequently, equations involving
creep strains or the Reuss Constant will be taken in incremental form,
as required.
( 1} Ref. 8.
22
f
STR ESS
Figure 5.
Continuous vs Incremental
Stress-Time Relationship
----t
----�
4. Determination of Creep Strains
The fundamental relation,
correlating stresses with creep
{ 1}
strains, is obtained through the Reuss incremental plasticity theory_
This theory correlates plastic strain rate with stress; thus
. . . (39}
or equivalently
. . . (40}
where
D.e ��
1
=
€/'
1
•
D.t
. . . (41}
In Equation 40, the time increment, D.t, has been incorporated
implicitly into the Reuss Constant, K, as will be evident subsequently.
From Equation 4 0, the incremental plastic strain components are derived.
D.e"
X
=
KS'
X
;
D.e"
y
=
KS'
y
'
·
D.e" = KS'
xy
xy·
•
•
'
'
(2)
where s ' S1' and s
are the deviatoric stress components
given
x
xy
y
-------
(1) Ref. 8, p.38 and Ref. 7, p . 2 19 , e q. 16.
( 2) Ref. 9, p. 16, eqs. 56-58.
•
(42}
as
23
=
SI
X
s
y
=
I
s
l
u
y
3
X
xy
I
•
.
•
•
•
•
.
•
•
2 f1 - f1
:f
xy
3
l
y
f
X
I
(43a)
(43b)
(43c)
5. Determination of Reuss Constant
To determine the Reuss Constant, K, for the first time incre ment, bt , Equations 4 2 are inserted into Equation 35
1
after
taking
the square root of Equation 35, and considering its incremental form,
))
" 2 112
+ b.exy
\' xy
in which the first time increment, bt , is
1
+le "
e
II
X
also,
y
= e
II
xy
= e
II
I
•
•
•
(44)
= 0;
II-
0
2-
Taking the square root of equations 42
be�'
1
Equation 44 becomes
2
=
2
K S.' 2
1
from which is obtained
K =
---- -t:.fi!;
fi(:s 2 + 2 s
I
X
-
sI
y
+
--------:�
I
X
s
y
I
+
)
1/2
sI 2
xy
•
•
•
(45)
24
Still unknown is the quantity,
6Jii, which is derived from the uni­
axial creep law (Ref. Equation 31} and its modification in terms of
stress - strain invariants found in Equation 38.
The incremental form
of Equation 38 is
•
.
.
•
•
•
(46}
Equation 46 is substituted into Equation 45, resulting in
K=
(47}
Equation 47 is the expression for the Reuss Constant to be used for
the first time increment.
F or the second and subsequent time interval, the expression
6�,
for the Reuss Constant involves �he second creep- strain invariant,
(Ref. Equation 35).
The increment to this parameter; viz.,
must be determined without resorting to the Reuss Constant.
I�
This is
attempted by rewriting Equation 38 in: this form
•
.
.
•
•
•
•
•
•
(48}
Differentiation of Equation 48 with respect to time g ives
o CU
-vI'�
at
2
• ••
t
=
k
•
A
"f3T2
.:>' c..
•
•
B J3/2 ·J2
e
•
.
t
k r:u
= -v L,
k-1
t
2
(49}
(50}
25
Substitution of Equation 50 into Equation 49 yields
� J?:
at
Ul
then .�v 1
2
=
2
[
]
B J3 I 2 J 2 1 I
k
k /3i2 A·e
= ------.---,�;-:--:----1/ k - 1
2
!: R
t
becomes
�
g2
=
.
[� ]
.. .(51)
(j_ J?: )
.at
·�t
2
Insertion of Equation 51 into this expression yields
flfili A·eB J3I 2 J21 1I k
t-JI; k
[�j! l/k -1) J •
0
...(52 )
�t
Equation 52 g ives the incremental value of the second creep strain
invariant,
r;'
for the second and subsequent time intervals.
An expression for the Reuss Constant, K, for the second and sub-
sequent time intervals may now be determined analog ous to the procedure used for determining the Reuss Constant for the first time incre ment. Therefore, Equation 44 is expanded, which yields
JS � [
+ �
/2
ll2
= e
X
11
11
+ 2e • �e +
X
X
( )
�e11
X
2
2
+ e11 + 2e11•
y
y
(I
(
�e11
y
) ]112
2
II
2
II
II
II
A
A
A
A
+ ue • ue + e11 + 2e • ue
+ ue
x
y
xy
xy
xy
xy
•
•
.
(53)
26
Substitution of Equation 42 into Equation 53 g ives a quadratic in K
2
aK
+
bK
+
...(54)
c = 0
where
2
a = s'
X
+
b = 2 e11 s
'
X X
2
c = e11
X
+
s' 2
y
+
+
s'
X
2e11 s'
y y
2
e11
y
+
s'
y
·
+
II
1
e e1
X y
+
s'2
xy
+
e 11 s'
y X
+
2
e11
xy
. ..(55)
e11 s'
X y
+
(�
2 e11 s'
xy xy
+
�
2
... (56)
�)2
...(57)
wherefrom
K =
-b
+
J
2
b - 4ac
. .. (58 )
2a
Due to its nature, the Reuss Constant, K, must always be positive.
Therefore, only the plus value of the root in Equation 58 is considered.
The magnitude of creep strain, e�1 , at the end of the second and
subseq uent time interval,
strain increment,
1
� t, is determined by calculating the creep
�e�1 (Ref. Equations 42 ) , for the time increment
1
considered and adding the increment to the magnitude of creep strain
existing at the beg inning of the time increment.
6. Integ ration of the Equivalent Load Function
The expression for the equivalent load density, q
e
(Ref. Equa­
tion 25) , requires integ ration of creep strains across the plate thick ness. Therefore, a continuous function for the accrued creep strains
over the plate thickness,
,
z,
must be determined.
The variation of
27
creep strain (F igure 6) is assumed in the form of a power series .0)
"
e
(z)
"
= e
(B=l)
an
• tJ
.
•
.
(59)
where
/3 = z /(h/ 2)
z = -h/2
-z
---- ---
II
-e.
I
'----- z = +h/2
F ig ure 6.
�
{3 =I
Distribution of Creep Strain Across Plate
To determine the exponent, n, the calculated values for e�'
and
1/3=1 / 2
II
are inserted into the following equation
e.
1/3=1
II
II
= e.
e.
1/3=1
1�=1 / 2
where
'
( 1) Ref. 7, p.22 3.
•
II
e.
1[3=1
II
e.
1f3=1
•
.
•
( 60 )
28
or equivalently,
•
•
•
( 61)
II
II
and e.
have been determined from Equations 42.
wherein e.
1 �=1
1� =1/2
The integ ral expressions shown in Equation 25 are evaluated as,
l h/2e11
-h/2 1
.
•
II
zdz = e.
1{3=1
•
f
h/2
n
� zdz
- h/2
Chang ing variables from z to {3 based on the relation {3 = z/(h/2),
results in Eq uation 62.
. .. ( 62)
or finally,
(
h/2 II
II
h
e. zdz = e.
2(n 2+ 2)
1
1
i
h/2
{3 =l
I
)
•
.
.
( 63)
Equation 63 permits evaluation of the integrals in the equivalent load
expression (Ref. Equation 2 5).
'
The second derivatives in Equation 2 5 will be converted to ex­
(l)
For this
pressions obtained from the finite difference theory.
( 1) Ref. 10, pp.63-69.
29
purpose the central finite difference method is employed, g iving the
following equivalent operators:
=
2
o (F }
xy
2
oy
2
o (F )
xy
oxoy
=
=
2F
F
+F
xy
(x+l)y
(x- l) y 2
D.x
F
- 2Fxy + Fx( y+l}
x(y-1 )
2
D.y
-
F{ x- l)(y- 1) + F{ x - l)(y+l) - F( x+l)(y+l) - F(x+l)(y- 1)
F igure
D.xl::.y
7 illustrates a g eneral g rid point located at i
©
I
I
Q
©
I
y
@
=
•
.
.
•
0
•
•
•
o
(64a)
( 64b)
(64c)
(x, y).
X
© t
I
b.y
�+
b.y
�
I
@_l_
�6•+6•4
Fig ure 7 . General Notations for a Grid Point and its Vicinity
The finite difference operators represented by Equations 64a, b, and c
are presented in sketches a, b, and c, F ig ure 8.
The complete ex -
pression for the equivalent load (Ref. Equation 25} may now be assem­
bled, the integ rals evaluated, and the differentials taken as indicated.
30
I
2
a
(F
xy
2
ax
Y
1•�- -�
I
y
6x
I·
X
I
I
G)
... I ...
I
6x
@--I
.... I
------�-
I
--
2
(F
a
x Y)
2
ax
__
_
Y
--....
-��� .
-
®
1
Q
1
x
--
-(DY -
--
a.
-
b.
--
-
-------� X
l--�-1' "-�-y
(Fx
y
\tj)
)�-(
/
'
1,
)
\:tY
, -,
�-,
1---1
'-�'
'-..
)---1
'-.._/
T
I
I
I
I
I
1-
/
F ig ure 8. Representation of F inite Difference Operators
c.
31
The equivalent load expres s ion for a particular plate point yields
the intens ity of the dis tributed load at that point; however, the equations
! determining deflections and moments ( Ref. Equations 5 through 8 ) are
bas ed on inputs providing concentrated loads at the res pective grid
points .
Therefore, the dis tributed equivalent load mus t be converted
into dis crete concentrated forces acting on the g rid points .
This is
accomplis hed by integrating the dis tributed load function over the area
of domain for the res pective g rid points .
The s pecific integ ration
technique us ed will depend upon which routine is mos t compatible with
the computer s ys tem employed.
The equivalent loads are applied to the fictitious plate at the
end of each time interval, LH.
At that moment, the g eneral magni­
tudes of deflection are identical between the fictitious and the actual
plates .
On this bas is , new magnitudes for moments , stresses , s trains ,
invariants , and the Reus s Cons tant may be determine d for the actual
plate as outlined earlier in this s ection.
C.
COMPUTER ADAPTATION
The equations des cribing the elastic and plastic behavior of a
s imply-s upported rectangular plate were programmed for s olution on
the IBM-360, Model 50 dig ital computer.
in the Fortran-IV language, vers ion
H.
The program was written
A s implified flow chart of the
'program is g iven in F ig ure 35, and a complete lis ting of the F ortran
:s tatements is g iven in the Appendix.
The program could be recon­
s tructed by punching the s e statements into s tandard 8 0 - column IBM
·cards .
32
The computer prog ram, which was designated " P LATE," com­
prises a main portion and six subroutines.
The total leng th of the
' program, as compiled in machine lang uag e, is 35, 768 bytes. (Each
byte contains 8 binary bits.) In addition, several standard library
subroutines are required for operation of the prog ram.
When these
are loaded, the prog ram occupies 51,051 bytes in the core of the com­
puter.
The total capacity of the Model 50 is 462 , 000 bytes, and thus
the program uses only a fraction of the capacity of the computer.
The program was arrang ed so that multiple cases may be run,
one after the other.
At the start of each new case, all new data may
be supplied, or else the input data for the previous case may be re­
called.
The user may specify new values for as many or as few of
these input parameters as he wishes.
These new values, plus all of
the unmodified input parameters from the previous case, will be used
in the computation of the next case. Since the computer does not pause
for inspection of the results between cases, the user must specify in
advance all of the data for each of the cases which he wishes to have
run as a sing le computer job"
There are four types of input data card decks.
The first block of
data consists of cards containing numerical values for each of the 45
different influence coefficients that are used in the matrix of 62 5 coef 1
ficients specified for the Algebraic Carry - Over Method of plate anal ysis.
The second block of data consists of 9 cards containing infor -,
mation reg arding arrang ement and conditions of symmetry of the 45
33
distinct influence coefficients in the matrix.
These 18 cards are a
permanent part of the prog ram, and should not be revised by the user.
The third block of data consists of 10 cards containing identifica­
tion labels for e ach of the 50 input data locations that are used by the
program.
These cards are also a permanent part of the program.
The·fourth block of input data is specified by the user, as required
to define the problem.
The first card of this block must be a title
card, containing any desired description of the case to be run.
This
card is followed by as many cards as are needed to define the input
parameters for the first case to be run.
The last card in this set must
have a dig it punched in its first column.
If a
1
( one ) is punched, the
input data will be saved for recall at the start of the next case. If a
2 is punched, the input data will be used in the current case, but will
not be saved for recall.
If a 9 is punched, the data will be saved for
recall, but the current case will not be run.. The presence of either
a
1 or 2 punch terminates the loading of input data and starts the
computation of a case.
Each successive case must start with one title card, followed by
as many numerical data cards as desired.
The last data card for each
case must have a 1 or a 2 punched in its first column.
A more de­
tailed description of the data cards is g iven in the Appendix.
The program has been desig ned to apply specifically to square
plates with loading s applied at any or all of 2 5 points located on a
,s
x
5 g rid, with a grid spacing equal to 1 I 6 the length of a side of the
34
square.
The deflections, bending moments, stresses, and strains are
calculated at each of these 2 5 points.
The program could be modified
to apply to rectangular plates, and a finer grid spacing could be used
if the appropriate influence coefficients were supplied.
Modifications
to the main program would be minor, but the subroutine, PREETA,
would have to be rewritten.
The printed output from the computer has been made more lengthy
than would ultimately be required for a production program. The extra
output has been found useful in checking out the validity of the numerical techniques used in the computations.
!
Unwanted output could be
eliminated easily by removal of cards from the source deck of subroutine, WRITEA / WRITEB.
!.
V.
.A.
EXPERIMENTAL EFFORT
EQUIPMENT
A conceptual layout of the test fixture is shown in Figure 9. Shown
in Figure 10 is a photog raphic view of t4� plc:tte creep test arrang ement.
The fixture was custom-built for the plate size to be tested, and so con­
structed that measurements could be taken of the lateral deflection at
each of the 4 9 intersection g rid points.
For reference and calibration
purposes, the lateral plate deflection, if any, at the half - length of each
support edge could also be measured. A threaded stud and nut in each
corner
of the fixture prevented the specimen Qgrners from lifting off
the support edges during testing.
F ig ure 11 is a schematic diag ram of the p ower and instrumenta­
tion system.
The test fixture contains an upper andlower set of heaters,
each having a variable transformer affording separate power control.
By properly adjusting the power control variable transformers,
the
applied wattag e to each set of heaters during a test could be regulated
to about 9 50 watts (Figure 11) . Six LeedE! and Northrup, Model 8 69 0,
Millivolt Potentiometers provided the means for monitoring the tem­
perature at six points of the plate surface.
For the sake of economy,
an open-loop temperature control system was installed.
B.
SPECIMEN
The test specimenwas a 14 - in. - square plate of 0. 1875 in nominal
thickness.
The size of the specimen was determined from the length
35
GAUGE REF. PLANE
PARALLEL TO SPECIMEN
SUPPORT PLANE
INDICATOR EXTENSION
TOP COVER WITH
GAUGE LOCATION
HOLES
SPECIMEN
14 in. x 14 in.
HEATER
ELEMEN T S r�" �()
THERMAL
INSULATION
w
DEAD W EIGHT
Figure 9. Conceptual Layout of Test .Fixture
w
0'
37
FIXT URE FOR
COVER WITH
THER MA L
INS ULATION
VOLT AND A MPERE
MET ERS
PLAT E
SP ECIM EN WITH
TH ERMOCOUPLES
FIXTUR E
FRA M E
HOL ES FOR
GA U G E STEM
MILLI VOLT
POT ENTIOMET ERS
UPPER
H EATER S ET
A DA PTER FOR
W EIGHT HANG ER
VARIA BL E
POWER
TRANSFORMER
TH ERMAL
INS ULATION
.. •
Figure 10.
Plate Creep Test Arrangement
MILLI-VOLT (mv)
POT ENT IO M ET ERS
FOR THERMOCOUPL E
R EADOUT
I
TO LOWER
HEAT ER
�
A
�
v
VOLT AND A MP E R E
MET ERS
LINE 220v,l cp , 60 cps
�
v
HEAT ER R ESiSTANCE
UPPER
23.3Q
LOWER
26.8Q
VARIA B L E POW ER
T RANSFORMER NO.I
FOR UPP ER H EAT ERS
Figure
1 1.
TO LOWER
111111
HEATER
I
v
VA RIABLE POWE R
T RANSFORME R NO.2
FOR LOW E R HEAT ERS
Schematic Diagram - Power and Instrumentation
w
CXl
39
of the available s alvage heaters , which in turn, es tablis hed the s ize of
the fixture.
The plate thicknes s , finally s elected for tes ting, was as ­
certained from the res ults of preliminary computer runs .
Aluminum is a preferred tes t material becaus e its high-s trength
alloys are renowned for dis playing creep characteris tics s imilar to those
of ferrous alloys but, at a cons iderably lower temperature.
Since the
important objective in this effort was to analytically predict the creep
deflection of a laterally - loaded s quare plate, bas ed on the uniaxial
creep characteris tics of the tes t material, aluminum, Type 7075- T6,
was s elected.
The availability of extens ive uniaxial creep tes t data is
a prerequis ite of any candidate tes t material, and a literature s urvey
confirmed that aluminum, Type 707 5-T6, would mos t s atis factorily
meet the neces s ary requirements .
P hotographs of the two tes t s pecimens , ready for ins tallation into
the fixture, are s hown in Figure
1 2.
The thermocouples (T / C) were
arranged in a certain pattern that permitted a cros s -check of the indi­
cated temperatures between the two s pecimens at certain locations ,and
yet would facilitate effective utilization of the few available thermo­
couple s ens ors in thos e areas where creep behavior was expected to be
mos t s ens itive to temperature variations (Zone 2 , Ref. Figure
1 2).
Several thermocouple mounting techniques were tried before the
mos t promis ing one, illus trated in Figure
1 3 , was s elected.
This
mounting was s imple and economical, and s howed potential for main­
taining good cons is tency.
Some improvement could be made to the
mounting by s lightly peening the edge of the well - hole.
40
SPECIMEN I W I TH GAUGE ASSEMBLY
(INSTALLED INTO TEST FIXTURE)
CORNER HOLD-DOWN HOLES;
"'l/2in. dia. (4) PER SPECIMEN
SMALL CENTER HOLE
(0.093in.) FOR WEIGHT
HANGER ROD. BOTH
SPECIMENS
SPEC I MEN 2
Fig ure
1 2.
Test Specimens With Gaug e Assembly
41
HIGH TEMPERATURE
FIBREGLA SS TAPE
I I
I
....
rvl.50 in.--•JIIo---1......
"'
HIGH TEMPERATURE
INSULATION
. 50
---1
I
LIGHT ELASTIC
P RELOAD
ON WIRES
I
""I. 50 HOLE DIA .
HOLE DIA.= JUNCTION DIA. ± "'.002 in.----j
Figure
1 3.
f.--
Thermocouple Mounting
42
The the rmo couple junction c ombination was Chromel-Alumel. The
junction was w e lded by dipping it into a m e r cury bath of '"'- 5 0 -volts
ele ctr ical potential .
The high tempe rature fib r e gla s s adhe s iv e tape
s e cur ely bonded the the rmoc ouple wire s , and retained adequate adhe ­
s ion for the intended purpos e after expo sure to ,....._ 6 0 0 ° F fo r about 1 0 0 hr.
C.
CALIBRAT ION AND PRE LIMINARIES
Nume rous p r e liminary te sts w e r e conducted to calibrate the fix ­
ture and the gauge a s s embly.
Calib ration efforts may be divided into
two group s , concent rating on the me chani cal and the thermal a spe ct s .
The main obj e ctive s of the me chanical calibration w e r e t o : ( 1 ) de ­
termine the a c cura cy of the fixtu r e , (2) e stablish te chnique s fo r taking
readings conducive to good repeatability, and ( 3 ) a s ce rtain optimum
" ze ro" s etting of indicator gauge .
The primary obj e ctive s of the the rmal calibrati onw e r e to : (1 ) ve rify
output of the thermo c ouple s and temperatur e s indicated by readout in ­
strument s ; (2) dete rmine effe cts of various the rmo c ouple mounting
te chnique s upon output of the rmocouple s ; ( 3 ) confirm the steady- state
tempe rature di stribution a c r o s s spe cimen and fixture for a given wat­
tage input to the heat er s ; (4 ) ob s e rve range of temperature variations
acr o s s specimen at constant p ow e r s etting due to line voltag e fluctua ­
tions , change s in plate emi s s ivity c oefficient , and other cau s e s ; and
( 5 ) e stimate the highe st sugg e sted operating tempe rature for fixture.
The calibration r e sults r e lating to the s e obje ctive s a r e pr e s ented
in the following pa ragraphs .
43
l . Me chanical Calibration
The m o st important requirement r e garding fixture a c curacy
i concerns the dimensional relation betw e en gauge r efe rence plane and
i
;
specimen suppo rt plane.
Thi s relation wa s calculated by dir e ct m ea -
sur ement , us ing the indicator gaug e a s s embly s pe c ially made for the
plate c r e ep te st. The re sults of thi s dimens ional inspection a r e s hown
in Figure 1 4 .
The r e in, the b e st "bas e - line " r epr e s ents an ideali zed i
I
spe cimen support plane , paralle l t o the gauge refer ence plane for
establishing the z e r o s etting of the indicator gau g e . T o obtain the be st
repeatability when taking defle ction r eading s , a uniform procedure
involving s equence of te st points to be measured and attitude of gauge
bar with reference to the fixture frame was e stabli shed and follow ed .
2 . Thermal Calibration
C o r r e ct emf l evel of the thermocoupl e s and outputs of the potentiom eter w e r e a s certained with an industrial p r e c i s ion (mercury)
the rmomete r .
F o r thi s purpo s e , the junction o f T/ C 4 wa s t i e d t o the
bulb of the the rm om eter with a fine wir e .
The the rmometer wa s then
ins e rted into the t e st fixture w ith its bulb re sting a gainst the spe cimen
surfa c e .
Figur e 1 5 shows the temperatur e of T/ C 4 a s being ,..._ l 0 ° F
higher than the tempe ratur e indi cate d by the the rmometer . Thi s minor
dis crepancy m eant that pos s ib ly a low e r r efle ctivity coefficient exi sted
for the the rmocouple junction than for the the rmomete r bulb.
Since
1
I
radiation is the p r edominant m e chanism of heat transfer , the deviation
obs erved for that particula r a r ea i s readily explainable Consequently I
.
,I
1
- ---------- -------------·---------
+ DEVIATIONS
(in.)
D
+
�
INITIAL ARBITRARY IND ICATOR
SETTING L INE
/
I I
�
8
V///////4
I
I
12
•
I
2
I
4
V///////11
I
6
I
�
8
Figure 14.
C OR N E R OF T E S T F I XT U R E
AREAS TO BE USED FOR ZERO
SETTING OF IND ICATOR GAUGE
0.002
DEVELOPED SPECIMEN SUPPORT EDGE ( i n .)
10
=
0.004
BEST BASE -L INE
t'L.{.(L]
M I D - LE N G T H O F S P E C I M E N
S U PP O R T E D G E
I
10
I
12
�
I
2
�
- DEVIATIONS ( in.)
I
4
I
6
..-
�
I
8
I
10
I
12
!:LLLL.d
•
I
2
I
4
I
6
I
�
8
I
10
I
12
•
I
2
I
4
I I
6
�
Dimensional Ba s e - Line for Creep T e st Fixture
>+>­
>+>-
45
800
700
6 00
60
T H E RMO METER
Iz
0
a..
(/)
(.!)
z
II50 w
(/)
T/C 4
T/C 2
(..)
<(
T/C 5
a::
<(
>
T/C I
3 00
-9-
LL.
LL.
0
500
400
2
>
0
C\J
C\J
LL.
0
w
a::
:::>
I<(
a::
w
a..
�
w
I-
0
lO
T/C 6
T/C 3
U PP E R H E AT E R
(VARI A C I )
L O W E R H E AT E R
(VA R I A C 2)
20 0
I OO L-------�--L--�L---�
200
1000
4 00
600
800
( WA T T S / H EA T E R )
( A F TE R E O U I L I B R AT I O N OF T E M P E R AT U R E )
Figure 1 5 . The rmoele ctric Calibr ation fo r Aluminum 2 0 24 - T 3
46
4 6 0 i n.
.
6 . 40
�
� 1 4 in . �
T Y P.
>
w
a:::
::::>
!;i
AV E R A G E O F
'
TIC s I , 2, S 6
{ W/ N ± 5 ° F )
400
a:::
w
a..
�
w
1- 300
5 0 (!)
/
/�
'
A-/
{UPPER
/
----- V A R I AC 2
,
"
z
I­
I­
w
(f)
/
/
1z
0
a..
0
(\J
(\J
/�
(..)
<[
VA R I A C I
H E AT E R )
40
{ LO W E R H E AT E R )
1 00
o �-----L----�--._--� 30
300
400
500
6 00
700
800
900
1 000
W AT T S / H E AT E R
{ A F T E R EQ U I LI B R AT I O N OF T E M P E R ATU R E )
Figure 1 6.
Thermoelectric Calibration for Aluminum 7 07 5 - T 6
a:::
<[
>
47
in the sub s equent t e s ts , the temp e r atur e derived from a thermoc ouple ' s
emf r eading was ac c epted as being cor rect.
Shown in Figur e 1 7 a r e s e veral thermoc ouple mounting tech ­
nique s , evaluated during the early t e s t runs .
Generally, in a nearly
i s other'mal r egion, the the rmocouple attachment m etho d yielding the
low e s t t emp eratur e i s c onside r ed to b e m o s t accurat e .
This judg e ­
ment i s b a s ed o n the as sumption that the pr edominant heat transfer
m echani sm i s the one di s cus s ed above . For ac curacy and c onsistency,
the op en-well technique wa s p r ef e r r ed for attaching the thermocouples
to the t e s t specimen ( Figur e 1 3 ) .
Thi s type of installation was us ed
throughout the actual test runs .
The effecti ve.nes s of the thermal insulation in the te st fixtur e
was inv e s tigated. For this purpo s e , two the rm o c ouple s wer e install ed
at the heater -to -insulation ( T /C 5 ) and at the insulation - to -fixture ( T /C 6)
interfac e s , r e spec tively . The temp e r atur e l e vels w e r e obs er ved at the s e
locations and, wh en c ompar ed with the specimen temper atur e , provided
1
a m e asur e for the fixtur e s the rmal effici enc y . P r e s ented in Figur e 1 8
i s a c one eptual c r o s s s ection of the t e s t fixture and the the rmocouple
locations e s tabli shed for thi s pr eliminary t e s t . The prevailing plate
t emperatur e , TP L ' i n g eneral, r ep r e s ents the average o f s eve ral ther ­
mocouple s located about the mid - and c enter - r egions of the plate . The
purpo s e of T /C 3 was to monitor the effect of the plate support edg e on
the sp ecim e n temp er ature in that ar ea.
Given in Figur e 1 8 a r e empiric al equations c orr elating the tem ­
p e r atur e s at thermoc ouple locati on s , No . 3 , 5 , and 6, with the nominal
48
H IGH TEMPERATURE
F IBRE GLASS TAPE
L IGHT INSULATED
COPPER WIRE ( 0.005 in.)
�
(2) SMALL THROUGH
HOLES (�0.02 in.)
TI C
L EAD
I
�
�
"'
I 1
0.04 in.
VffM
_
SPECIMEN
INCONSISTENT e m f OUTP U T
POSS I BLY D U E TO
" EL ECTRICAL L EA KS "
T HROUGH T I E - W IRE
DOES NOT ASSURE GOOD
CONTACT B E T W E EN
TC - JUNC TION AND PLAT E
H IG HLY INCONSISTENT e m f
a) TA P E - O N
--'j r
b) T I E - O N
0.38 in.
"-' 0.30 in.
t
t
I
1 -- "' 0.20 in.
"' 0.60 in.
I
I
�WOj
e m f TOO H IG H DUE TO
LOCAL RADIAT I V E
H E AT A BSORPT ION
c ) I NSULAT E D W EL L
Figu r e 1 7 .
l l r(JUNC.
I
:
I
1
I
d i a. +
0.002 ln.)
1 - 1/2 HOLE
dio.
""
e m f IS CONS I S TENT
d ) OP EN WELL
Evaluated The rm ocouple Mounting Te chnique s
l
49
A P P L I C A B L E E M PIR I CAL EQUAT I ON S
T
H
=
2 . 3 3 TP
L
-
1 378 x
.
TF I X T. = - 1 3 5 + 3 . 0 8
I-
..J
Q.
10-3 TP 2 (° F )
L
�
TP L 2. 2 2 x i0-3 TP (°F)
2.0
1
.
0
..J
Q.
.......
..,:
I-
.......
J:
u:
X
I-
II
a:::
I-
0
1 .5
200
400
600
8 00
S P E C I M E N T EM P E R AT U R E T
PL
( °F )
1 000
Figure 1 8. Temperature Distribution Acros s
Fixture and Spe cimen
.
5
II
C\1
a:::
50
plate temperature , T P L .
The s e expr e s sions a r e a ccurate to within
±5 % for the tempe ratur e range of a Type 2 0 24 - T 3 a luminum plate ,
.
2 00 ° F
<
T P L < 7 0 0 ° F.
When Type 7 07 5 - T 6 aluminum alloy was te sted
under invariant c onditions , the specim en tempe rature wa s "'"'9 % lower
than that for the othe r mate rial.
to oxidation.
Thi s is due to it s higher r e s i stance
T he reflectivity coeffi cient of a material having greater
r e s i stanc e to oxidation r emains higher which ac counts for the lower
temperature of the latte r spe cimen.
The s e equations are applicable
to a T ype 7 0 7 5 - T 6 spe cim en by fir st multiplying the a ctual plate tem­
peratu r e , T P ' by the fact or , 1 . 09 , and then entering this adjusted
L
value into the empiri cal tempe rature equations given in Figure 1 8 .
The evaluations of the the rmoelectric te st fixture calib rations
shown graphically in Figures 1 5 and 1 6 s e rved to predict the appr oxi ­
mate wattage and variac settings required to a chieve a d e s i red speci·
men temperatu r e .
T o a cquir e temperatur e contr ol i n the final t e st s ,
,
the varia c s w e r e p r e s et ( Figure 1 6 ) and then thr ough fine adjustments ,
the desired nominal plate temperatur e was reali z e d.
This procedure
was c ompleted "-'24 hr after startup and was follow ed with the applica ­
tion of the test load.
Since the tempe rature contr ol system wa s the
open-loop type , no adjustments w e r e made to the variac s etting s dur ­
ing the actual t e st run.
The primary cau s e for the temperatur e fluc ­
tuations w e r e variations in the line voltag e ( 2 1 0 to 2 1 5 volts ) .
The
a s s ociated fluctuations in plate temperatur e did not exc e e d ±7 o F.
51
3 . T emperature Distribution A c r o s s Spe cim en
T o enhance c onfidence in the te st r e sult s , a thorough knowledge
of the temperature distribution a c r o s s the s pe cimen is ne ces sary.
P relimina ry te sts w e r e conducted c onc entrating on thi s a sp e ct . A s a
result of the s e t e s t s , a spe cimen wa s divided into thr ee the rmal zone s :
a s indicated in Figure 1 2. Z one 1 i s bounded by a circle having a diameter
of "- 5 . 5 0 in. , and repres ent s the a rea of the plate whe re only minimum
var iations in temperatur e o c curr ed.
The nominal plate tempe rature ,
T P L ' i s defined a s the tempe rature of Zone 1 .
Zone 2 c ompri s e s that
part of the specimen showing s om e nonuniformity in temperature dis ­
tribution.
In this area, the tempe ratur e dropped s lightly toward the
spe cimen suppo rt edg e s .
The main effort t o determine temperature
fluctuations a c r o s s the s pe cimen wa s concentrated on thi s zone be ­
cau s e of the potentially significant effe ct that a nonuniform tempera­
ture distribution in this area c ould have on the cr eep- induced plate de­
flection. For this reas on, six of the available the rmocouples were
pla ced in Z one 2.
Zone 3 included the support edg e s and adjacent
area s ; the temperature in this re gion was .-.. 5 to 2 0 o F lower than that
. in Z one 1 .
Minimal effort wa s s pent to obtain temperature informa ­
tion on Z one 3 because of its l ow - st re s s levels and insignificant c on­
tribution to the c r e ep r e spons e at the plate cente r .
The actual locations o f the the rmocouples for Specimens 1 and 2
are shown in Figure 1 9 .
The s e thermocouple locations a r e at lattice
points diffe rent fr om but symmetrical to the points indicated in
52
9
SPECIMEN
N o. I
"'-' 1 2 . 2 5 i n . D I A
1:
,
____ 1 . 75 i n . ( T Y P I C A L )
._. .,.
8
A
7
.-
T
4
-I
.....
S P EC I M E N
N o. 2
I
8
7
6
5
4
3
I
L
I
\
�
•
T/C·6
2
Figure 1 9 .
�
•
�
•
•
\
T/C·2
/
�
E
\
v
-
�
v
�
•
ZONE I
_/
\
/J SEUD PGPEOSR T E D
!
T/ -3
�
v
'
)
v
ZONE 2
� '---
•
�"'
T/C-1
�
f'
H
T/C-4
· -
�
1
1 .75 in.
I
TYP
S I M P LY
G
�
1 4 in.
REF
I
J
F
'
�
NE 3
1
T/C-2 T/C-5
\
�
Z O NE 2
D
�
H
_t
I
ZONE I
I
�
G
�
T/C-1
� I"--
c
\
\
I/
I
/
-
•
-.. I
8
A
F
T/C-5
\
2
9
E
�
3
\
3
�
j.....--
•
---;_I
6
5
I D E N T I F.
D
T/C-4
"' 5. 5 0 in. D I A .
GRID
C
./
8
--I
1 4 in. R E F
/
J
;
:
NE 3
Thermocouple Ar rangem ents (Actual)
53
Figure 1 2 . A plot of i s othe rmal r e gion s for the test spe cimen is shown
in Figure 2 0 . Thi s figur e applie s t o Type 7 07 5 - T 6 aluminum alloy in
the tempe ratur e rang e of 5 0 0 to 6 0 0 o F . No attempt was made to gene rate expe rimental data as the ba s i s for a s imila r graph on T ype 2 0 24-T3
aluminum "
Deviations from the nominal plate temperatur e for thi s rna -
terial c ould be twice the magnitudes s hown in Figure 20.
Selection and pr oper balancing of the magnitude s for the test loa d
and temperatur e w e r e of crucial importance s inc e only two spe cimens
were available .
La cking applicable data, the following approximate
procedur e was u s e d to e s tabli sh "ball - park" magnitudes for the test
load and the t e s t temperatur e .
A s suming that the d e s i r ed av erage c r e e p rate a t the center of
8
•
=
max 0 . 0 0 1 in. /hr, the load - st re s s - strain- defle ction r e lations at the cente r of a simply supported, cent rally - loaded, ela stic
squa re plate with P ois s on ' s ratio = 0 . 3 0 are ( 1 )
the plate is
(
W
a
o-m ax = 0 . 1 4 3 5 2 4. 3 3 log 2 r
t
0
2
= 0 . 1 27 Wa
8
max
Et 2
o-
1
=
=a
=
(] 2
max
fromwhich
E'
max
( l ) Re£. 1 1 , p 2 0 3 , case 3 1 .
=
+
z" so)
(psi)
(in. )
E E" max
1.3
(p s i )
l . 3 a max
E
(in. /in. )
54
4.0
6.0
5.0
7.0
1.0
/
2.0
ZONE 2
3.0
I
4.0
5.0
I
6.0
ZONE I
------+oo<--- <t_PL
y
Tx y
T xy
=
R xy
=
TPL
Figur e 20.
=
=
R x • T p L ( O F)
y
WHERE
GENERAL SPEC IMEN TEMPERATURE (°F)
TEMPERATURE AT CENTER OF SPEC IMEN (°F)
EXPERIMENTAL CORRECT ION FACTOR
T empe rature Di stribution A c ro s s Spe cimen
55
By sub s tituti on of
a-
max
and
8
max
into
E
i s obtained
(in. / in. )
or in terms of s train rate
€
max
= 5 6. 0 x 1 0 - 3 • t • 8
max
, and 8
a r e , r e spe ctively, the ela s max
max
tic str e s s , the strain, and the deflection at the plate c enter . Defini In the s e equations , CJmax
,
E
tions and magnitude s for the paramete r s in the s e equations are :
W = load at cente r of a squa r e plate { lb )
E = Y oung ' s n: odulus , ,..._ ? . O x 1 0 6 p s i for aluminum at 5 0 0 ° F
a = side s of s quare plate ( 1 4 in. )
r 0 = radius of area under concent rated l oad,
t
(J
8
E
=
w
(in. )
plate thickne s s (in. )
= str e s s at cent e r of plate ( p s i )
= defl e ction at center of plate (in. )
= strain at cente r of plate ( in. / in. )
Continuing with the a s sumption of ela stic plate behavior and
letting t = 0. 1 87 5 in. and
8
0
=
max 0. 0 0 1 in. /hr , the strain rate at the
center of the plate becom e s E max l O . S x 1 0 - 6 in. / in.-hr.
=
A review of R eferenc e s on c r e ep test results ( 1 ) sugge sts that
·
a c r eep strain rate of about 1 0 x 1 0 - 6 in. / in.- hr is a s s ociated w ith a
s tr e s s of ,...._ 5 , 00 0 p s i for Type 7 0 7 5 - T 6 aluminum near 5 0 0 ° F. A s sum ing that the initial ela sti c str e s s at the outer fiber in the cent e r of the
plate i s ,..._ 3 0 % highe r , i . e . , 6 , 5 0 0 p s i , the plate load, W , b e c om e s
( 1 ) R e f . 1 2, p . 8 8 3 , Figure 2 8 .
56
2
t o­
W =2 max
=
0 . 1 87 5 2 X 6 5 0 0
2
=
l l 4 lb
The load w eight and tempe ratur e , applied in the a ctual test of
Spe cimen 1 , w e r e obtained from the evaluations in the preceding a s sumptions .
The load weight and tempe ratur e for Specimen 1 w e r e :
Nominal c oncent rated load a t cent e r o f plate - W = 1 00 lb
Nominal test tempe rature - T = 5 5 0 ° F
To dete rmi ne the load w eight and temperature for the actual
test of Spe cimen 2 , an app r oa ch s imilar to that used for Spe cimen 1 ·
wa s followed , how eve r , with the a dded advantag e of including the test
results fr om Sp e cimen 1 . The load weight and temperature for Speci men 2 w e r e :
Nominal c oncent rated load at cent e r o f plate - W
=
9 0 lb
Nominal t e st temperature - T = 57 5 ° F
D.
A CQUISITION AND EV ALU AT ION OF DATA
1 . Developm ent of Defle ction Mea s uring System
Development of the data a cquisition te chnique was gove rned by
both e c onomic and te chnical c onside rations .
Li sted h e r e are c riteria
that influenced the s el e ction of the app r oa ch and method to be followed
for monitoring the c r e ep behavio r of the specimen.
In o r de r of their
te chnical importance , the s e a r e : ( a ) s ens itivity o f gauge a s s embly,
( b ) rep eatability of mea surement s , and ( c ) short time span r equi r ed
to take one c omplete s et of r eading s .
C o s ts of the gauging devi ce, as
well a s that of the fixture prope r , w e re to be held to an absolute
57
minimum .
Ba s ed on the s e considerations , a te st fixtur e was pr oduced
( Fi gur e s 9 and 1 0 ) requi ring only a s imple gauge ba r a s s embly ( Fig ­
ure 1 2 ) a s additional har dwar e to repre s ent the data acquis ition system.
B e caus e of the nature and the conditions g ove rning the te st proj ­
e ct , no definite pe rformanc e requirement s w e r e specified for the test
hardware.
How ev e r , development and d e s ign of the test fixture and
gauge a s s embly w e r e strongly influenced by the proj e cted or anticipated
inte ra ctions p r evailing und e r test conditions between human, me chan­
ical, and envir onmental factor s .
2 . P e rformance of Gauging Sys tem
Num e r ous t e s t s w e r e c onducted to determine the range of per ­
formance characteristi c s for the gauging system under ope rating con­
ditions .
The nominal s ens itivity of the gauge a s s embly is 0 . 0 0 0 1 in.
per divis ion.
Thermal effe cts in the gauge stem extension (Invar ) of
,...._ 0 . 0 0 04 in. w e r e obs erved.
Anothe r s ource of unce rtainty in plate
deflection m ea surem ent wa s the gap between gauge stem s leeve and
pos itioning hole in the fixture c over plate.
Thi s e rr o r was minimized
by ave raging the deflection value s taken fr om tw o opp osite diametrical
position s in the cove r plate hole.
The combined unce rtainties in mag ­
nitude of the plate defle ction due t o the s e cau s e s w e r e e stimated a s
±0. 0 0 0 8 in.
No va riation in the nominal gauge depth, correlating the
gauge refe r ence plane w ith the spe cimen support edg e s , wa s detected
for any tempe rature level of the fixture .
58
3 . R e cording of Data
The functi ona l data a cquisiti<:m form shown in Figure 2 1 wa s
designed to facilitate r e cording the information on plate behavior ,
the rmal environm ent , heating system , time data , etc .
of
,..__ I 0
A tim e span
min permitted monitoring plate defle ction and r e cording the
data at the plate p oints indicated in Figure 2 1 .
The magnitude of the
tim e span s e lected fo r r e cording r eflects the c ompr om i s e betw e en the
minimum numb er of plate points to be monitored and the a c ceptable
incons i stencies in data a s the plate c r e ep deflection progre s se d.
The gauge a s s embly was s et to r ead 0. 2 0 0 0 in. , applied to a
pe rfectly flat spe cimen; howev e r , becaus e of variations in fixture and
spe cim en, reading s taken at plate points just prior to application of
the test load often differ ed slightly from the original gauge s etting. A
spe cific cau s e for diffe r ence s in defle ction r eadings among mutually
symmetr ical plate point s is a c ertain mismatch betw e en the grid sys ­
tern of the s pe cimen and the location of the hol e s for the gauge stem
s leev e s through the fixtur e c ove r. T hi s mismatch i s unavoidable s ince
the exact r e lative diffe r ential the rmal expans ion for fixture c over and
spe cimen vs thei r r e spective s eating areas is not known. The ma gnitude
of the adve r s e effe ct, cr eated by the mismatch on data c onsi stency,
is approximately p r oportionate to plate defle ction.
Defle ction at a plate point is given by the diffe rence between
initial and sub s equent test gauge reading s . A s et of 29 plate point s , a s in­
dicated in Fi gure 2 1 , was u s ed c ons istently in monitoring plate deflection.
59
DATA A C Q U I S I T IO N F O R M
ENGI N EE R I N G - 5 9 8
T H E S I S PRO J E CT :
------
CREEP BENDIN G
O F S Q U A R E P L AT E
S A N F E R N A N D O V A L L E Y S TA T E C O L L E G E
J UNE - J U LY , 1 9 6 7
G
7 1
I
3
2 ��----�--4--+---�-+----�4--��+---�--
IL _ _
l�
--
1
s E C IM E N S U PPORT EDGES
_/
T H ER M O - ° F
CO UPLE ---+---+---4---4-�----�4---��
mv
DATA
HEAT I N G DATA VA R I A C :
-a-
U P P E R LOW E R
S E TT I N G
V O LT S
= GAUGE R EADI N G
( i n .) A T P L A T E
PO I N T
N UM E R A L S IN
T I M E DATA
DATE
D ATA T I M E
AMPS
STAR T I N G T I M E
WAT T S
E LA P S E D T I M E
LINE
VOLT A G E
H R . M I N.
e
= GAUG E R EA D I N G
( in.) AT S U PP O R T
M ON I TO R I N G P OI N T
T E M P E R AT U R E
( N O M I N A L T/C I )
M ATE R I AL :_ A LU M I N U M , S PE C I M E N D I M E N S I O N 8 N o .
D AT A TA K E N B Y :
W I T N E S S :.
Fi gure 2 1 .
OF
LOAD A T C E N T E R
O F P LATE
Data Acqui sition Form
X
lb
X
60
40 Evaluation of Data
The approa ch taken t o evaluate
involv e s two step s .
te st data at a given plate point
Fir s t , the a ccrued plate defle ction i s det e rmined.
S e c ond , the deflection repr e s enting a plate point is calculated by apply­
ing the lea st squa r e s method to the apparent deflection of other gauge
points , r elated to the fir st point by symmetry.
The repre s entative deflection value s thus de rived from the data
a cquis ition she ets a r e given in Table s I and II for Specimens 1 and 2,
re spe ctivelyo The ten plate points for which defle ctions are listed c over
all othe r plate points by symmetry.
Data from Table s I and II will be
used sub s equently for c ompa r i s on with analytical r e sult s .
TABLE I .
Plate Elastic
Deflec P oints>:< tion
w'
t
EXPER IMENTAL DEFLEC TION OF SPEC IMEN l
Tim e Elaps ed Sinc e Load Application (hr ) t
1 1 , 00
2 1 ,50
2.50
3 1 .25
26 . 00
7 . 25
36 . 00
w
w
w
w
w
w
w
w
w
w
w
w
w
w
9 . 5 0 . 7 1 0 . 3 1 . 5 1 1 . 3 2 . 5 1 2 . 7 3 . 9 1 2 . 6 3 . 8 1 3 . 2 4. 4 1 3 . 8 5 . 0
48. 1 5
w
w
14.0 5 . 2
25.9
II
II
II
"
II
II
II
II
B-2
8,8
C -2
1 6. 2
1 7 .. 5
1.3 19.2
3.0 21. 1
4 . 9 22. 6
6.4 2 3 .7
D -2
20 .0
22 . 4 2 . 4 24 . 5
4.5 26,7
6.7 28. 8
8.8
3 0 . 1 10 , 1
3 1.5 1 1.5
32.5 1 2.5
3 2.8 12. 8
E-2
21.8
23 . 9
2. l 26.2 4.4 28.7
6.9 3 1.2
9 .4 3 2 .7 1 0 . 9
33.7 1 1.9
35.4 1 3.6
3 5 .7 1 3 . 9
C-3
28.8
30 . 1
1.3 33.3
4 . 5 3 6. 7
42 . 2 1 3 . 4 43 . 7 1 4. 9
45. 1 1 6 . 3
45 . 9 17 . l
D-3
3 6. 5
40 . 4
3 . 9 45 . 2
8 . 7 49. 6 1 3 . 1 5 4. 2 1 7 .7
5 6 . 5 20 . 0
5 8 . 8 22. 3
6 0 . 5 24. 0
6 1 . 6 25 . 1
E-3
38.8
44. 4 5 . 6 48 . 9 2 0 . 1 54. 9 1 6 . 1 5 9 . 2 2 0 . 4
6 2 . 3 23 . 5
64 . 4 2 5 . 6
6 6. 8 2 8 . 0
6 8 ; 3 29 . 5
D -4
48 . 5
55 . 3
6 . 8 6 1 .4 1 2 . 9 67 . 9 1 9 . 4 7 4. 9 2 6. 4 7 8 . 3 29 . 8
8 1 . 6 33. l
83.5 35.0
8 5 . 8 37 . 3
E -4
5 2. 7
6 1 .7
9 . 0 6 8 . 8 1 6 . 1 7 6. 0 23 . 3 8 3 . 6 3 0 . 9
9 1 . 3 3 8. 6
9 4. 2 4 1 . 5
9 6 . 8 44. 1
E-5
57 . 3
7 1 . 5 1 4. 2 7 9 . 1 2 1 , 8 8 4 . 7 27 . 4 9 6 . 8 3 9 .5 1 0 1 . 5 44 . 2 1 06 . 1 48. 8 1 0 9 . 2 5 1 . 9 1 1 2 . 6 5 5 . 3
7 . 9 40.4 1 1 .6
7 .5
87 . 8 3 5 . 1
8.9
25 . 1
25 . 5
9.3
9.7
>:< See R ef . Figur e 2 1
General Data : T est load at plate c ent e r , W = 1 0 3 . 5 lb
t - w = total d efl ecti on x 1 0 3 in.
T est tempe ratur e , T 545 o F
w 11' = elastic d efle ction x 1 0 3 in.
Specimen material : Type 7 07 5 - T 6 Al ( c lad )
w = cr eep deflecti on x 1 0 3 in.
Specimen dim ens ion s : l 4 x l 4 x 0 . 1 87 5 in.
Exc ept for E - 5 , all elastic d eflection value s a r e bas ed on extrapolations applied to r e sults of
c omput e r Run l45C C ; E - 5 i s an expe rim ental data p oint .
=
"'
......
TABLE II.
Ela stic
Plat e
Defl
ection
P oint s ':<
w' t
B-2
6.0
0.50
w w
7 . 1 1. 1
II
EX PERIMENTAL DEFLEC TION OF SPECllvl EN 2
T ime Elapsed Sinc e L oad Application (hr )t
24. 2 5
20 . 5
1 5 . 25
38.5
10.3
w
w
w
w
w
w
w
w
w
w
9 . 0 3 . 0 8 . 5 2 . 5 1 0 . 5 4 . 5 1 0 . 1 4. 1 1 2 . 7 6 . 7
II
II
II
II
II
48 . 00
w
w
15.8 9.8
II
C -2
1 1 .3
1 2 ,4 1 . 1 1 5 . 8
4. 5 1 6 . 6
5.3
18.2
6.9 18.6
7 . 3 2 1 . 7 1 0, 4
2 5 . 3 1 4. 0
D-2
15.0
1 5 .5 0. 5 20. 2
5 . 2 2 1 .7
6 . 7 23 . 2
8 . 2 23 . 5
8 . 5 27 . 9 1 2. 9
3 2. 0 1 7 . 0
E-2
1 6. 6
1 7 . 1 0 . 5 2 4. 4
7 . 8 24 . 6
8.0 26.4
9 . 8 27 . 2 1 0 . 6 3 0. 7 1 4. 1
3 4. 5 1 7 . 9
C -3
2 1 .4
22.4 1 . 0 29. 8
8 . 4 3 2 . 1 1 0 .7 34. 1 12 . 7 35 . 5 1 4. 1 40. 4 1 9 . 0
44. 4 2 3 . 0
D-3
29.2
2 9 . 7 0 . 5 3 9 . 8 1 0 . 6 4 2 . 5 1 3 . 3 45 . 2 1 6 . 0 47 . 0 1 7 . 8 5 3 . 3 34. 1
5 8. 1 2 8 . 9
E-3
3 1.0
32.5
1 . 5 4 1 . 8 1 0. 8 47 . 5 1 6 . 5 5 1 . 1 20 . 1 5 2 . 5 2 1 . 5 5 9 . 0 2 8 . 0
64. 2 3 3 . 2
D -A
40 . 0
4 1 .7
1 . 7 5 6 . 0 1 6 . 0 6 0 .7 2 0 .7 65 . 1 2 5 . 1 6 6 . 4 2 6. 4 7 5 . 5 3 5 . 5
80. 1 40. 1
E -4
43 . 5
45-. 8 2 . 3 6 2 . 9 1 9 . 4 6 8 . 2 24.7 7 3 . 4 29 . 9 7 5 . 4 3 1 . 9 8 5 . 0 4 1 . 5
9 1. 5 48. 0
E-5
47 . 2
5 5 . 0 7 . 8 7 3 . 0 2 5 . 8 7 9 . 0 3 1 . 8 85 . 2 3 8 . 0 88 . 0 4 0 . 8 9 8 . 5 5 1 . 3 1 05 . 4 5 8 . 2
- ------ --------- �
----
=
>!< See R ef . Figur e 2 1
Gener al Data: T e st l oad at plate c enter , W 9 0 . 5 lb
T e st temper ature , T 5 7 2 o F
w total deflecti on x 1 0 3 in.
t
Specim en mater ial: T ype 7 07 5 -T 6 Al (clad )
w e lastic defl ection x 1 0 3 in.
Specim en dimensions : l 4 x l 4 x 0 , 1 87 5 in.
W11
c r eep deflection X 1 0 3 in,
Except for E - 5 , all ela stic deflection value s a r e bas ed on extrapolations appli ed to r esults of
c omput er Run 247 C C ; E - 5 i s an experim ental data point .
-
=
1 =
=
=
"'
N
VI.
A.
DISC USSION AND C ONC LUSIONS
OBJEC T IVES AND APPROACH
The primary obj e ctive of thi s effort was directed toward verify ­
ing the expr e s s ions d e s c ribing the c r e e p behavio r of a plate for the
particular load and boundary c onditions s pecifie d.
A s ec ond obj ective
was to ob s e rve the effects upon the end r e sults by applying c ertain ap­
proache s and methods in the quantitative evaluation of the basic math ­
ematical equations .
The latter obj e ctive was accomplished by g en -
e rating and c omparing experimental r e sults with analytical r e s ults .
The analytical treati s e of a c omplex phys ical phenomenon often r e ­
veals alternate r oute s for evaluating the mathematical model of the
problem.
Typical advance decisions that may b e required when evalu -
ating mathematical expr e s s ions involve s e lection of the type of com puter subroutine mos t suitable for the problem, deciding how to best
r epres ent a complex equation by a finite difference model, s e le cting
suitable lattice s pacing s , nodal o r g rid point arrang ements , and other
c onsiderations .
Quantitative analytical r e sults c ould be affected s ig ­
nificanfly by the adequacy of the s e de cision s .
For the c r e ep problem und e r study , decisions c o r r e s ponding to
tho s e listed above w e r e made ; namely , to sub stitute a complex surface
deflection equation, Equations l , 2 , and 3 with the algebraic carry - over
method ( l ) that e mployes a finite difference type of deflection equation ,
( 1 ) Ref. 3 .
63
64
Equations 5 thr ough 8.
Further , a finite differ enc e method was u s ed
to s olve the c r e e p strain integ rals in the e quivalent load intensity
equation, Equation 2 5 .
Als o , a c r e ep law , Equation 3 1 , which en­
g ende r ed expre s sions for both primary and s e condary uniaxial creep
was s e lected from s eve ral alternate exp r e s s ions .
Cons e quences of
the s e deci sions are dis cus s ed lat e r .
B.
EXPERIMENTAL R ESU LTS
The purpo s e of c onducting an experiment paralleling the theoreti ­
cal treatment of this creep problem was to obtain an indic ation of the
c onfidenc e level that may b e attached to the analytical results , e s pe ­
cially conc erning the influence of the applied evaluation technique s .
A more general justification for c onducting the experiment was to ob ­
s e rve ove rall effects of the actua l , r ather than the ideal experimental
c onditions upon the quantitative r e sults .
The experimental r e sult s , evaluated earlier , are pre s ented in
Tables I and II for Specimens 1 and 2 , r e s pectively.
Initially , Spe ci ­
men 2 was t o s erve a s a backup for Specimen l , but was not r e quired.
The r e fo r e , Specimen 2 was te s ted unde r c onditions differ ent from
tho s e u s ed for Speci men 1.
Table s I and II give the total plate deflec ­
tions for the given g rid points and the time laps e s after load s were
applied.
Exc ept for g rid point , E -5 ( Figure s 1 9 and 2 1 ) the value s
,
for ela stic (initial ) plate deflections , w ' , c ould not be measured due
to the limitation s of the data acqui sition system.
The se values were
dete rmined by extrapolating sub s equent plate defle ction point s , w to
65
tim e , t
=
0 , and c onside ring the analytically pr e di cte d defle ctions vs
time rel ationships fo r given grid points immedi ately following the
11
applic ation of the l o ad . The cre ep deflections , w , of the plate c on stitute
the differenc e between acc rued total plate defle ction, w , and elastic
plate defl ection , w ' , for given grid points .
Factors that may affect the numerical r e sults of the plate creep
r e s ponse ar e identified and g r ouped by pr obable magnitude of effe cts .
P r onounce d Effe cts e s tim ated at
1)
::; ±- 1 Oo/o
II
of w at plate c enter
Nonuniform tempe r atur e distribution ac r o s s plate
(Figure 20 )
II
Noti ceable Effect s e stimated at :s; ±-5 o/o of w at plate center
2) Gaug e r e ading e r ro r s due to the rmal and fixtu re effe ct s
3 ) Fluctuations in voltage of supply line
4) Effe ct of c l adding of s pe cimen
5 ) Adequacy of extr apolation pr ocedure used to dete rmine
11
the e l as tic plate deflection , w
II
P o s s ible Notic eable Effect s e stimated at .S±-2% of w at plate
c ente r
6 ) Flatn e s s of s pecimen
7 ) Po s s ible r e sidual stre s se s in s pe cimen, stemming fr om
cutting ope ration
The pe rformance of the c re e p fixture was s ati sfacto ry thr oughout
the te st within limitations expe cted.
B as ed on the r e s ults of perform -
ance t e st s , the maximum rec ommende d te st spe cimen temper ature
was - 7 00 ° F for a c o r r e s ponding exte rnal fixtur e tempe r atur e of - 3 50°F.
66
To m aintain thi s ope r ating level , 1 2 00 to 1 3 0 0 w atts of power w e re
c onsumed p e r s et of he at e r s (Figur e l l )
.
The as so ciated tempe r ature
o .f the he ater element s ( T / C - 5 , Figure 1 8 ) w as �1 0 0 0o F. By impr oving the thermal insul ation of the fixtur e , a temper ature of 8 0 0 t o 9 0 0° F
c ould probably be obtained on the specimen while still maintaining an
exte rnal fixtur e temper ature of �3 50 ° F . The ele ctrical r e s i st ance of
the upper and low e r heater sets at o pe r ating tempe r atur e w as e st ab ­
lished a s 2 3 . 3 ohm s and 26. 8 ohm s , re s pe ctively.
F e atur e s or improvem ent s that could be included or added to th e
type of c r ee p te st fixtur e c ons ide red ar e :
l ) Automatic re c or ding s ystem t o monitor plate defle ction,
thereby allowing dete rmination of the plate creep behavior
immedi ately aft e r applying the load
2 ) High tempe ratur e strain g auges to det ermine str ain magni ­
tude s at the pl ate surface
3) Clo s ed - l oop ther rr1al c ontr ol s ys t em to minimiz e vari ations
in te st t e mper ature s due t o fluctuation s in supply l ine voltag e
Quantitative analytical re sult s ar e b as ed on a s et of experimental
c onstant s , A , B , and k, that appe ar in Equation 3 1 .
Thi s equation de -
s cribes the creep character is ti c s of the m ateri al under uni axial t e st
conditions .
Ther efo r e , · complete expe r im ental suppo rt c ould be g ained
by including a s er i e s of uniaxi al c r e ep t e s t s to determine the s e c ons tants for the mill stock us ed in the actual t e st s .
C.
ANALY TICAL R ESU LTS AND COMPU TER NOTES
Pr oblems enc ounter ed in thi s ar ea of the effort that c ould affect
the quantitative r e sult s may be divided into three broad c ategorie s :
67
( 1} theor etic al s etup of the pr oblem, ( 2 ) m ethods of evaluation, and
( 3 ) material pr ope rti e s.
1 . Theoretical Setup of the Problem
B ased on a lit e r atur e survey, the Algebr aic C ar ry - Over Method
(1)
(A. C. O . M. )
was chos en to calculate s e ctional bending m oments and
lateral defl e ctions due to a gener al s ystem of forc e s acting laterally
upon a simply supported plate .
Whil e thi s method of evaluation i s
much s impl e r t o u s e than the c or re s ponding clas si c al equations g iven
by Navie r , L evy, or Ritz (Equations 1 , 2, and 3 ) , it has s ome dis ad vantag e s , e s pe cially notice able i n thi s applic ation.
stem from the rathe r
The difficultie s
c oar se grid-line s y stem of 1 / 6 the plate width
and length s pe cifi e d for the A. C. O . M .
This provid e s for 2 5 grid point s
whe r e concent r at e d l o ad s m ay be applied o r whe r e moment s and d e flections may b e c al culated.
By ne c e s s ity , evaluation of the e quiva-
lent load intensity function (Equation 2 5 ) is al so limited to the s e few
The r efore , if the q - function prove s t o be relatively
e
c omplex ( Figur e 3 4} , the few plate points for which q e -valu e s are
plate point s .
available do not permit adequate u s e of highe r order curve fitting
technique s .
This limitation nece s s itated that the time -dependent
equivalent load s b e c alculated thr ough integr ation of the q - function
e
derive d fr om a le s s accurate curve fitting te chnique , the reby, ad ver sely affe cting the the or etic al c re ep re sults .
2.
Methods o f Evaluati on
A comparis on of preliminary the o r etic al with experimental r e ­
sult s indi cated that the the oretical l ate r al cr eep re s pons e of the plate
( 1 ) Ref. 3 .
68
was too low.
The pr obable caus e for this w a s found in the initial der -
ivation of the e quivalent concent r ated load at a given g ri d point; thus
p
lJ
e. .
=
b.x
•
fly
•
lJ
qe . .
(lb )
. . . ( 65 )
Bas ed on a r eview of the q -value s , a mor e r ealistic c on centr ated load
e
was defined for the c enter of the plate
(lb )
. . . (66)
Thi s load r epr e sents the volume of a pyramid of base length , Za, and
apex height , q 1 3 . The definition of the par amet e r , a, is indicated in
Figur e 34 and A i s an e stimated adjustment factor . It s magnitude i s
clo s e to unity and inc re ase s the par amete r , a , until the volume s of
the tw o s e parately shade d are as depi cted in Fig ure 34 ar e approxi mately e qual .
The refor e ,
A
adju st s the time -dependent c oncentr ated
load , P 1 3 , at the plate center to a magnitude corr e sponding to the r e ­
sult expe cted as though a highe r order curve fitting te chnique had be en
applied to the q -values .
e
The magnitude of the c onc entrated load, adj ac ent to the plate
center is define d as
(lb )
. . . ( 67 )
Thi s l o ad is re pre sente d by the dotte d outline in Figure 34. At the re maining grid point s , whe re the q e -value s ar e quite small , the con ­
c entrated load was determine d by E quation 6 5 . The final c omput e r
runs for Spe cimens 1 and 2 ar e b as e d on values for
A
o f 1 . 0 6 8 and
69
1. 045, res pectively.
A compar is on of the res ults with exp� riment al
data s hows g ood conformity, and s upports the s election of thes e A values as well as the approach for determining the equivalent con centrated loads .
The computer prog ram is s et up to calculate pertinent param eters at pres cribed time intervals ,
�t. .
1
The as sociated command
inputs consis t of three parameters , i. e. , initial time interval,
multiplier factor,
R
=
�t
i+ l
/ �\ ; an d the time-s top limit, T.
�t ;
1
In the
final computer runs , the values for the s e s pecific parameters were:
�t
1
= 0.02 hr ;
R
=
1. 5 0; T
=
50 hr. V arying �t
1
and
R s hows that for
the type of problem cons idered, quantitative res ults remain cons is t -
< �t 1 < 0.10 and l.O < R ± 2.0. F or r eas ons of
economy, the low ranges of � t and R s hould be avoided. Combina­
1
ent for rang es of 0, 0 1
tions of the hig h r anges for these parameters may res ult in an exces s ively larg e final time- s tep.
Exploratory computer run s s how that
the final time- s tep s hould be well below 50% of the time- s top limit , T,
preferably in the vicinity of � 0. 35 T, otherwis e the res ults may be
inaccurate or erratic.
When planning computer adaptation of a complex problem,
much emphas is s hould be placed on liberal print- out provis ions . This
feature will be of g reat value when checking the program for proper
functions or attempting to locate flaws . F or routine us e of the prog ram, the time cons uming print - outs may be curtailed.
70
The s e c r iteria, dete rmine d through trial and er r or , show that
various fallacie s may exi st in the compute r adaptation of c omplex
mathematic al expr e s sions whi ch could adver s ely affe ct the quantitative re sult s . Ther efo r e , mere exi stenc e of a valid mathematic al model
doe s not insure quantitatively ac cur ate re sult s - much di s cr etion and
jud g ement must b e exer c i s e d when planning c omputer ad aptation.
3 . Effe ct of M at e ri al Propertie s
The quantitative analyti cal solution of a c r e e p pr oblem i s usually
bas ed on exper imental dat a, and c onse quently , analytical r e sults may
vary within an envelo p, analog ou s to the s c atter band formed by ex­
perimental test point s .
Fo r this study , thr e e expe rimental material
c onstants , A , B , and k, we r e r equi r ed to de s cr ibe the uni axial creep
characteristics of the t e st mater i al .
Factor s o r c onside r ation s that may affe ct the creep respons e
o f the t e s t mat e ri al unde r giv en str e s s and temper ature ar e :
1 ) Effe ct of thermal hi story , e s pe cially , time elapsed at
elevated temper atu r e , prior to applic ation of load
2 ) F abrication his tor y , e specially , the rolling te chnique s
involved
3 ) Type of loading te chnique u s e d in uniaxial c re ep test ( c on ­
stant load or c onst ant str e s s )
The value s for the c r e e p c onstants , A , B , and k , u s e d in the analysis of
Spe cim ens 1 and 2 , are s een as functions of temper atur e in Figure 4.
71
However , the s e value s w e r e obtaine d fr om structural shape s , r ather
than fr om plat e / sheet stock.
The refor e , the se c r e ep const ant s may
not be fully repr e s entative for the t e st material us ed. Shanley ' s uni ­
axial c r e e p data ( l ) show s a small s c atte r b an d of about ± 5 o/o str es s at
600° F.
V ariation by only ± 3 % of the c r e e p c onstants , expe cially, con -
stant, B , was shown to affect the compute r plate c r ee p r e spon s e by
- ±l Oo/o.
Although no explicit data c ould b e found on the s catte r b and
for the c r e e p c onstant s , a value of ±3% was c on side red as r epr es entative , and c o r r e s pondingly a ± l Oo/o band of unc ertainty was attached to
the nominal the o r etic al plate c re e p r e s ponse finally presented.
D.
C OMPARISON OF EXPER IMENTAL WITH ANALY TICAL RESULTS
In studying the c r eep problem c ons ider e d here , the lateral plate
creep deflection w as the par ameter of primary intere st. Shown in Fig ure s 2 2 and 2 3 ar e the analytic al an d expe rimental cr eep char act e r istic s o f Spe cimens 1 and 2 , r e s pectively.
The grid points for which
data ar e given c over the entir e plate by symmetry. At most grid point s ,
. the experimental elastic defle ctions ar e initially higher than the analytically pr edicted value s which may be due to one or a c ombination of
the following :
1 ) The elastic m odulus of the cladding materi al , aluminum
all oy , type 7 07 2 , - 0 . 0 0 4 in. I side , may be c onside rably
lowe r at te st tempe r ature than it is for the substrate material'
( l ) Ref. 1 3 , Chap. 1 7.
72
E X P E R I ME N TAL ( TA B L E I )
- - - A N ALY TI C A L (COM P. R UN 145 CC)
E LA S T I C D E F L E C T I O N
0 E XP E R I M E N TAL
.J. A NALY T I C AL
.
U PL A T E G R I D P O I N T , R E F. F I G . 2
�
z
0
1u
w
....l
lJ,.
w
0
w
1<(
....l
a.
....l
<(
a:
w
!;i
....l
20
Figure 2 2 .
T I M E ( hr )
30
40
50
Experimental v s Analytic al Defle ction - Specimen 1
73
----
E X P E RI M E N TAL { TA B L E li )
- - - A N ALY T I C A L (COMP. R UN 247CC
E LA S T I C DEFLECT ION
0 E X PE R I M E N TA L
• A NA LY T I CA L
D PLATE G R I D POINT ( R E F. F IG .2)
w =
1u
LLI
...J
LL
LLI
0
w
let
...J
!l.
h/2 AT C E NT E R�
--
--
.060
...J
et
ei
l­
et
...J
.050
20
30
40
50
T I M E ( h r)
Figure 2 3 .
Expe rimental vs Analyti c al Deflection - Spe cimen 2
74
2 ) The effect ofthe nonuniform temper atur e distribution acr o s s
the plate (Figure 20 ) i s not refle cte d adequately in the r epre ­
s entative mean tempe r ature , T , s ele cted for the compute r
s olutions , T
=
5 4 5 and 57 2° F , r e s pectively for Specimens
l and 2 .
Since c r e e p behavior r athe r than elastic plate behavior w as of pri mary intere st, thi s di sc repancy was not pur s ue d further .
A simple c om parison of expe rimental vs analytical c r e e p deflec tions c on s isted of super pos ing the expe rim ental and analyti c al elastic
deflection points , and ther e after , dire ctly ob serving the differ enc e in
lateral plate c r e e p defl e ction ove r time for any given point. On this
bas i s , good confor mity existed betw een the two methods for up to � 20 hr.
Beyond this tim e , at s ome g rid points , the analytic al cre ep deflection
began to outrun the as s o ci ated expe rim ent al data curve.
After �45 hr
the maximum di s c r e pancy in c re e p deflection w as �0. 0 0 9" ? 1 5o/o of the
creep deflection ac crued at the c ente r of the plate.
Likewi s e for Spe cimen 2 , a slightly bett e r c onformity b etw een
experimental and analyti cal r e sults was ob s e rved ( Figur e 2 3 ). Since
Spe cimen l was the prim ar y te st spe cimen , furthe r quantitative eval uation or comparis on of analytic al or expe rimental re sults was lim ite d t o thi s s pe cim en only; how ever , h ad a s et of g r aphs been c r e ated
for Spe cimen 2, s imilar char acte risti c s would hav e been evident .
Shown i n Figure s 24 through 27 ar e the analyti cal plate deflection s
at s pe cified plate s e ctions and as a function of time . Supe rpo s e d on
·
I
I
.010
[ill
.020
I
i
.030
.
0 40
@)
I
--j
P L AT E CE N T E R L I N E
.050
.060
-
- t =O
.070
5.1
. 08 0
. 09 0
.100
.110
1 .5
I=
<w
�- w� )
= (w
E
-w
�)
- (w
A
-w
�)
1 1 .6
1 7. 5
( i n .)
�
E X P E R I M E N TA L P O I NT S H O W I N G R E L AT I V E
D E V I AT I ON I N C R E E P D E F L E C T I O N T O
A N A LY T I C A L R E S U LT S ; S P E C I M E N I
D G R I D PO I N T ( R E F. F I G. 2 )
26 . 2
. 1 2 0 �----�--�����1 .0
2.0
3 .0
40
5.0
7.0
6 .0
x ( in.)
3 9. 4
.
w ( i n.)
Figure 24.
Plate Deflection ,
w
(in. ) at
y =
L/ 2
-.1
. U1
I
�
[2]I
.010
P LAT E C E N T E R LI N E
.0 20
--l
I
.030
.040
. 0 50
.060
.070
. 0 80
l=
. 090
. 1 20
w ( in . }
f:
- w
�}
= (w
E
E)
- w
- (w
A
- w
A)
(in . )
E X PER I M E N T A L POINT SHO WING RELATIVE
D E V I AT I O N I N C R E E P D E FLE C T I O N T O
A N A LY T I C AL R E S U LT S· ; S P E C I M E N I
. 1 00
. 1 10
<w
J:
D G R I D POIN T ,
I
l.u
· -
.....
I
.....
R EF. F I G. 2
!
2.0
Figur e 2 5 ,
I
-
-
3.0
I
..
....
I
4.0
Plate Defl e ction,
I
-
-
5.0
w
(in. ) at
I
-
-
6 .0
y =
I
- -
7.0
I
x
-
J •
•
( in.}
LI3
--J
"'
79
the se cur v e s are experimental data points showing the r elative deviations in creep deflections as defined abov e .
The o r dinate s e gment , I,
b etween an experimental data point and the as s o ci ated defle ction curve
is given as
(w � - w� ) (w E - w �) - (w A - w �)
=
(in. )
. . . (68)
:wherein the sub s c ripts , E and A, r e fe r to exper imental and analytical
value s , r e s pectively. Prime , double -prim e s , and no - prime notations
r efer to elastic , c r ee p, and total lat e r al plate deflections , re s pe ctively.
The value s for the se deflection par ameter s have be en obtained from
Figur e 2 2 .
Studying plate deformation a s time progr e s s e d r evealed that the
c oncentr ate d l o ad at the pl ate cente r g r adually changed the shape of
the plate to a form r e sembling an inve rted pyramid ( Figure 24 ) , o r
pear - shape (Figur e 27 ) .
E.
R ESULTS O F ANAL Y TICAL STRESS , ST RAIN , AND LOAD
R ELAT IONS
In Figur e s 28 through 3 4, r e spe ctively, ar e the analytically d e -
veloped str e s s , strain, and equivalent l oad intensitie s f o r dis c rete
,plate points or ar e as as indic ate d .
T o contr ast the strong effe ct upon
total elastic and plastic str ain s by stre s s , a thorough gr aphical c om ipar i s on betw een s t r ains and stre s se s at the are as o f highe st stre s se s
:( plate cente r ) w ith the adjac ent g r id point 1 2 i s pre sente d in Figur es
'2 8 thr ough 3 1 .
80
�
, --­
1 2 �--�----�--�----�--�----�
l------l-
I I l------l-
10
�/(
�
� --�+---�----+---_,
t----t-
1-------+--+-�
tL
=e
e'� = e yq---1------i
9 1----+----ll-
-
c:
.......
7
t::
·�
�
z
6
t
=
�
y
�
,8 ='1/2 -
''-"- -----� -�-� e � =e yB = I _
£1·
�_j::_·i:.
I
�-�
-+1---+VI
L �-q��
-, l"'- I r
--+-
4
,8
�---+--++--+/
�
'
�
,
5
y
-+----+---+----t
f--
r/
I
1---1------+-1---+---t-'-
�
--+-+-----+---+---+-1
-� � =e.
.
.;__--+---+---+---+---+X-
10
Figur e 2 8.
I
---- --=----+---+----i
,
'
-A
r2
I
I
�0
�
y ,8
� �
�-"---i------i
��..
.. � - -�
�---��
30
40
T I M E ( hr }
Str ain v s Time at Plate Cente r
50
/3 = 1
h/2
z
:
.090
PLATE OUTER F I B R E
.080
�
.070
.s::
.060
c
/3
z
=
:
1 /2
h/4
u)
tn .050
I
�
I,
�,.....7',..... .>,7',
- z
i:
1-
.......
.040
t =
�
t = 26.2
!;i .030
...J
a.
/3 = 0
= 0
z
I
7',
39. 4 (hr)
1 7. 4
.020
.01 0
-
P L ATE M ID PLANE
1000
Figur e 2 9 .
2000
3000
4000
5000
fy , (psi)
S TR E S S ,
fx OR
Str es s v s T im e at Plate Center
00
.-
82
y
7
L
�
¥+
5
c
..
y /3 = 1
/. /
V4 �
1-.
<t
0
z
<(
a:
l­
UI
3
- - ·
=
e"
- --
-
··
v
0�! -10
Figure 3 0 .
�
I
�--�:.
ye y/3 =
�/t
0
--
�-I - --
__.-'
2
__... �
4�-�--(J
c
'
v
�
'
6
.;
..
7-1-
I
20
�
,.
,
ly
, /-·
y
Y-'
-�- I
e
"
y /3 =
1 /L - - -- - - -
,_ _ _.
TIME ( h r)
30
40
Str ain vs Time at Grid Point 1 2 (R ef. Figure 2 )
{3 : I
z =
h/2
PLATE OUTER Fl BRE
{3
z
(/)
(/)
=
�
1/2 � .050
= h/4 �
:I:
1-
o4
.
.
. ,- 7 ....,,...:>'
,....7'-ss....7
I
,. ........
.. ...
:>'
t : 1 1 . 63
t : 26.2
I
39.4 ( h r )
PLATE M I D . PLANE
f3 = o
z =0
1000
Figure 3 1 .
2000
3000
STRESS , fy (psi)
Str e s s vs Tim e at Gr id Point 1 2 (Ref . Figure 2)
00
w
84
Figure 2 9 show s that the maximum stre s s r eduction at the outer
fib r e and plate c ente r o c cu r r e d after - 1 2 hr.
Ther eafter , the s tr ain
hardening effect b e c ame sufficiently pr onounced to cause the stre s s
in that ar ea t o s lowly ri s e again.
By c ontrast, the str e s s in the y ­
dir e ction at the outer fibre of grid point 1 2 did not decre ase , but c on ­
tinued t o inc r e a s e ste adily ( Figu r e 3 1 ).
Primarily , this incre ase was
due to the c onside r ably low e r str e s s level exi sting at that point ; how ­
ever , he r e the inc re asing str e s s le vel was not attributable to w o r k­
hardening effect s but to the continuing plate deform ation following
str ain compatibility .r equir e ment s .
This als o explains the g radual
inc r e ase in s tr e s s level s at the plat e quarter point s ,
z
=
h/ 4, for both
grid points unde r s tudy.
Figur e s 3 2 , 3 3 , and 3 4 illustrate the analytically determined
time - dependent equivalent load intensity functions for di s cr ete plate
s e ctions as indicated.
qe ( p s i )
.10
D
.08
G R I D P O I N T ( R E F . F I G. 2 )
.06
.04
.02
0
-.02
[Q
I
I
==
:_ -.
--.;.,__
�
-!_
-. 0 4
-.0 6
-. 08
--
t
-�--
=
1 7. 4 7
......__ ____;..._____
-.10
2
Figur e 3 2 .
3
4.
5
Equivalent Load Intensity, qe at y
7
6
=
1t
( in.)
L/ 6
00
U1
q� ( p s i)
- 4. 0
D
G R I D POINT
0
I
( R E F. F I G . 2 )
-3.0
-:- 2 .0
t = 1 7. 4 7 ( h r )
t = 7. 75 (hr)
t = 3 . 42 ( h r )
t = 1 . 50 ( hr )
-1.0
�
I
I
I
0
2
Figur e 3 3 .
3
4
5
Equivalent Lo ad Intens ity ,
q
=
7
6
e
at y
x
(in.)
L/3
00
0'
Qe
{psi)
22
20
18
D
16
G R I D P O I N T, R E F. F I G . 2
14
12
10
@]
8
I
I
�<$Q�
I
· '
4
o
@]
I
6
2
I
,.
I
..
1
-2
-4
2
Figure 34.
3
4
5
6
Equivalent Load Intensity, q . at y
e
7
=
x
( in.)
L/ 2
00
-.]
88
R E FERENCES
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She lls , 2nd Edition, McGr aw - Hill B ook C o . , New Y or k { 1 9 5 9 ).
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The ory and Table s , " Engine ering R e s e arch Bulletin , Okla­
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1 1
7 . Lin, T. H . , Bending of a Plate With Non- Line ar Strain- Hardening
Creep, Colloquium on C r e ep in Structure s , July 1 9 62 at
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pp. 2 1 5 - 2 2 8 { 1 9 6 2).
11
8. Hill , R. , Mathematical The ory of Plasticity, Univer s ity Pre s s ,
Oxfor d , London { 1 9 5 0 ) .
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Maste r s The s i s , Univer sity of Californi a, Lo s Angele s {June
1 963).
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Gr aw - Hill B ook C o . , Inc . , New Y o r k ( 1 9 5 4) .
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89
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1 6 . B r a s s , J . , 1 1C r e ep Stre s s es of a Rigid Fram e.' 1 Mas t e r s The s i s ,
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Lo s Ang eles ( May 1 9 5 9 ) .
'
90
BIBLIOGRAPHY
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Str e s s Field i n Met als at Initial Stage of Pl astic Deforma­
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A. E . , Hende r son, J. , and Matter , V . , Th e Enginee r ,
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Sons , Inc . , N ew York .
" B ending of Symmetri c ally Loaded Circular Plate s With
Arbitrary Cr e ep Char act e ri sti c s , " Gebhart, C. E. ,
Master s The s i s , U C LA (May 1 9 6 1 ) ·
' ' Non - Line ar Creep Bending of a Square Plate , " Ganoung ,
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C r e e p Buckling o f R ectangular Section Column s , Libove , C . ,
NACA- T N - 29 5 6 ( June 1 95 3 ) .
Aeros pace Struc tur al Metal s Handbook, Vol. II, W ei s s , V. ,
E dit o r ; Pr e pared for Wright - Patte r s on Airfor c e Bas e ,
Syracus e Univer s ity R e se ar ch Institute , Syr acus e , New
Y o rk.
APPE NDIX - C OMPU TER DATA
The extent and obje ctives of the c om pute r data included are to
pre s ent a s ample print- out of the comput e r r e sult s and to pr ovide the
b as i s for r e c on structing the c omput e r evaluations .
The fir st five
pag e s of r eproduc ed c om puter print - outs illustr ate the parameter s
monitored and the format of pres entation.
s ists of five columns of t abular data.
The actual print - out c on­
The mi s sing columns ar e r e ­
l ated to the fir st and s e c ond c olumn s shown b y symmetry.
The s e
initial data pag e s are follow ed w ith a copy of the input - options com­
pilation.
It
lists all information spe cified for the problem c al culated.
Compilation of the main c om puter pr o gr am is pre sente d in the
five page s immediately following those mentioned above . The s e data
give information on the o r g aniz ation and c omputational detail - step s
of the compute r prog r am .
The final eight page s are print - out re pro ­
ductions that give the detai l s on the six subr outine s embodied in the
main prog ram.
Figure 3 5 shows the c omputer flow chart .
It is hoped that the computer data included here will gr e atly facil ­
itate generating a de c k of input and c ontrol c ar d s r equire d t o evaluate
the pr oblem which w a s pre s ented.
91.
D AT A FOR
T I., E
STE P
3 . E '7 2 4
- · -· · '"· � ---- ·
=
C. 5
CO 'II S T A N T S, K , F O R B E l A =
4 . 6 6 5 7 4E - C �
4 . 5 6 '7 8 9E - C 9
.
- 4 . 6 5 7C4E -0 9 .
· - ··- - - · - - -- -- · --4 �-56<; ti"i;E' -Ti;
5 . 2 5 4 9 BE -09
4. 7 3 6 6 5E - C S
4 . 6 5 7 C 3 E -09
4 . 5 6 'l e 7E - C S .
·�-·--- -4:T6-5 74E- c� - - · -4·� -56 9.8 8 E �-09 . .
R EU S S
C AT A
4 � 736 � i t: .:.=J 9 ;
stRICi N-,
0.5
-DE- ) � p p , F C R BE T A =
1 . 7 8 1 9 3 E -) 6
<; . 0 7 7 6 9E -07
4 . 2 5 6 2 4 E -3 6
3 . 0 9 9 3 CE - C 7
1 . 3 4 1 3 8E -06
=-W""6'1· 5·6E=-cr-···-s··� ·z 7T6at:-:=:ot-···
i �ifs.i 4 2 E�o s
3 . 0 6 8 9 7E - C l
4 . 2 5 6 2 3 E -:l 6
1 . 3 4 l 3 8E -06
3 . 099Z5f' - C 7
.1 . 7 8 1 84 E -06
3. 0 6 8 9 7E - C 1
9. C 7 7 6 5 E -07
_
_
_
_
_
__
_
_
__
--·
---.
·
'"
--·
·
·
-··
_ _____ ____ _ ___, _ ______ _ _ _ ____ ____ ----- - ·
1.0
IN C R EM E N TAL C R E E P S TR A I N , DE > � P P 9 F C R BE T A =
3 . 6 3 5 8 3 E -0 6
7 . 3 <; 5 7 9 E -06
l • 0 5 1 54E- C (:
·
-----,8
ii-.-.2 1 7 1 3-l:·=--C7--- ;; ;-6T44 8E=-a6· · ·· -y�-3 2T9ifE �35 -
- 1 . 5 2 444E- C 7
8. 2 1 7 C CE - C 7
-- ·--- -- · ·- r.-d5-7 54F.::: t E
I N C R H1 E' H AL
z : 3 i;-5 7 8E ;_06- -
6 . 6 1 6 8 4 E -J 5
l . 8 2 1 9 7 E -::l 5
3 � 6 3 5 B 3 E -0 6
0.5
CR E E P S TR A I N , OE , '1 P P , F CR B E TA =
3:-os 9 3.5F.<'it· · - ··:.:t �-}6.i4.5E:.oa3 . 0 6 902E i n
1 . 7 8 1 841:: - ( 6
---- ---·- -- -- 9-: oi 7 5 <ii: � cT
9 . 0 7760E - o 7
3 . 0 6 9 02 E - C
I N C R EM EN T AL
2 . 4 6 6 3 8E -0 6
4. 6 1 4 4 7E - 0 6
1
1 . 3 41 3 8E -06
4 . 2 5 6 2 4E - 0 6
C 3 4 i 3 8E ::.. o6
3 . C � 9 3 2 E -0 7
5 . 2 7 i 6 B E -H
1 . '3 5 1 4 2 E -3 5
s . 21 l 7 5 Hl7
- 7 . 3 6 7 4 8 E -0 8
CREEPTTR A I N-;-OE�PP;FGR-- BE"T A �-- ·y�o··- -·· - ---
1 . 0 5 ? 5 5E - C 6
2 . 3 9 5 7 6E - C 6
-------- 3�6::f!n'!3F n· - · - ·
2 . 3 9 5 7 6E - C 6
5 6E - C 6
5 7�
. 0-�1�
--------
S . 2 1 72 7E -07
4 . 6 1 4 4 8E -, 06
f� 8 2T 9 8E-..:o5
4 . 6 1 4 4 8E -06
8 � 2 1 7 1 8E -07
T I'I f= S T E P =
- 1 . 5 2 4 4 3 E -J 7
2 . 4 5 6 3 8 E -J 6
6� 6 i 6 8 4 E -:> 5
2 o 46 6 4 0 E -0 6
- 1 . 5 2 44 3 E -0 7
3 . E S2 4
0.5
F C R BE TA =
l . !:l 5 6 1 9 E -1 2
1 . 9 8 63 6E -06
1 . 9 8 6 3 6E - C 6
3 . 0 9 1 6 � E -1 2
. L 92 0 49E -06
6 . 9 1 0 2 5 E -1 4
- 3 . 6 92 2 1 E -1 5
l . Q 7 1 0 3E - 1 2
- 1 . 08 1 7 2 E-12
- l . S Z C 4 9E -06
- 1 . 9 R c 3 5E - C 6
------·---- .- -=· z : 1r2 1 fi 7r: ..:: n-- - :.:y� <i s 6-3-u _;o& ·
-- -r : 0 6 4 1 1 E .:. u
I N CR E'4 E� T AL
4 . 7 3 6 '> 6 E -J 9
5 . 2 5 4 9 8 E -:l 9
1 . 6 9 9 7 9 E :<l 8
5 . 2 5 4 9 7 E -Q 9
1. C .
R EU S S CJN STAN T S , K , F O R B E lA =
. · e; . C 74 7 5E-_o9
4 . 3 1 8 4 3 Hl 9
- ---- ----· - - · .. . . -· a�T76;(4F c �
B . 1 7 6 81 E - 0 9
1 . 1 9 1 3 9 E -J 8
6 . 0 7 4 7 3E - C S
4 . 2 4 9 8 6 E '-0 8
1 . 1 91 39E-08
4. 8 1 842E - C S
----------7
�
.
i
f
74
6
7 1 E .:-·c-e;---- - -·€:T768oE=ti9- · -- -T� FH 3 A E -o s· ·
6 . C 7 4 7 3 E -09
8 . 1 7 62 51' - C S
4 . 8 1 8 4 2 E -0 9 ;
---u�·c-R EME�TA[ · rR.EEP.
FO R
CR E E P
S TR A I N , . D E > 'I P P ,
2 . 8 2 1 8 7E - C 6
1.0
I "l C-R· ·E'I-· EI'-J-T AL CR HP S TR A I N , D E n P P , F C R B E T A =
·
-- · · ·- - . . f . o i 5 4 5f :.:
. 5. 3 3 1 84E -C6
2 . 1 5033 E-12
C5
5 . 3�1 81E- C 6
6 . '7 5 8 1 5 E - 0 6
1 . 3 6 2 4 5 E -l l
- 4 . 6 3 1 82 E - 1 4
2 . 1 9 2 HF - 1 2
3 . 87247 E-12
-------- --- ·= 5-:-TffaoF
= u - - -.::: 6 :- s 5 liT 3 E: -·o6- - :.:4;3· 4oo<i E
�i 2
- 1 . 0 l 5 4 5F - C 5
- 5 . 3 3 l 82 E - 0 6
-2 . 1 6 4 8 B E - 1 2
-ElWoN·E� T � N i(x ,
·-·
1 . 4 C O l l E CO
B A S E D - O N f1 E nP_P_
1 . 7 9 4 8 8E
OC
1 . 4 0 6 6 '>1" C C
1 . 7 8 2 45E 00
·-----r;c�;;c;cz E:--·-cc - --z-�-i26T6f: - oo·
1. 4 0 6 6 Sf C C
1 . 7 8 2 44E 0 0
1 . 7 8 4 8 <;[ C C
1 ... 4- C O l l E 0 0
-- -·-- ·--·· ·--- . ··-- · · ·-
B A SE D O N D E H P P
1 . 4 0 6 6 9E 00
1 . 7 8 4 8 8E C C
-- - - -T;t;;iO r.:c·cc·- --- · r;·re·z·,.-s.E= - o a · ·
·
2. (' <;7 8 5E 00
1 . 0 .� 8 <l 2 E C C
1 . 4 i1 0 1 GF - C C
1 . 7 B 2 4 5 E 00
-· -·- -- - - i� -7
i3 4 8 Rf C C
i. 4 C 6 6 9 E 06
E:X P O \J EN T ,
- - · · - ·· -
N YY ,
T , 'II X Y ,
E XP O \J EN
----
-· ----- -
--- -r:-fi4"i4ct:___cc--· · 1. 4z•rscf ·c o
BA SED
ON
1 . 4 2 4 5 Ct
. - - i . 4 2 4 5 CE
1 . C: 2 5 6 2E
1 . 8 4 1 4 CE
D E ) 'I P P
CC
CC
C(
CC
1 . 85 723E
3 . 6 4 9 C3 E
1 . f! .5 7 2 3 E
l . 42 4 50 E
00
00
00
00
l . Q2 8 9 2 E 30
2 . 'l 9 7 8 5 E ::JJ
2 . J978 5 E J O
1 . J 2 Fl 9 2 E 0 0
i : 83 7 5 1 E j j
3)
n
03
OJ
1 . 04 9 0 3 E O J
1 • .J 4 CJ O 3 E
2 :� z s o & E
1 . 33 7 5 1 E
2 . 22606 E
1.�'J 2 %9 f j :J
2 . 1 3 'H 4 E 3)
5 . FJ J 8 3 7 E 3 :>
2 . 00440 E 3 3
l . 0 2 45 6 E 0 3
--· ·-·�" ... .. -·- · .
SAM PLE PRINT - OUT FOR S PECIMEN ),
LOAD AT CENTER = 10 3 . 5 lb; TEMPERATU RE = 545°F
TIME t = 11 .62 hrs ; PRECEEDING TIME STEP " DT" = 3 .89
hrs .
-.!)
N
DATA
T I '1 F =
FOR
EQU I V IL EN T
LOAD
l l. H71
I N TE N S I Ti f � , QE
-2 . 6 6 7 0 9 E -0 3
- 5 . 0 6 .� 1 5F - Cl
. - 1 . 0 2 5 94 E - C l
-4 . 0 3 99 6 E -:l l
2 . 3 l 7 5 2 E :) J
- 4 . C 3 9 9 5F -Ol
- 1 . C 2 5 9 3 E -Ol
.:: s; 015 174t :,. 03 ·
·-::. f. 0 6 9·0-SF .:: C2
- -
EQlJ IV AL fN T
.:. 2.
CD N C E � TR A TE O
L C AD � ,
4 4 5 9 SF. - C l
- l . C· l 7 1 CiE - C l
-l
. · - -- - .
�! OM EN T S ,
M XX
OJ
-4 . 0 3 1 5 1 E -05
-6 . 7 6 6 8 4 E
00
00
1 .672 3 9 E J2
-6 .-7 6 6 8 4 E :) :)
- 1 . 0 1 7 1 3 E -Ol
-3 . 4 7 3 R 3 E -O S
2 . 6 2 3 6 6E - 0 2
3 . l 8 7 2 B E -0 2
2 . 6 2 ">, 6 6E - C 2
4 . '72 3 7 8E -02
6:"t 9 4 2 3f: .:.oz
6 . 1 9 4 2 3 E -J 2
8 . 4 2 i 1 5 E -0 2
4. 'l 2 3 7 8E -02
6 . 1 9 4 2 3 E -0 2
l . 4 2 R 8 l f: - C 2
2 . 6 2 3 6 6 E -02
3 . 1 8 7 2 9 E -0 2
7 C G C7F 0 0
·4; iJ4 1 42"E ···· c c . -·l: O�')f 4ZF. 0'1
1 . 2 2 0 C;> E Ol
4 . 1 1 8 3 �E 0 (
l . C 51 43 E Cl
4 . 0 4 1 2 5E c c
·s . 6 <i 9 <i 9E oo
2. 3 9 5 7 7[ c c
00
2 . 1 43 0 3 E : H
W
- 6 . 7 6 6 84F
- 2 . C .2 .2 4 1E
- 2 . 02 2 4 9E
1 . 4 2 8 E!l'' - C 2
- --- · . . - . 3 � i8 72FE- (:"� - ·
2 . 6 2 � 6 5!:' - C 2
"1 0 M EN T S ,
7 C 9E -O l
-
-4 . 0 3 9 9 8 E -J l
:::z· . 6 6 6 '> l E -33
00
- 2 . ;> M 4 7f- - C �
... - - ---- --- --· · - ·:.:: t : ol 7 l 6F'- C l
- 2 . 4 4 "- C l E - C l
D EFLE C T !ON s,
Cl
P
•.
2 . 3 9 5 1"3E
5.
cc
9 . 1 5 9 75 E
4 . 3 99 1 6 E
CC
l . 0 5 1 42 E
01
1 .22002 E
01
CC
2 . 1 4 3 C3 E
01
4 .3991 6 E
01
5 � f: q 9 9 2E
CC
l . C 5 1 43E
CC
4 . 04 1 3 4 E
l.22J02E
4 . 1 1 848 E
)1
2 . 3 9 5 7 CiF
01
"Mo�E"�E�- r;.;·xv -- --
r. c
c . t, 9 c 4 EE
4 . 8 3 2 7 5[ c c
2 . 5 2 0 C 3E - C {
- 4 . 8 3 2 7 5F C C
- 4-� M1 4 2 E
oo
T CI T AL
..
:J J
-
--
E X X, A T BE lA =
z. 4 0 ' !: eE - c 5
3. 3 4 1 6 i E - C 5
.
.
A T R F: l A =
4 . 11 0 7 3 5E - C 5
6 . 6 A 3 ? 2E - C 5
S T R A IN ,
·l .
5 . 0 9 4 0 5 E -0 4
3 . 9 2 2 5 5E - C 5
1 . 6 5 4 3 5 E -04
8 . 8 2 7 4 9 E -J 4
4 E -o 4
4 . e C 7 4 CE - C 5
l . 2 8 64 2 E: -04
2 . 2 1 l � :) E -) 4
LTe2 s e"E =-c:s------i;-nr,ni-4 t =o4 ___ --- s� o94.o
EVY,
AT
c-: 5 ·
B E l A ;,
6 . 4 3 2 1 8E - t; 5
i :- i. 3-s-7sf. �.:.- c 4 !: . 4 3 2 0 31.: - C 5
2 . 4 C 3 74':- C 5
S T R A IN ,
EYY,
A T
-.
3 . 3 4 1 61 E -0 5
1 . C 5 4 <; 0 E -04
1 . 9 6 4 2 8 E -0 5
8 . 2 7 1 1 4 E -0 5
1 . C 5 4 9 2 E -04
8 . 2 7 1 1 3 E -0 5
3 . > 4 1 54E -05
1 . 96 4 3 8 E -G 5
z�-5'47 CTE:---c4___ ____4:4T3TsE=-o4· ·-
. --
CO
2 . 5 2 0 0 3 E -0 6
6 . 7 2 0 0 9 E ..:.:J 6
00
- 2 • 5 2 0 0 3 . E -:J 6
CO
l � 3 4 4 C 2 E -0 5
- 4 . 6 4 4 8 8f 00
.
8 3 2 76E
·
1.0
BE lA=
. P O B 5E - C �
6 . 6 8 3 2 2 E -0 5
3 . 9 2 8 5 5 E -0 5
_____ ____ _ 4r:;;
_
s c 44E·=-c 4 · · -- -z�Tcii-s rE--=o4 -- -- · - r:-654""3-s E:-:.oii-- ·
2 . ? 7 1 5 EE - C 4
5 . 0 9 4 C 5F: -04
8 . 8 2 7 4 9 E -:l 4
2 . 1 0 Ci 8 3 E - C 4
1 . 6 5 4 3 5 E -0 4
1 . 2 8 6 4 1E - C 4
4 . P d7 4 eE - c 5
TOTAL
S T R A IN ,
---
--
2 . 2 n: 5 7 E -ci 4
2 . 1 0 9 8 1 E -C4
2 . 40 3 6 & - C �
T O T AL
. ...___ ...
1 . 1 3 5 7 9 E -0 4
2 . 5 4 7 0 3 E .:.0 4
-
1.0
Z 8 6 44E ::.04
EXY;
..
-- · -
TOT AL
C. 5
6 . 4 3 2 1 8 ( -0 5
1 . 0 5 4 CiOF: - 04
1 . 9 6 4 2 8f' - c 5
4 . 4 1 3 7 5 E -0 4
8 . 2 7 1 741: - 05
3 . 3 4 1 4 4E - C 5
1 . 0 5 4 Ci2 E -C4
2 . 5 4 7 0 2 E -0 4
.... . 2 . 4 03 ? CE ..::-c-5- · - 6 � .43.2 C 9E -05 - · - · - r:U"580 E -o·4
S T R A I '� ,
..
1 1 . 6 :-1 7 1
----
-
E XY,
A'l
6 � 6 i' 3 c aE .;:.b5
-
G. 5
!1 E l A =
-
-
- 9 . c o 4 a �E - c 5
- 4 . 6 <; 5 5 7E - l l
9 . 0 () 4 8 Sf - C 5
1 . 2 4 7 7 5E - C 4
- e . 6 5 4 80E -05
-1 . 2 5 2 1 5 E-lD
- 2 . 5 C 4 3 CE - 1 0
- 5 . 4 7 8 1 7 E -1 l
-
E . 65"it 7 9F -05
<; . C C 4 8 7F -0 5
; -· · r:e:i'iir_At:__ _sl'iri-i iN ·� -EX"v;- iiTT('IA"
- 2 . 4 9 '> 5 1 E - C 4
- l . 8 0 0 9 7E -04
- 1 . H 0 0 9 7E - C 4
- 9 . 3 9 l l 4E - l l
2 . 9 4 0 0 4 E -0 6
_l�'::'>2.��_:_()_�
1 . 8 0 0 9 7f - C 4
-5 . 4 6 0 0 7 E-06
-----
!SAM PLE PRINT - OUT CONTINUED I
3·:,rz s 76 E .:.os
- --= f� 24 i7"" 5-E'-=-4-- · ·-=-.;-:--c c4a6r:.: o!'r--- - · =:t.-.:·6-9s
5 t E:-=ir --c
--
..
4. 832 75E
4 64 4 8 8 E
-4.
- 6 . 6 <l 6 '• f'E C C
-··--·------�-· - ·--···· -·- ·-··· · . ..____ - .
..
00
--
01
5 . 7 0 C C 7E
9 . 1 5 S 7 1E
-
--- - ·--- - - --
2. 14303 E J l
9 . 1 598 7 E J J
·4 ; ii 8 3 7 E J O
M YY
-- - -- " " - -..· --- -----:z�-3 9 5'1"3F - rc
S TR A IN ,
T O T AL
- 5 . 0n 7 2 2 E -03
- 5 . 0 6 13 5 ?E - C 3
TI�E =
FO R
.
- l . U 8 ' H l E' - C2
- 2 . 6 6 n f! 7E' - C 3
-- - - -- -- - ·
DATA
·---··---..-.
_____
..
._
·
--
- 1 . 1 3 C 96 E -04
-5.
C 0 86i E -1 0
4 . 69 5 5 7 E-l l
1 . 0 1 7 3 1 E- l 0
· - -----
---
--
- -- -- -
-9 . 3 9 1 1 4 E - l l
-2 . 5 0 4 3 1 E - l J
-l � 0 95 6 3 E .,;.l J
9 . 3 9 l 1 4 E -l l
!� .!:1.2.22!!:_-:g�---- ��'>1.!! g:_l 0
l . 1 3 C 9 6f - 04
_ ___
-..D
LV
1 - 1 . 6 3 7 ,1.
.
·-·-�
I'L AS T I C S TR A IN , F X XP , A T BE 1 A =
G. 5
2 . 2 8 1 0 7F. - C <;
6. C 6 4 C l t: -05
1 . 1 6 3 3 4 E -0 4
2 . 3 8 7 9 4 E -0 4
3 . 2 1 ? 8 �E - C 5
l . 0 0 l 5 8F - 04
E . C 3 4 4 8E -05
3 . 7 9 2 1'> 2 E -0 4
l . 9 8 2 4 "3E - C �
3 . ? l 2 6 eE - C <;
1. 0 0 1 5 9E -04
2 • 3 8 7 9 4 E -:J 4
·-· ----- · -----·- ··- · :t� ·et ro: - c ; - - -6 ; ·c6H2 E' .::os- · ·· - · r. o &·n s E
.::o 4
z
DATA
EL A S T IC
fr R
T l >I E =
S TR A IN , E X XP , A T B E l A =
1.0
i . l 'l 8 A BE -04
4 � 4 4 'I 4 SE - C 5
6. 3 7 6 8 4E - C 5
1 . 94442 E -04
l . 5 52 1 5f -04
3 . 9 6 5 l lE - t:: 5
2 . l 3 l & 3 E -0 4
4 . 4 3 6 4 8 E -0 4
5 . 2 0 5 7 4 E -0 4
- -- l. ·; c;4 ''4s E· :: o.
l • l 9 A 8 6E -04
. .. '4 ;'4"36 4 1 t:.:.:J 4
- 0. 5
2 . 2 8 1 0 7E - C 5
3 . 2 1 2 8 5E - 05
l . C 0 1 5 8E - 04
6 . G 6 4 0 1 F. - C 5
-·----- ---··-- ---T; b-6334·E .:. t4 - .. · ·- :z·�3 8 7 G4E - o4
1 . 0 0 1 5 9 F -04
6 . C 6 3 P 6�- C 5
3. 2 1 2 7 BE - G5
2 . 2 � 1 1 4E - 1: 5
1 . 9 8 2 4 3 E -0 5
8 . 0 3 44 8 E -0 5
. 3 . 7 9 2& 2 E -3 4
fl , 0 3 4 4 7 E -Q 5
l . 9 8 2 5 3 E -0 5
----- · - ··6:3 76 s ·cE: = C5
4. 4 4 3 5 5E- C 5
r:C i\ s tl c ·sTRA IN ; t v vo , A T R E lA =
1.0
S TR A IN , E V Y P , A T B E l A =
6 . 3 7 6 84 l -05
4 . 4 4 '1 4 �E - C 5
----- ---·- ·· --l.TGii i'iiiE :.. 64-- -· . i� 94 4 42 t: :.:.o4
4. 4 3 6 4 8f -04
2. I 3 H 3E - C 4
1 . 1 G R 8 5 E - C4
l. 9 4 4 4 5f - 04
-- -- ....
- . . 4. 4 4 3 6 :\E - G 5
t . 3 7 6 7 dE -05
3 . % 5 l l E -0 5
C 5 5 2 1 5 E -j 4 .
5 . 2 0 5 7 4 E -0 4
1 . 5 5 2 l 'i E -04
3 . 9 & 5 3 1 E -0 5
A S T I C - -S TR A -IN , E X YP , A T 1\ E lA =
0. 5
-EL-- --··· ·-- · :.: r;-3538 '3 -c4- - - -.:;;� n·5 7n·=a
·
s · · ·· =-4; T43.6 il E:_ i t
·
F
- l . 2 5 46 'i E - l O
- <; . 3 5 92 7 c -05
- 9 . 7 6 5 7 7t' - C 5
- 4 . R 3 7 2 6E - l l.
- 2 . 5 02 B () E - 1 0
-5 . 5 6 9 6 4 E -l l
4 . 3 8 9 7 9 E -l l
<; . 3 5 9 2 5[ - 05 .
9-� 7 b '5 7 7F. - t 5
1 . 0 2 5 7 6 E -D
G . 7 6 5 7 8f: -C5
1 . 3 'i 3 fl 3F - C 4
.
_
- 2 . 8 2 1 0 3F - C 4
- 1 . 9 8 4 1 7 £::. -04
- l . G 'l 4 1 7F - C 4
- 1 . 9 5 7 5 0E -04
. - 5 ; C C 0 0 6F - l 0
:_ 9·. f. 8 4 4 S E - 1 i.
1 . S '5 7 5 CE - C4
l . S 8 4 1 7F - C I•
l ._�84_1 7E .- 04
--- --- ---- _ _ _ _ _ _ f? . R_2_00} E ::_f.4_ _
D A -T A
--�--
S T R ES S ,
-
FOR T I M E =
---- ·---- -- - -·
F X XP ,
S T R E S S , F X XP
. .... . . .
..... .
,
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- 9 . 4 9 7 5 Cl E - l l
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I S N 00 3 1
ISN 0 0 32
I S N D0 3 3
ISN O Q 34
I S 'l 0 0 3 5
_
D A TE
6 7. 2
F O RT R A N H
CS/.3!> 0
- -- .. - -..�-�- ·-..·--� - - -· · --· ·-·· -· - · -M A l N , O P T =OD , li N E C NT =ft2 , s OU R C E, E B C D I C, N O L I S T , D E C < , L J A D , M A P , ' O E J I T , I D
A TO M I C S I N TE R NA H C NA L
- NA M E =
C O M M O N / /E T A C 5 , 5 ,5 , 5 ) , P I 5 , 5 1 , W DX X 1 5 , 5 l , w D'I' Y I 5 , 5 1 , w DX V I 5, 5 1
A B 000020
A 8 000D30
1 , M X XI 5 , 5 1 , M 'r VI 5 , 5 1 , � X V 1 5 ,5 I , E l( X 1 5 , 5 , 2 1 , EY f ( 5 , 5 , 2 I , fl( Y I 5, 5, 2 1
A B 000040
2 , E X XP I 5 , 5 , 2 1 , E 'I 'Y P I 5 , 5 , 2 1 , E X Y P I 5 ,5 , 2 1 , EX l( P P I 5 , 5 , 2 1 , EY Y P P I 5, 5, 2 1
A B 000050 .
3 , E X YP P I 5 , 5 , 2 l ,FX X P I 5 ,5 ,2 i ,FYY P I 5 ,5 ;z i ; i=i � p{ 5 , 5 , 2 i ···· . ... . -AB 000060
4, S X XP I 5 , 5 , 2 1 , S 'r Y P I 5 , 5 ,2 1 , S XY P I 5 ,5 , 2 I , J2 P 1 5 , 5 , 2 1 , I 2P P I 5, 5, 2 I
AB 000070
5, R Kl 5 , 5 , 2 1 , D E X X P P I 5 , 5 , 2 1 , D E Y Y P P I 5 , 5 ,2 I , DEX f PP 1 5 , 5 , 2 I
AB OOOOBO
6 , E N X XI 5 , 51 , E N V Y I 5 ,5 l ,E NX Y I 5 ,5 I , D E NX X 1 5 , 5 I , 0 ENY Y ( 5 , 5 I , D EN XV I 5, 5 1
C O MM O N / I N P l T S / T I T L E { 1 8 1 , P L OAOS 1 2 5 1 , E , MJ , C A , C B, Cl{ , H, L X , L v , X X , Y Y
A B 00 0 1 0 0
A B D OO l O l
1 R., EA
TIN
T S TO
P , D TI N l T , D T MU LT , PR I NT , F I L L i l ) I
L I*4T , lfC
, t :X , t v -- . -- ·- . . .. - - - - - -- -- -- ... - -- . -- - - -- - ----- -- - --. A B DOD1 D5
A B 000200
c o M M O N / SE IIE N S / k i 7 , 7 1 , PH I I 7 ,7 1 , PS I I 7 , 7 ) , 0M E I7 , 7 1 , Q E I 7 , 7 1
A B D002 1 0
D I M E N S I O N . P P I 2 5 1 , lo l\ ( 24 5 1
A B OD0212
D I ME N S I O N D � l 7 , 71
AB 000220
E Q UI VA L E NC E I P i l , l ) , P P i l l l , 1 11 1 1 , 1 1 , WW i l l l
A 8 DOD230
E Q U I VA L E NC E I T A U ,F I l l l 2 l I
tbMM fiN ill ME RII'i" ME ,b t , � P Rf llii' ;M_P___
A B D002 5 0
A B 000300
R E A L * 4 J 2P , I 2 P P , � X X , �Y Y , �X Y
1 0 C A l l P R E E TA I E TA I
AB 001000
50 C A L L SE l lP
A 8 0 0 1 500
A 8 0 0 1 90 0
DO 9 C N = 1 , 2 4 5
.
90 WWI N I = C . C
A8 001910
Dci 9 5 N = i , 5 o ·
.. -iis o t oooo·· -· --- ---· - 9 5 P L O A D Sl N l = C. 0
AB 0 1 0005
AB010010
1 0 0 C A L L R E AD
A B 0 1 0 D 2 0 ..
T I ME = 1 1 NI 1
D T = 0 TI N I T
A B 0 1 00 2 2
MPR I N T = 0
o\ 8 0 1 0 0 2 4
6T6o2 ·6· --·· -- - - M P ;, .PRTN T. c : i.
- - - · · ···· - - - -- - �� 8
--DO 1 1 0 J = l , 5
AB 0 1 0040
00 1 1 C I = 1 , 5
AB 0 1 00 50
P l l , J I = 0. 0
A 8 0 1 0 0 52
DO 1 1 C M = 1 , 2
o\ 8 0 1 0 0 5 5
E X XP P I I , J , M I = O . D
A B 0 1 00 6 0
·
·
· E vvi>-i> I I' ; J � M r- ·;· a:·a· --·A·s o i oo7o____ __ _ __ __ __
E X YP P I I , J , M I = 0. 0
A 8 0 1 0 080
1 2P P I I , J , M I
= 0. 0
A B 0 1 00 8 5
1 1 0 C O N T I N UE
A B 0 1 0 1 00
C A P = C A * SQR T I 1 . 5 1
A B 0 1 D l 80
C B P = C B *SOR T I 1 . 5 1
A B 0 1 0 19 0
·
c t<. i = ·l . C /C K- . . ..
· - ---A·s- o i <l"zoa - ··· - -- - D X = L Y. / 6.
A8 0 10220
COM PILATION OF COM PUTER
MAIN PROG RAM
...!)
00
-- fsi'Coo 36
ISN 00 37
I S 'l 00 3 8
i> V = L Y / t .
o x s = D > *D X
I S "' 0 0 3 9
I S "! 0 0 4 0
ISN 0041
. -ISN . 0042
ISN 0043
I S 'l 0 0 4 4
I S N 00 4 5
ISN 0046
ISN 0047
I S\j 00 4 3
I S "l 0 0 4 9
IS til 0 0 50
IS'l 00 5 1
IS tll 0 0 5 2
I S N 00 5 3
I S N OT 5 4
ISN 0055
ISN 00 56
ISN 00 5 7
I S N 00 5 8
· ·-gN · oo 5 9
I S N 0 0 60
I S N 00 6 1
ISN 0062
I S N 00 6 3
ISN 0064
---rs�r ·o-o65
ISN 0 0 6 6
I S N 0 0 67
isN oo6a
ISN 0069
-
- ---·T
0070
I S N <fo-71'
s'N
rsN 00 1 2
. J�-'l_ 00 7 3
I S N 0 0 74
ISN.... 00 7 5
___
--
O Y S = D ' *D �
0 XD Y = D X*O 'I
D X Y 4 = 4 . c •o x •D v
UM M U = 1 . 0- M L
UP M U = 1 . 0 � M lJ
UM M U 2 = LMM L * L P M L
0 = E *H *H *H / 1 1 2 . *UM fooi U2 1
OM = D * LMML
C O N A = 0 l!*D '1 /0
C ONS = � . / I E +H '*l l
C ONC = -C O NB * U P M L
C OND = E / U M M U 2
C O N E = -C O NO * LM M L
C O NH = H *H / 2 .
C O N J = -C O ND
C O N K = C O N J * L M M L *2 .
c o N P = D x•o v/ a.· ·· · · · ·
ADD E O S E PT
C O N S U N T S B A S E D O N C RE E P M U = 0 . 5
C ON J = - E / 0 . 7 5
C ON K = C O N J
DO 1 7 5 N = 1 , 2 5
1 7 5 P P I N I = PLOAD SI N I
. f O NA X = t o NA /T. U
C O N A D = C O NA X-C O N A
D O 1 e 5 1 =2 , 6
DO 1 E 5 J = 2 , (:
D W( I , J I = C. C
D O t e O K,l , 5
- ----- · - --· - -- -- -- o o 1 e c i_ ;1 ;5 - 1 80 D W( I , J I = O \l ( l , J l + E T A I I - 1 , J-l , K , U * P ( K , l l
1 8 5 O Wl I , J ) = D \1 1 1 , J l *C C I\ A O
D O 1 e o; N =1 , 2 5 .
1 8 9 P P I N I = C. C
WR I TE I t , 1 8 � I
16 ,
·ia6 -i'oR MA T - -i · o · · DE L iA- 11 ' 1
WR I TE I t , 1 e 7 1 I I D \1 I I , Jl , I =2 ,6 1 , J =2 , 6 l
1 8 7 F O R M A T ( 1 5 X , 1 P 5E 1 5 . 5 1 ·
C R E T � N P O I N l F OR NE X T P A S S T H R OUGH L O O P
190 D O 1 S 5 N=l , 2 5
1 9 5 f>_P (_NJ = f'P ( N l. + P LOAD S I N I
I PROGRAM COMPILATION CONTINUED I
1967
AB 01 0230
A B 0 1 02 4 0
A B 0 1 0 2 50
A B 0 1 02 5 5
A B 0 1 0260
A B 0 10 2 7 0
A B O l OZ BO
A B 0 1 02 9 0
A B 0 1 0300
A B 010310
A B 01 0320
A B 0 1 0330
A B 0 1 0 340
A B 0 1 03 5 0
A B 0 1 03 6 0
A B 0 1 0 370
A B 0 1 0 3 80
A B 0 1 0390
.
- A B O i 64cici - ----- -------- ..
A B 01 0500
A B 010510
AB 0 10520
A B 0 1 0 52 4
A B 0 1 05 2 6
. 8 0 1 0 53 0
A B 0 1 0 53 2
A B 0 1 0 53 4
A B 0 1 0 53 6
A B 0 1 0 54 0
A B 0 10550
. A B 0 1 0 5 6 0 - - - - -- - A B 0 1 0570
A B 0 10580
A B 0 1 0 59 0
AB 0 1 0595
A S 0 1 0600
f8 o 1 o6 1o-­
A B O l0620 A B 0 1 0630
A B 0 1 0 89 0
A B 0 1 0900
A B 0109l0
.
...0
...0
-- ·-----rN<J"O"f6
r
-
IS� 0 0 7 7
- iSN 0 C l 8
I S � OC 79
I S N --00
0080
--· - TsN
81
---
I 96.Tdi·fiTN.LE
____
-
.
CO�P UTE E L A S T I C D E F LE C T I O N S C O RR E S PONDI NG T O POI NT L O A D I N GS
DO 2 1 5 I =2 ,6
0 0 2 1 5 J =2 , �
WI I , J I = 0. 0
0 0 2 1 C 1< = 1 , 5
_ ____
I S 'I 0 0 8 2
00 8 3
- ISN
- .. . . .. . -
---- no - 2"i c.. L :;i ;·s- ·-·
2 1 0 WI I , J. I = h l l , JI + E TA I I -1 , J-1 , K , ll * P I I< , l )
2 1 5 W I l , J I = !oi l I , Jl *C C N A
COMP UTE P A R T I A L DE R I �A T I IIE S OF TH E D E F L ECT I ONS
220 00 2 2 5 I =1 , 5
IS'I 0 0 8 4
I S N 00 8 5___ _ ·-· -- - D O 2 2 5 J = l , 5
--:rwo
--ts"N·oc;-a6
·i•.-c·;,; ·-w-, Y+"i ·; j+ 1; :.:· ..-, i"+C,
J+1 1
··
lsN oo a9 ·
I S 'l 0087
I S N 00 8 8
WD X X I I , J I = I II I I , J+ l i - TW CwC+ W ( I + 2 , J+l l i i DX S
WD YYI I , J I = l iol l + l , JI - T W G WC+ W ( [ + l , J+2 l l / Df S
-2 2 5 wo x v c i , J l = c \lc i , J > - �o c i , J+ 2 > -w o + z , J I +w l r +2 , J + 2 l l t ox'f 4
CiJMP UTE M O ME N T S A NO E LA S T I C ST RAI 1\S
2 3 0 DO 2 3 5 l =l , 5
·
I S N 0 0 90
.
·
- --lsN"-cio9C___ __ _____66 . .2 3 5 .. J":;i �-�f . --·· ---- -- -- - - - · - ·
fsii-- -ci() 94
ISN 0092
I S N 00 9 3
--
M X X( [ , J l
-D * ( IID XX I I , JI + IIU*WDYV ( l , J ) )
M VY( I , J I = -D * I IoiD VV I I , Jl + I'IU *W D X X ( l , J i l
M XVI i , j )
O M *WD XY ( I , J, =
C O NB *I ,. XX I I , J I -MU*MYY I I , J l l
E VY( l , J , U = C O NB * I IIVV I I , JI -MU * MXX I I , J l l
----·---·----Exv(T�- J,- ff · .,; - C'tiNc -;. M xvH� · J f - ---- - - - - - - - - .. - - - - - - ...
E X X I I , J ,1 1 *2 .
E XXI I , J , 2 1
I S N 00 9 8
E YYI I , J , 2 1 = E Y Y I I , J ,1 1 *2 •
I S N 0099
- e xvc i , J, 2 i = E x v n , J ,u •2 �
--- isi4--oioo -- I S N 0 10 1
. 2 3 5 C O"' TI N lE
C.O �P UTE TR UE S TR A I N S ---A F TE
R C-·RE
---E-P .
- - -2----=c.;;24ci --D o -z4·s ·M-;l.·i ·-; l-;:;:
o-;:;:
---iSN:--:;O:-;
I S N 009 5
ISN 0 0 9 6
09T·
=
E X XI ( , J , l )
=
SNii
---I-
=
-·-
. . ---ISN --0 10 4· -· - - - · . - ·· · - - ·e00 2 4 5 J =l , �
X XP ( I , J , M I = - E X X I , J ;MI -E
I S 'l ofo 5 I
X X P P I (, J , M l
E VYP I I , J , M I = E VY ( I , J , Ml -E VY P P I I , J , MI
ISN 0 106
E XYP I I , J ,M I = E X V ( I , J , MI -E X Y P P ( l 1 J , Ml
I S N 0 10 7
;-;
=:-:
---1 S N 0 10 8
--·FxxP ( l ;J" ;M I-;;·c c i\if*-iE X X P -ii-�-j-.--l!f+ MJ *F;; v· P-1 � � j ; ·M l -,
F YVP ( J , J , M l = C O I'.D * I EYV P I I , J , Ml + MU *EX X P ( I , J , M l l
ISN 0 1 0 9
F XVP ( I , J ,M I = C O I'.E *E XY P I I , J , Ml
N 0
1 10 -· ·--· -I Ss'.r
oTii
- -- ·-· · - -- . s Xxi> ( i- J�M
,
I . = I F X X P li ' J , ,., + F X X P ( l ' J , Ml -FU p ( I . J , � I I / 3 .
I
S V YP I I , J , M I = I F VV P U , J , ,.I + FVV P I I , J , MI -FX X P I I , J , M l l / 3 .
ISN 0 1 12
ISN 0 1 1 3
g_yp_!.J..!..!!..!. � l. ":...E.�_Y."PJJ. _,,I ! M I ··- ___ _ ___ _ . . . .
ISN 0 10 3
D O .2 4 5 1 =1 , 5
___
_____
__
l PROGRAM COM PI �ATION CONTINUED I
_
A B 0 1 0920
A B 0 20000
A B 0 2 0040
AB 0 2 0 0 5 0
A B 020055
AB 020060
A B 0 2 00 70
A B 0 2 0 0 80
A B 0 20090
A B 0 2 0 1 00
A B 0201 1 0
A B 020120
AB 020130
A B 020140
A B 0 2 0 1 50
A R 0 2 0 l 60
A B 0 2 0 1 70
A B 0 2 0 1 80
A B 0 20 1 90
A B 020200
A B 0 2 02 l 0
A B 020220
A B 020230
A B 020240
A B 0 20250
A B 020260
A B 0 2 0 27G
A B 0 2 02 80
A B 020290
A B 0 2 0 30 0
AS 020310
A B 020320
A B 020330
4 8 0 20 340
AB 0 2 0 3 5 0
A B 0 2 0 3 60
A B 0 2 0 3 70
4 8 020380
A B 0 2 0 3 90
li B 020400
A B 020410
AB 020420
......
0
0
SN-5T F._______ __ ___ _ __ __
TEMJ --;: F Xi(i> iT;J;i'.l * *Z:iTV'f p(i ; J � Ml * *Z + i FX X P I 1 . J , M I - FY V P ! I, J , Il l I * * 2
I
I S N 0 1 1 '5
J 2P ! I , J , M I = TE M J /3 . +2 . *F X Y P ( I , J , M I * *2
2 4 5 C O N T I N LE
ISN 0 1 1 6
T
C A L L F O R O lJ TP l l OF D A T A R E LA T E D T C T I J'! E
DO 2 5 0 1 =2 , t
ISN 0 1 1 7
DO 2 50 J=2 , t
ISN 0 1 1 8
- ----Tsr'ioTi9___ _ _ -250" wiT� Jy -; - � n ; Jf-i- o w n ·; Ji
IF I M P R I N T . E O . O I C A L L wR I TE A
ISN 0 1 20
I S 'II 0 1 2 2
I F ! D T . L l . C . O l G O T G 1 00
C J M ? UTE VA L UE 5 OF THE RE l; S S C O I\ S T A NT F O R T H E T I M E S f E P
3 0 0 D O 3 c C 1'1 -= 1 , 2
ISN 0 1 24
I S 'II 0 1 2 5
D O 3 t 0 1 =1 , 5
I:lcf- - 3 i': c----iS!if-bT26_ __
Ff� �- -
---
-
-
T WO A = S X XP I I , J , f' l * *2+ S Y Y P I I , J , Ml * *Z + S X X P I I , J , M i t: S Y Y P ( I , J , II I
T WDA = ( T wOA+ S X Y P I I , J , Ml * *2 1 *2 •
TE M P = C A P *E X P I C B P * S Q R T I J2 P t l , J , fo'l l l
I F I T I ME . E O . 0 . 1 G G TC 3 5 0
.
_ R l 2P P = SQ R T I I 2 P P I I , J , fo'l l
____
- --- --TE F F ;;;- (Ri Z P P-/ TE ;.. P I * *C J< I .
ISN 0 1 34
D R I 2 P P = C K *D T *R I 2 P P / TEF F
TWOC = I R I 2 P P+ O R I 2 P P I * *2 -I 2 P P ( l , J , M I
ISN 0 1 3 5
--- - · B X= -F X X P P ( [ -, J , M I * I S X X P ( I , J , foii + SX X P I I , J , M I +S Y Y P ( l , J , M I I
- -- - g ij '6I36:- - - B Y= E Y YP P ( I , J , Ml * I SYY P I I , J , I"l + SY Y P ( I , J , M l +S X X P ! I , J , M i l
ISN 0 1 37
I S N 0 1 38
B = B X+B Y+- 2 . *E X Y P P ! l , J , M I * S XY P I I , J 1 Ml
R-i<. iT, J M I ="i �A-i-- S OR T I B * B-i- T w c il •t W OC I l / T W O A -- rs·N·· of3if - - GO TO 3 � 5
I S N 0 1 40
3 5 0 R K ( I , J , M l = TE M P *D T * *C J</ S O RT I T W O A l
I SN 0 1 4 1
- ---- --· g-ij -Oi 4-;;!" - - 3 5 5 [) E X XP P ( l , J , M I ·:, R K ( [ , J , M J *S X X P ! I , J , M I
D E Y Y P P I I , J , M I = R J< ( I , J , M I * SYY P ( l 9 J , Ml
I S N 0 1 43
P ( I ,--J , MI
SXY
R K I I , -J- -··, M I. . *
l ---=- ---·
Y Pi"P
E X'f
I S N 0 1 4 4 -·- ----- --3 6(; D orii
.
NIiE:I ·,-J-, .II-TsN'-6'145·-C
c o M P UTE N E W T I ME A ND l.POA 1 E C R E E P S T R A I NS
4 0 0 T I ME = l l ME + O T
ISN 0 1 4 6
Do 4 t o M -= 1 , 2
- ---- - tsN o i 47
ISN 0 1 48
DO 4 1 0 1 = 1 , 5
I S N 0 1 49
DO 4 1 C J -= 1 , 5
·-rs'N- oTso _____ _ _
··-E"Xxi> rff ,-j� i-1 1' -;--fxXPfi i f; j---; MI +-iJEi x f> i>f:l-.�-::;, 1-l -i --E V YP P I I , J , ,. I = E Y Y P P.I I , J , ,.l + DE Y Y P P ( I , J , M l
I S 'II 0 1 5 1
E X YP P I I , J , M l = E X V P P I I , J , M l + O E X Y P P ( l , J , M I
ISN 0 1 5 2
. - TE M I - ;, (X XP P I I � J , Mi >tE X X P P I I , J , Ml + EY V P P I I , J , M I * FV f P P I I, J , '1 l
-- -·-EN. -()1 5 3 - ..
TE M I = TE M I + E X X P P I I , J , Ml * E YY P P I I , J , M I + EX V P P I ! , J , M I ., EX Y P P I I , J , II I
ISN 0 1 54
____ mL.9 kS_s__ ___ --- ------------ · - - I�P.J'JJ. !.J � � , :o_ __ Jp•t +-.T E M L _ __
_
-----
--
ISN 0 1 27
ISN 0 1 28
iS !>i- O i 2 9 -- -- I S N 0 1 30
1 32
SN -0
-- ··--yI s·N"
of33
-
-
- ----
,
-
- -
--
--
-
-
-
--- -
-
--
-
-
I PROGRAM COMPILATION CONTINUED I
A B 020430
11 8 0 2 0 4 4 0
A B 0 20450
AR 0 20460
1\ B 0 2 0 4 6 2
1\ !3 0 2 0 4 6 4
A B 0 2 04 6 6
AB 020470
A B 0 2 0 50 0
A f\ 0 3 0 0 0 0
1\ 8 0 3 0 0 1 0
1\ 3 0 3 0 0 2 0
AB 030030
/\ !3 0 3 0 0 4 0
A B 0 30 0 5 0
<\ 8 0 3 0 0 6 0
A B 0 3 0 0 70
11 8 0 3 0 0 80
A B 0 3 0090
AB 0 3 0 1 0 0
A B 030l l0
AB 030120
AB 030130
A B 0 3 0 1 40
A B 0 3 D l 50
A B 0 3 0 1 60
1\ 8 0 3 0 1 7 0
1\ 6 0 30 1 8 0
t. 6 0 3 0 1 90
AB 0 3 0 2 0 0
A B 0 3 02 1 0
A B 0 40 00 0
AB 040010
A B 040020
AB 040030
<\ 8 0 4 0 0 4 0
1\ 8 0 4 0 0 50
1\ 8 0 4 0 0 60
A B 040070
11 8 0 4 0 0 7 2
11 8 0 4 0 0 7 4
A B 0 40 0 7 6
,_..
-0
,_..
ISN 0 1 56
·
---
ISN
ISN
IS�
I S f\1
. .I S N
ISN
ISN
ISN
ISN
0 157
0 15R
0 1 59
0 1 60
Ol61
0 162
0 1 63
0 1 64
0 165
· I s �f
ISN
ISN
ISN
ISN
ISN
ISN
ISN
ISN
ISN
ISN
ISN
O l 66
0 167
0 1 68
0 1 69
0 1 70
0171
0 173
0 1 74
0 1 75
0 177
0 1 78
·-rs-..ioit'f
.
··--
·a f72
· ··
I S N 0 1 80
ISN 0 1 8 1
ISN o i a 2
ISN 0 1 8 3
-·· 1sf,r-aTB4· ­
rsN o 1 8 5 ·
I S :-.!
ISN
ISN
ISN
-·
0 1 86
0 187
0 1 88
0 1 89
---··· i s·N' -o 1 90- ..
ISN 0 19 1
ISN 0 19 2
I S N 0 1 9j
I S N 0 19 4
I S )! 0 1 9 5
I S t\1 0 1 9 6
·
·
4 1 C C O N T I N LE
C O M P U TE E XP O N E N TS F OR C RE E P S T R A i t\ V A R I AT I ON T HR OU G H T H E PL AT E
450 DO 46C 1 = 1 , 5
00 4 6 ( J = 1 , 5
E N X X I I , J l = A L DG 2 1 E X X P P I I , J ,2 l , E X X P P I ! , J . l l l
E N VYI I , J I = A L OG 2 ( F Y V P P I I , J ,2 1 , EV V P P I I , J , l l l
E N X Y ( I , J I = A LOG Z I E X V P P ( I , J ;z i , E X V P P ( I , J , l l l
D E N X XI I , J l = A L OG2 ! D E: X X P P I ! , J , 2 l , D EX X P P ( I , J , l l l
O E N Y Y ( I , J l = A L OG2 1 D E V V P P I I , J , 2 1 , D E 'f V P P I I , J , l l l
D E N X V ( I , J I = A L OG2 ! 0 E X Y P P ( ! , J , 2 l , D E X V P P I I ; J , l l l
460 C O N T I N LE
C J M P UTE E Q U I 'vA LE I'i T L O A D 1 1-< TE N S ! T I E S DU E T O C R E E P
. 500 DO 5 1 0 . I =1 , 5
DO 510 J=l , 5
E X XN =
I O E X X P P ( l , J ,2 1 / 1 0 E t\X X ( I , J I + 2 . ! 1
I D E V Y P P ( I , J ,2 1 / ( D E I'<V Y ! I , J I +2 . 1 1
E Y YN =
I D E X YP P ( ! , J ,2 1 / I D E t\X Y ( I , J J +2 . 1 1
E X YN =
P H i l I + 1 , J+ l l = C O NH * I E X X II+ . 5 * EV Y NI
... ...
. . . . P S I ! I + i , J+ l l .;, C O I'<H * I E YY II+ . 5 * E X X NI
OM E ( I + l , J+ l l = C O NH *E X Y N
5 1 0 C O N T I N LE
I F I M P R I N T . E Q . O J C A L L liR I TE B
5 2(1 0 0 5 2 5 ! =2 , 6
00 5 2 5 J=2 , 6
P H I D [f ; . fPHTfl �i-;J) - Pt-i' Ii i: � jj - i>Hi l l � j l + PH I I I + 1 , J I l l DX S
P S I D D = I P S I I I , J- l i - P S I I I , Jl - P S I I I , J l + PS I ! I , J + l i i / OY S
O M E D D = l OME ( 1 + 1 , J+ l i - C M E I 1 + 1 , J - 1 I - OM E I I -1 , J + l l +OM E I I - l . J - l i i / D X V 4
O E I I , J· l .;, C O N J * I PH I DD+ P S I D D I + C C I'< K* C ME D D
5 2 5 C ON T I N LE
C O M P U TE C O NC E N TR A T E D L CAD S E QU I V A LE NT T O D I S T R I RU T E D L O ADS
.
. . 5 3b i:i ci 5 3 5 I ='i , s · - . . . .. . ···· · .
00 . 5 3 5 J = 1 , 5
P I ! , J I = P I I , J I + D XD V *I.J E I I + l , J+ l l
. 5 3 5 C O N T I N LE
1'\ 0 D X = Q E I 4 , 4 l *F l l l l l i / I CE I 4 , 't i - I.J E I 4 , 3 l l
P l 3 , 3 1 = P I 3 ,3 1 -D XD Y *Q E ( 4 ,4 l * l l • -1 • 3 3 3 * 400l( * * 2 1
f> i 2 , 3 1 =· P (2 , 3 J -o xo v *Q E I 3 ,4 1 * t A oox -::i � 5 1
P l 3, 2 1 = P l 2 ,3 1
PI 3 , 4 1 = P ( 2 ,3 1
P l 4, 3 1 = P ( 2 ,3 1
C A L L FOR S I Z E O F N E X T T l II' E S TE P
C A L L F I ND O l
G O T O l 'it
END
I END OF PROGRAM COM PILATION I
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11 5 4 0 0 5 3 0
1 E ( b, 2 5 l I P P I N l , 1\1 = 3 0 1 , 3 2 5 )
;,< K i l l: l 6 o 2 0 l T l Mt.
E L A S T I.C S T k A l N ,
Fllk M A l" 1 ' 0
Wk l lt. ( o , 4 8 ) 5 1
WK 1 T E ( 6 , 2 5 1 I P P I N l , N = 3 2 6 , 3 5 0 1
WK I H : I b , 4 8 1 B 2
Wt U I E ( 6 , 2 5 ) I PI' ( i\j l , N= 3 5 l , . H 5 1
A \3 4 0 0 5 4 0
l o , 4& 1
�IK. 1 1 1:: ( (> , 5 0 )
I '0
I' Uk M A T
( t.o , 2 5 J
WK l TE: (
( 6, 50)
6. 2 5 1
E L A S T\ l C
S TR A I N ,
( ' ()
AB400550
A B 4 0 0 5 60
E XX P ,
AT
1:\ E T A= ' , F6 . l l
11 5 4 0 0 5 90
A d 40 0 6 0 0
11 5 4 0 0 6 1 0
EYY P ,
AT
1:\ ET A = ' , f & . l l
Wk I l i:: I o , 2 5 1
( 1U
fUK M A T
A B 40 0 6 2 0
11 54 0 0 6 3 0
I P P I N l , N = 3. 7 t. , 4 00 1
11 8 40 0 6 4 0
ll 2
AB400b50
I Pl' ( l\j ) , N= 4 0 1 , 4 2 5 1
ELASTIC
STR A I N ,
AB400660
EXY P ,
AT
B E T A = ' , F6 � 1 l
;,K l T E
( 6 , 54)
AB400 5 7 G
A 5 4 0 0 5 BO
tl 1
I a , 52 1 B l
w i U T E ( b I 2 5 ) I P P I N l , N =42 6 o 4 5 0 l
Wk I T E ( 6 , 5 2 I ll 2
Wk I T E ( b I 2 5 1 I P P I N I , N=4 5 1 , 4 75 1
l·iid T E 1 6 , 2 0 1 T I ME
Fu R M A [ ( ' 0
5. T K E S S , f X X P , A li
WR ! T I:: ( 6 , 5 4 ) 6 1
>* I ll:: ( 6 , 2 5 ) t I' P ( N l 1 N = 6 2 6 , 6 5 0 1
f UR M A l
WK l l E
5o
tll
T Ll T A L
A B 4 0 0 3 90
rJrl I 1 i::
Wk l i E
00o4
ISN
WK
wR I T I::
L:>N 006 5
l S N 0 0 66
LS N
I •u
WK l
48
;'1 X V
T I M£
10 T A L
I 'u
o o 42 l I H
6 , <: 5 1 I P P ( I� )
o o 4 2 l L> Z
6 , 2 5 I I I' P L.J l
251
6, 2U l
6 , L 5 1 I P P i iol l , N =2 2 o , 2 50 l
b t 4 4 1 t> 2
WK l T I:: l o , 2 5 l I P P ( i\1 ) 1 N = 2 5 l 1 2 7 5 1
TOTAL STRAI N , E X Y ,
fllk i•\ A T I ' 0
WK l hc
L S N UU 5 l
I
Ot
A B400680
A B 4 0 06 9 0
A B 40 0 7 0 0
11 5 4 00 7 1 0
II S 4 0 0 7 2 0
ll E T A = 1 o f b . l l
A B400 7 5 0
A B 40 0 7 60
I P P I N l 1 N= 6 5 1 , 6 7 5 1
F YYP ,
A B 40 0 7 7 0
A T BETA=' ,F 6 . l l
I SUBROUTINE "A"
11 8 40 0 7 3 0
A B 40 0 7 4 0
ll 2
S TKt S S ,
A 5 40 0 t> 7 0
CONTINUED
I
AB400780
I
i
I
1-
I
.......
0
0"
LS N 0 0 1 �
L S N 00 7o
I S N uon
bN
J0 74
b N 0 0 7 di
L S N iJU /9
Bl
I P P I N I , N= 6 76 , 7 00 1
A B4 0 0 790
A B 4 0 0 8 00
W i< l l t
( 6 , 56 )
1:1 2
A B 40 0 8 1 0
I 6 , 25 )
I P P I N I 1 N=70l 1 72 5 l
'0
( 6, 5 8 1
S TR t S S ,
1:1 1
hR l l E
58
I
0.0 8 0
iili. l TI:
Wi U l E
( 6, 25 )
lSN
OO d l
LS N
UO B 2
lSN
0084
WR l lE
WR 11 t
( 6 , 581
I 6, 25 1
00.63
fl.)R K A T
WR I T E
I 6 , 20 )
6 0 FOR M A T
I '0
1 6, 251
ISN 0085
L S N O O <Jb
WR I T E
hN
W K l T.i:: I 6 , 2 5 )
FOR M A T 1 ' 0
lSN
LSN
l. S N
l..S N
Wk l l t
62
0090
0 0 '1 1
LS N 0093
L > N u094
ISN
WK i l E
0087
l S N 0 0 13 8
LS N O O d �
1I
( 6 ,. 5 6 1
l o, 251
lSN
LSN
!
Wk i H :
Wk l l E
U092
0096
0095
l ::. N
1 S t�
0 10 1
0 10 3
1 S N 0 10 4
l.S N 0 1u 5
LSN
0 10 2
lSN
L :. N
L:. N
bN
0 10 6
O lu 7.
O liJ B
O W9.
lSN 0 1 10
lSN
bN
l.S N
ll l l 1
0 1 12
l S N 0 1. 1 3
6, 621
6, L51
( 6 , 621
ITE
I 6, 64 1
� br 251
I 6, 641
wk 1 r i: 1 6, 2 5 1
lE I 6 , 2 0 1
F U K. I'. A T I ' 0 .
wK I T E ( 6 , 6 6 )
wK l l E ( 6 , 2 5 )
W K l l. E I b , 66 1
WR i T [ ( 6 , 2 5 1
WI U
o6
A f BE T A = ' 1 f 6 . l l
82
A8400850
l P P i i� l 1 N = 7 5 l 1 7 7 5 1
T l ME
A B 400870
A B 40 0 8 9 0
Bl
A8400900
AB400860
S X XP
AT
A B 400880
BE TA= ' ,f6. l l
82
I PP I NJ , N = 7 7 6 , 8001
AB4009 1 0
AB400920
( I' P ! N l , N= 8vl , B 2 5 l
S Y YP
I PPINl
82
AT
AB400930
B E T A = ' , F6 . l l
A B 40 0 9 4 0
Bl
AB4009 5 0
, N = 82 6 , 8 5 0 1
A !\ 4 0 0 9 6 0
A B400970
AB400980
A 8 4 0 0 9 90
I P i' I N l 1 N = 8 76 , 9 0 0 1
1:1 2
A B 4 0 1 0 DO
I P P I N I , N= 9 0 1 , 9 2 5 l
T l ME
AB40 1030
1:1 1
I N VA R I A N T ,
J2 P ,
A B4 0 1 0 1 0
AB40 1 0 2 0
A B 40 1 0 4 0
AT
BE T A = ' , F 6 . l l
Bl
AB401050
A B 40 1 0 6 0
I P P I N l , N = 92 6 , 9 5 0 1
B2
AB401080
AB40 1 070
( l' P I N I , N = 95 1 , 9 7 5 1
A B4 0 1 0'JO
R l: l lJkN
wl H l E:
EN T K Y
1 2 U F O R �I A T
UO fOR M A T
WR I T£ 1:1
1 6. 1201
or
D A TA
AB40l l00
A B 40 l l l 0
A B 't 0 l l 20
f UR
( 6 . 1 30 1 8 1
1 6 , 2 5 1 I kP I N l , N=
1 6 , 1 3 0 1 1:1 2
WR I I E
Wk l T £
WK l l E I
WR I T E
I 'U
I ' 1
6, 2 5 1
AB400820
A l:l 4 0 0 8 3 0
A B 40 0 8 4 0
I P P ( i� l 1 N= 72 6 , 7 5 0 1
I 6 , 2 5 1 . I P P I N I 1 N = 8 5 l , 13 7 5 1
FOk �; A T. I ' 0
S X YP A T B E T A = ' 1 f 6 . l l
WR I T E
WR I T E:
009.9.
l S N 0 1.UO
WR I T £
Wk
L.:> N 0 0 9 7
L S N 0098
l t> 1 6 0 l
wR llt I
nR I T E I
lok l l [
o4
( o , 60 l
F XY P ,
T I ME
S TE P
f\ E U S S C O N S T A N T :> ,
I R P I N I , N=
I
1 '
26 ,
Kt
=
' 1F l 2 . 4 1
F OR
B E T A = ' , F6 . 1 1
251
501
SUBROUTINE "A" CONTINUED I
A B 40 1 1 30
A 8 40 l l 40
A840 1 1 50
A B 4 0 1 1 60
A B 40 1 1 7 0
A B4 0 l l 8 0
I
I
1-
I
......
0
-.1
---------�--�-----
I S. N
l H 14
l !:> J�
U l l5
lJ2
>>k i l t ( b , 2 5 J
" f( ! T E ( o , 1 3 2 J
0 L lb
0l l7
I. ::. N 0 1 1 8
l :'. N 0 1 1 9.
l !:> N u 1 20
l ::. N U l L 1
lSN
bN
lSN
L S I�
l !:> N
1 !:> 1"1
l.S N
U U4
1J6
0 1 <: 6
1!:> N O U 7
L.!:> N
138
1 34
l.S N l H J :i
l S N O l.J b
L !:> N O U 7
L S N 0 l 38
LS N 0 l.:l9
140
l.S N 0 1 3 ::.1
L!:> N
0
142
FOR M A T
( 6 o l 34 1
( oo25l
( 6, 1 341
( 6, 2 51
( 6, 120)
( '0
vjk I T C: (
••K i l t (
"k L TE (
ovR l l t
6, 1361
6 , 25 )
o. l 36l
( 6, 2� 1
tiK l l t (
FOR M A T
( 10
wR I I t
( 6 , 1 40 1
i�R I T E
( o, 2� 1
1 ' 0
h.JK M A T.
Wk i I t (
Wi d T t I
1 4 4 r U R ii, A T
wK I I.E
TE
l NC KI:: M t N T A L
C R I:: E P
( RP ( Nl ,N=
B2
I P. P ( i� l , N =
�1 ,
(10
( 6 , 1 4Z l
o , 25 1
6 , 20 I
1 '0
B1
IH
I NC K I:: i�t N T A L
! <t B
Wk ! TL
f'UR M A l
liR 1 T �
0 15 2
WK i T I::
I S N 0 1 53
W H l T E:
l.S N
( 6 , 25 1
I 'U
HIR
A B 4 0 1 290
B ET A= ' 0 F 6 . l l
B2
A B 4 0 1 3 00
AB40 1 3 10
11 8 4 0 1 3 2 0
A B 4 0 1 3 30
I:: X P U N E: N T o
NXX ,
ti A S E O
I: XP U NI:: N T ,
NYY ,
ON D E X X P P 1
O I: Y Y P P ' I
ON
D I:: X Y P I-' ' 1
! R P I N I , N = .3 0 1 , 3 2 5 1
t: X P U NL N T ,
NXY ,
I
B A S � O ON
l:l A S � O
A B4 0 l 340
A B 40 1 3 5 0
11 8 4 0 1 36 0
A B 4 0 1 3 70
Ao40 l 3So
A B 40 1 39 0
A B 4 0 l 40 0
A B 40 1 4 1 0
A B 40 1 4 2 0
AB40 1 43 0
A B 40 1 44 0
N l , N = .3 2 6 , 3 5 0 1
T I ME
C UM UL A T I VE
CKE E P
( 6 , 14o l
wK l T I:
D[XY PP,
( K P ( N I , N = 1 76 , 2 00 1
( b o 2 5 1 I P P i i� I , N = S 2 6 , 5 5 ll l
( 6 , 1 46 1 B 2
0 1 48
S T kAI N ,
B1
WR I T E
bN
AB40 12BO
( kP I N I ,N=1 5 1 , 1 75 l
WR I T E
L!:>l• 0 1 4'S
l !:> N U l �U
lSN 0 1 5 1
II H 4 0 1 2 40
A B 4 0 1 2 50
A B 4 0 l 2 60
CREEP
I S N 0 1 46
0 1 4 7'
H ET A= ' , F b . l l
A B 40 1 2 7 0
LS N
wR I TE
146
( b , 25 1
F OR
! k P ( N J , N= 1 2 6 , 1 5 0 l
( 6, 25 1
1 '0
.
O I:Y Y P P ,
82
FO!i. h � T
0 14�
LS N 0 1 44
l,:) j··
fl l:l 4 0 1 2 30
S TRAI N ,
I R P I N l ,N=101 , 1 2 5 1
( KP(
A l:l 4 0 l l 90
AH40 1 2D O
A f.\ 4 0 1 2 1 0
CkEtP
( 6, 1441
0 14 2
B f T A= ' , f'6 . l l
76 , 1 0 0 1
I NC R I:: MI:: N TA L
Wk I
u 1 43
F OR
t H\ 4 0 1 2 2 0
! 6 , 1 4 4)
LS N
D t:X X P P ,
75 1
ioR I T E
l S I� 0 1 4 1
ST RAI N,
81
b, l381
.( 6 , 2 5 1 ( KY ( N ) o N =2 7 6 , 3 0 U I
FG k M A T
wk l l E
l S N 0 1 40
is N
1 1 0
1"1R l l t
lS N U U2
'
I TE
wi d T l
wk 1 1 1::
o<i'<. ! T C:
" R I T l:
t o, 25l
F U R I"• A T
Wk
0123
0 1 28
1 !:> ;� O l 2<J
l !:> N O L IO
l !:> N O LH
( 6,
WK I H :
134
0 1 L£
O i LS
13 2 1
1 10
f' O K I"l t d
wk l l l::
S T RA I N ,
[XXPP,
FOR
B ET A= 1 , F6 . l l
BZ
I P P I N I , N =4 7o , 5 0 0 l
I P P I N I , N= 5 01 1 5 2 5 1
l: UM U L A T l Vl C R E E P
B1
I P P I N I , N= 5 5 1 1 5 75 1
C UM UL A T I VE C R E E P
AB40 1480
S T KA ! N,
t'Y Y P P ,
B E:T A= 1 , F 6 . l l.
A B 40 1 4 90
AB40 1 50 0
AB40 1 5 1 0
S T RA I N ,
I: X Y P P ,
( 6 , 1481
I SUBROUTINE "A"
FOR
AS40 1 520
AB40 1 5 30
( b , L� I
Bl
I P P ( N J , N = 5 76 , 6 0 U I
( 6 , 1 48 I B 2
AB40 1 450
A B 40 1 4 6 0
A 8 40 1 4 7 0
B1
FOR
B ET A= 1 , f'6 . l l
A B 4 0 l 5 40
A B 40 1 5 5 0
A840 1560
A B 1t 0 1 5 7 0
,
A E\ 40 1 5 8 0
CONTINUED I
r
......
0
00
I.!:> N 0 1 54
lSN 0 1 55
lS N O l �o
L.S N 0 1 5 7
l S N 0 1 58
L!:> N 0 1 :> 9
150
l.S N 0 1 63
1 54
U165
l :l6
l .S N u 1o4
1.S N
J,S N 0 1 66
l.S N O l 6 d
l S N 0 1 u9
LS N 0 1 6 7
1-I
I. $ i�
l .S N 0 1 7 0
0171
158
WR
1
TE
.l.S N
lSN
u 1 78
0 177
0 1 79
1 ' 0
Wk l T t:
l 6r 2 0 )
( 6, 1 5 6 )
I '0
fOK M A T
I o ,. l 5 8 )
1 6, 2 5 )
1 6 0 FOR M A T I ' 0
Wl\ l rt: l 6 , 1 o 0 l
wk l T E
162
l TE
Wi d T E
1 6, 251
wh
K E T URN
ENLJ
6,
( 10
fUf( M A T.
l.SN
Ull
25 I
( o, 25 l
LSN 0 1 80
0
I
WR I T E
WR I T E
w k J. n :
.O l /6
ON E X X I' P ' I
I RP I N l r N= 2 0 l , 2 2 5 1
. E X P O Nt: N T , N Y Y , 6 A S E G
UN
A B 4 0 1 5 90
A l:l 4 0 1 6 0 0
A B 40 1 6 1 0
AB40 1620
A \3 4 0 1 6 3 0
A l3 4 0 1 6 4 0
EYY PP ' I
AB40 1650
I K P I N I r N = 2 2 6 , 2 50 1
E: XP O N E N I , N X Y , B A S E O
1 6 , 1 54 1
( 6 , 2 5 1 ( R P ! N I r N= 2 5 1 r2 75 1
SE C ONlJ C KE E P S T R A I N
fO,l. M A T 1 ' 0
W R I T E l 6 r l 5 b l ti l
i-IR l l E l o , 2 5 1 I P P I N I . N = 9 76 , 1 00 0 1
1IR I T E
ISN
6,
I I' P l N l r N= o O l , 6 2 5 1
T L M t:
t: XP U N E N T r N X X r B A S E D
A l:l 4 0 1 6 60
I
AB40 1 6 8 0
AB4 0 1 6 90
.
I NV A R I ANT ,
1 2PP,
AT
H ET A= ' , f6 . l l
AB40 1 700
A840 1 7 1 0
A FI 4 0 1 7 2 0
82
I PP ! N l r N = l O O l , l 02 5 1
A B 40 1 7 3 0
T l ME
PHI ' I
AB40 1 75 0
A B 4 0 1 7 60
I
( PH
P SI
II
'I
I , Jl ,
l
1
A fl 4 0 1 7 4 0
=2 , 6 I , J=2 , 6 I
I I P S I I I , J I , I =2 ,6 I , J =2 , 6 1
OMEGA '
l t> 2 l
( 6, 25 1
A B 4 0 1 6 70
ON E X Y P I' ' I
WR I T E
WR U E
l S I� 0 1 7 �
LS N
25l
( 6, 15 2 1
f O ii. M A T
I S N 0 1 13
.l.S N 0 1 74
LSN u 1 75
o,
fUR l� A T. I ' 0
WK 1 T E ( 6 , 1 5 0 )
WR 1 1 E
0 1o l
0 162
1!:> N
I
I 6 , 20 1
WR l T E I 6 , 2 5 1
1 5 2 fOR M A T 1 ' 0
L!:> N O toO
lSN
WR 1 l t:
WK l T t::
I I UME
I
I , J I 1 1 =2 , o I ,
A B 4 0 1 7 70
A B 40 1 7 8 0
A B 4 0 1 79U
AB40 1800
A B 40 1 A l 0
AB401820
J =2
t6l
A !\ 4 0 1 8 31)
A B 4 0 1 840
A E! 40 1 8 5 0
A840 l860
I END . OF SUBROUTINE "A" I
I
1-
0
...0
l.. i:V I:: L
.!.
1-: t: b b l
O S /3 b 0
Li.J M P iL I51-( CJ P T liJ N $
lf· I
OOU3
l5N 000�
1!> 1'1 1.W u o
l !> N
f- UNL T I U N
uUUL
bN
l S i'!
OOOb
lSN
0 0 1.0
l :> N
ARG
f!
l S N 0 0 ()4
1, T IN
ISN
J. S I\j 0 0 13
U002
l.::. N
UOU3
1.� 1\j
00.04
uuuo
.L> N U v U ::.
iSN
1 S i..,
iSN
48 500050
= Li T *D T M UL f
If
l l H'i l . GE . T !> HJ P I
IF
uuuu
0007
UOU9
, MU , C A , C B , C K , H ,
LL I 1 0 l
, f:
LX,lY,
XX,
Y 'f
-=
�lf'k 1 N T + l
{ N Pk l N T . Gt . N P I
UT
= -U T
MPR I N T
=
0
I S U BROUTINE TO FIND" Dr�]
eND
NAMt=
SUb R .J u T li'� t
(. L, M i•\u N
CAL L
)0
!:>t
TUP
L i>. tl l l l
H l:: l l B I , P U T I N S I � O l
5 0 1 , SA V I N S I 5 0 J
L A b t: l
K L A u l N I N U , L A B E L , P U T l N S , SA VI N S , T l T L E l
R [ T UR N
l: N iJ
A B 3 0 1 000
A B3 0 1 0 1 0
AB30 1020
A B 3 0 1 030
MA I N , O P l =O O , L i NE C N T =4 0 r ::. OU I{ C E I E I:i C O ! C , N U L ! S T , N O D E C < , L
/ l NP U T S/ f l
u l l� t: N !> I U I�
k lA L * d
=
A B 30 0 1 0 1
A l:l 3 00 1 0 �
A B 3 0 1 0 40
A l:l 3 0 1 0 80
K l T Ui( N
NG
A H 3 00 0 0 0
A B 3 0 0 1 00
AB300250
/ l l M l R / T l ME , D T r MP R I N T , M P
UT
C.u MP·LL tK· U P T ! U I. ::, l ::. N
f i NDD T
/ I NP U T S /
iH: A l * 4 M U , L X', L Y
1.5 N
lSN
4 8 5 00040
A 8 50004�
MA I N , O P T = O O , L I N E C N T =4 U , � OU kC E , E UC D I C t NUL I S T , N O D E C< , L J A D , "' A P , � J E O I T , [ D
T l TL E t 1 8 1 , P L UA 0 $ 1 2 5 l
I T , T S TO P , o T l N l T , u T M U L T , P R I NT , F I
SUU�U U T l N I::
CUi��ION
M P .< Hll T
0009
U() 1.0
00 12
NAME=
uP T I O N S -
LUMMON
LS N
A B 5 00030
A B 5 0 00 9 0
L S N IJU.u 6
l !> N O U 0 7
0.0 0 5
ID
A8500020
i:: N U
1 5 N 0002
i.S N U iJ U .3
l.�N
1.0
K E T UK N
�� M P a i £ �
r,
67.3
1\ 8 5 0 0 0 1 0
S UBROUTINE TO
DETERM INE FUNCTION
FOR LOG TO BAS E " 2 "
= A ll !> ( X / Y l
= A L O(, A RG I / O . b 93 1 4 7
100 ALOG2 =
TE
� 8 500000
1 00
K l T URN
0007
I ::. N u u u 9
I
A L OG 2 l X , Y I
l G O TO
Y . f: <;. . O .
A L li G 2
D l\
MA I N , U P T = O O , Ll NE C N T =4 0 , S UU R C E t E BC D i C r NOL 1 S T , N O D E C < , L J AD , "' A !> , ' O E D I
N A MI:: =
-
F O K T K A "J H
S UBROUTINE TO
l
SETUP THE PROGRAM I
Ci A D , "' A P ,
� J E D f T , I il
A B 2 00000
A B 2 00 0 1 0
A B 200020
A B 200030
A B 2 0 0 1 00
A B 2 00 1 1 0
A B 200900
A B 2 0 09 9 0
i
1-
[
......
......
0
S TART AND
READ
VALUES
P R E PARE
PREPARE
AND
MATRIX
" READ" S U B­
R O U T I N E AND READ
MAT RICES
LABELS FOR I N P U T
TO ZERO
DATA S U M M A RY
�'
READ I N P U T DATA
S TA RTING
�Y'
C O M P U TE TOTALS S T RA INS
P O I N T FOR
NEXT CAS E
ey ,
S E T TO ZERO:
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C O M P U T E C RE E P S T RA I N I NTEG RALS
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