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Masters Theses Student Theses and Dissertations
1965
Finite difference solution for triangular plates Finite difference solution for triangular plates
Hao-Yang Huang
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Recommended Citation Recommended Citation Huang, Hao-Yang, "Finite difference solution for triangular plates" (1965). Masters Theses. 6691. https://scholarsmine.mst.edu/masters_theses/6691
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FI1ITE DIFFERE1CE SOLUTIOt
FOR
TRIAt,G ULAR PLATES
BY
HAO-YAr\G HUArG
.Jt,,_ .. ( , 'I • ' .• . •I , .
. )
A
THESIS
Submitted to the faculty of the
University of Missouri at Uolla
in partial fulfillment of the requirement& for the
Degree of
Master of Science in Civil Ergineering
Roll a, ivti ssouri
1965
Approved by
,/ ( r./-;:'t~'.; . (
I t 'l "I/ .,;:>
I- J··J " f__
-? l· ;,- <'A:..-(.~-.--;
.ABSTRACT
In this thesis the finite difference method is used
to solve triangular plates with uniformly distributed
trarsverse load. As a matter of convenience triangular
coordinates are used. The suprorting condi tior;s as well
as the shape of plate are varied. All calculations are
performed by digital computer. The patterns of differen
tial operators in triangular coordinates, the deflectiors
of plates, ar d the momer.t distributions are shown and
discussed. The results obtained for some special cases
are compared with published solutions.
I
II
ACKNOWLEDGEMENT
The author wishes to express his sincere appreciation
ar;d thanks to Professor James E. Spooner for his guidance
ar1d advice. He also is deeply grateful to Dr. Larry E.
Farmer for his contribution to this work. The author is
likewise indebted to Dr. Joseph H. Senne, Jr. and Dr. Billy
E. Gillett for their helpful instructions during the courses
of study.
III
TABLE OF CO~TENTS
PAGE
ABS'l'RACT • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • I
II
III
ACKI\OWLEDGEMENT ••••••••••••••••••••••••••••••••••••••••
TABLE OF CONTENTS ••••••••••••••••••••••••••••••••••••••
LIST OF FIGURES ••••••••••••••••••••••••• •-•............. V
LIST OF TABLES • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • VII
TABLE OF SYMBOLS •••••••••••••••••••••••••••••••••••••••VIII
I. IKTRODUCTION •••••••••••••••••••••••••••••••••••••• 1
II. REVIEW OF LITERATURE • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 3
III. SMALL DEFLECTIONS OF LATERALLY LOADED PLATES •••••• 4
IV. FI~ITE DIFFERENCE APPROXIMATIONS •••••••••••••••••• 6
A. FD, ITE DIFFERE1\:CE APPROXIMATIOt\S IN
RECTANGULAR COORDINATES •••••••••••••••••••••••• 6
B. FIJ\ITE DIFFERENCE APPROXIMATIONS IN
TRIANGULAR COORDI~ATES ••••••••••••••••••••••••• 9
C. BOUNDARY CONDITIONS FOR VARIOUS PLATE SUPPORTS • 17
v. ILLUSTRATIVE PROBLEMS • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 23
A. TYPES OF STRUCTURES AND THE SUPPORTING
CONDITIONS • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 23
B. SUBDIVISION OF PLATES •••••••••••••••••••••••••• 24
C. LOADING COk"'DITION • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 25
D. TECHJ\IQUE OF SOLUTION BY USING DIGITAL
COMPUTER ••••••••••••••••••••••••••••• • • • • • • • • • • 25
E. RESULTS • •••••••••••••••••••••••••••• • • • • • • • • • • • • 28
IV
VI. DISCUSSION •••••••••••••••••••••••••••••••••••• 39
A. COMPARISON OF RESULTS WITH PUBLISHED
SOLUTIONS • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
B. ACCURACY OF FI~- I TE DI FFERE:KCE ME'l'HOD •••••••
C. TREATMENT Olf BOUNDARY CO~lJITIOl-."S IN
TRIANGULAR COORDINATES • • • • • • • • • • • • • • • • • • • • •
39
42
44
VII. CONCLUSIONS•••••••••••••••••••••••••••••••••••• 47
VIII. APPEl\l])IX
IX.
VITA
A. FLOW DIAGR~ FOR COMPUTER PROGRAM •••••••••• 48
B. COMPUTER PROGRAM ·•••••••••••••••••••••••••• 49
C. RESULTS FOR CASE 1 • • • • • • • • • • • • • • • • • • • • • • • • •
D. ~ OPERATOR ••••••••••••••••••••••••••••••••
BIBLIOGRAPHY ••••••••••••••••••••••••••••••••••
•••••• . . . . . . . . . ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
63
66
67
LIST OF FIGURES Figure
1. Element of Plate with :the Applied Lo~ and Moment. ••
2. Equally Spaced Points Along the x~axis . . . . . . . . . . . . . 3. Central Difference Operators ···~···••••••••••••••••
4. Operat.ors ••• ~ ..• ~ .............................•...•
Page
4
6
8
8
v
5. Triangular Co ordi:pa t~ s • • • • • • • • • • • • • • • • • . . • • • • • • . • • • 9
6 .· Triangular Net • . . . . . . • • . . . . . . . . . . . . . . . . . . . . . . . . • . • • 12
7.
8.
9.
~Difference operator •••••••••••••••••••••••••••••
'\72 in Equilateral Triangular Coordinates •••••••••••
Point Designation for 4-'r:::7 ••••••••••••••••••••••••••
12
13
15
10. "VA- in Equilateral Triangular Coordinates • • • • • • • • • • • 16
11. Rectangular Coordinates as One Case of Trian~ular
Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
12. Point Designation for Boundary Corditions • • • • • • • • • • 18
13. Point Designation for Boundary Conditions in
Triangular Coordinates ••••••••••••••••••••••••••••• 19
14. Point Designation ~or Boundary Conditions in
Triangular Coordina.tes • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 21
15. Types of Structures Solved ••••••••••••••••••••••••• 23
16. Subdivision of Plate and Point Designation ••••••••• 24
17. Symbol for Operator 4 v ............................ . 18. Operator for Mx • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
19. Deflection and Moment Pattern, Case 1 • • • • • • • • • • • • • •
20. Deflection and Moment Pa.ttern, Case 4 • • • • • • • • • • • • • •
21. Deflection and Moment Pattern, Case 7 • •••••••••••••
22. Deflection and Moment Pattern, Case 10 • • • • • • • • • • • • •
25
27
32
33
34
35
VI
Figure Page
23. Deflection and Moment Pattern, Case 13 • • • • • • • • • • • • • • 36
24. Deflection and Moment Pattern, Case 16 • • • • • • • • • • • • • • 37
25. Deflection and Moment Pattern, Case 21 • • • • • • • • • • • • • • 38
26. The Relation between Moment and Poisson's Ratio ••••• 39
27. Check for Boundary Condition
28. Points tear the Plate Corner
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 41
45
VII
LIST OF TABLES
Table Page
1. Maximum Deflection, Maximum Moments, and their
Locations for Simply Supported Plates ••••••••••••••• 29
2. Maximum Deflection, Maximum Moments, and their
Locations for Built-in S~pported Plates ••••••••••••• 30
3. Maximum Deflection, Maximum Moments, and their
Locations for Regular Polygon ••••••••••·•······~···· 31
4. Comparison of Results with Published Solutions •••••• 40
5. Extrapolation 'J1o from "Js, 0~' and Jm •••••••••••• 46
TABLE OF SYMBOLS
D Plate stiffness
E Young's Modulus
F Dimensional representation for force * h Equal space interval in x-axis
L Dimensional representation for length *
VIII
Qx,Qy Shearing forces parllel to z axis per unit length
of sections of a plate perpendicular to x and y
axes, respectively
q Intensity of lateral load
Mx,My Bending moments per unit length of sections of a
plate perpendicular to x and y axes, respectively
Mxy Twisting moment per unit length of section of a
plate perpendicular to x axis
t Thickness of a plate
u,T,w Triangular coordinates
~,P The angles between v and u, a~d w and u, respectively
A. Equal space interval ir· u-axis
r,.>.... Equal space interval in v-axis
r~~ Equal space interval in w-a~is
~ Poisson's ratio
* The dimensional rep~esentation;for "stress"
is "Ft-2 ''.
I I1'I'RODUCTIO:r-..
Many of the present day problems with which the er'f!itoeer
is confronted lead to a two dimensional linear partial dif
ferential equation of the boundary value type. For only a
1
few simple mathematical shapes, e.g., circles, squares,
ellipses, etc., are exact solutions available. For those
problems of practical importance but for which exact solutions
cannot be obtained, the approximate methods of solution must
be employed. These methods may be based on series expansions,
or they may be purely numerical methods such as finite dif
ference, which is used in this ir1vestigation.
As an approximate solution, the partial differential
equation can be replaced by its fiPite difference equivalePt,
so that instead of haviPg to solve ore governing differential
equatior1, the problem reduces to that of sol vir.g a set of
simple simultaneous algebraic equations. This numerical meth
od has become particularly popular iT' recent years because
calculating machines have become available at moderate prices,
ard it also has the advantage of allowirg the actual work to
be carried out by technicians without a knowledge of hir!:her
mathematics.
A flat plate is a basic structural element of modern
engineering structures. It may be thought of as a two-dimen
sional eqivalePt of the beam. The flat plate, in ger·eral,
resists loads applied either trrnsversely or axially, ard
it resists these by means of direct stresses, shear stresses,
2
bending stresses, and torsional stresses. The complete deri
vation of equations can be found ins. Timoshenko's book (1).
Those equations are found to be linear partial differential
e~uations up to fourth order. Timoshenko used Fourier series
in solving most of the problems, but the finite difference
method has also been employed by him and others. However,
the application of this method has been limited to some
simpler mathematical shapes, and the solution of a general
triangular plate was not found in the literature.
3
II REVIEW OF LITERATURE
The solution of boundary value problems by application
of finite difference methods is an idea far from new. Rung e
first used this method to solve torsional problem in 1908.
Richardson applied it on the analysis of ,stresses in a dam.
In 1918, after commenting on Runge's and Richardson's works,
Liebamann advanced the procedure of iteration as a more
feasible means of effecting the n<~erical solution of dif
ference problems. This procedure is now the most commonly
used finite difference procedure, with Southwell's relaxa
tion procedure being the best known. Southwell published
two books ( 2) , ( 3) , , in 1940 and 1946 respectively, repre
senting the application of relaxation methods in the realm
of structural analysis. More information concerning the
literature of finite differen ce methods can be found in
Grinter's book (4). In recent years there have been several
books written on this subject by Fox (5) a nd by Salvadori
( 6) •
The a pplication of fi n ite difference methods to
plate problems can be found in the books written by Timo
shenko (1), Borg (7), and Salvadori (6). Very few articles
which deal with triangular plates can be found, especially
those of irregular shapes.
4
III SMALL DEFLECTIOrS OF LATERALLY LOADED FLATES
The following assumptions are made in deriving the
differential e(mations for the laterally loaded thin plate:
1. The material is homogeneous, isotropic, and elastic.
2. The load acting on a plate is Jl ormal to its surface.
3. The defl ec ti or· s are small ir comiJarison with the
thickness of the plate.
4. At the boundary it is assumed thAt the ed~es of the
plate are free to move in the plane of the plate.
The complete derivation of the differential e"uations
can be found in Timoshenko's book (1). Iositive shears,
twists, and moments acting upon any differential element of
the ~late are shown in Fig.l.
Qy
My
Qx
Myx+~dy y
Fig.l Element of Flate with the Applied Load and Moment
5
Applying Hooke's Law and the equations of equilibrium
to the free body of the differential element leads to the
following set of equations:
d,.Mx ~ Myx = Qx ~x -t- dY
~My d-Mxy ~y - dx = Qy
a Qx cl Qy a x + d y = -q
Mxy =- Myx =D( 1-.u ) d~ Mx =- D ( a2J +AI i!-} )
~ dY2
My =-D(~+Al ~~2)
v4J =~~+ 2 ~+~=+
where q = lateral load function in x and y, (FL-2)
( 1-1)
( 1-2)
( 1-3)
( 1-4)
( 1-5)
( 1-6)
( 1-7)
J= vertical deflection of any point in the plate,(L)
.u =Poisson's ratio; in this thesis ...u is chosen
as 0.15 E t3
D =plate stiffness - 12( l-d} , (FL2)
t =the thickness of plate, (L)
E =Young's Modulus, ( FL-2)
IV FI~ITE DIFFERENCE APPROXIMATIO:t\S
A. Finite Difference Approximations in Rectangular Coor
dinates
In any application of the finite difference method to
6
solve a differential eouation it is first necessary to find
the approximations, or replacemer•ts, for the derivatives
dj/dx, d2~/dx2, and so on, at a typical point of subdivision
in the range of the solution. Assume that along the x-axis
there are five equally spaced points as shown in-Fig.2.
r~h!!:.-· __,._____.,h..__-t'-__.h_ _ _ -... -h._ .....
~~~----~--~~----~------~---- X 11 1 Xo r rr
Fig.2 Equally Spaced Points Along the x-axis
Now, in
tion d i.e.,
~
the neighborhood of any typical point xo, the func
can be supposed to be expanded in a Taylor's series,
_ ~ ( x-xo) , ( x-xo) ;_,, ( x-xo) 3~, ( x-xp)4 -J""' a i ( 2_ 1 ) - oo 1" 1 ·' Jo + 2 ·' QC' + 3 ! r:;. t 4 ! Jt. + (n) ( dnJ) where do = dxn 0 is evaluated at point xo.
If x is set equal, in turn, to xo+h, xo-h, Xo+2h, and
x 0-2h, it is found that
..h! 3"('1-) + •••• +·4~ 0
( 2-2)
( 2-3)
~ = ~ -t- 2h ':t r ~ 4h2 "':t''-t- _ 8h3J"'~ 16h4 ~l4> __. ••• aYr OP 1' (}P ' 2! o., 3_, v 1 4! 0'' 1
( 2-4)
~I = < _ 2h -:t' + 4h2J"-~ :r'''+ ~ ::?;~+J_ ••• ol O" 1! 0" 2! , 3 I O" 4 ! cro)
( 2-.5)
By subtracting Eq.(2-3) from Eq.(2-2), it is found that
( 2-6)
where all the terms containing third or higher powers of
"h" are included together as O(h3). Solving for (J: gives
~ ' = Jr - J.t + 0 ( h2 ) do 2h
Similarly, by adding Eq. ( 2-2) and Eo. ( 2-3) gives
For the finite difference expressions for J: and
( 2-7)
( 2-8)
"114-) ()" ,
Eqs.(2-2) through (2-5) are combined to give the following:
from which
~(4-Joo -
( 2-9)
( 2-10)
( 2-11)
7
Fi~.3 shows the mathematical molecules of derivative
expressions in terms of central differences with the corres
ponding order of error (e = O(h2) ) in the derivatives.
2h J I
h2 J"-2h3J"' =
h4 J'4) =
11 1 0 r rr
~ 0----0---Q)
,----,.- --(!)
CD----6-®-8-<D
Fig.3. Central Difference Operators
When two dimensional problems are considered the
central difference operators ir; the y direction have the
same form as in the x direc~ion, except the space interval
"h" must be replaced by the space interval in the y direc-
tion. In Fig.4 some patterns of operators for derivatives in
two dimensions are shown where the space intervals in the
x and y directions are the same.
• • • ())X 0 y = 4li:Z"
y
4 •••
" =h4"" Fig.4 Operators
8
B. Fi n ite Difference Approximations in Triangular Coor
dinates
9
There are several different systems of coordinates
which can be used to cover a two-dimensional reg ion, such
as rectangular, skew, polar, and triar1gular coordin ates.
The rectangular coordinate system is the most commonly
used system. If the shape of the boundary is trian gular
it is more convenient to employ triangular coordinates.
The direct derivation of differential equatiOJ~ s for a
plate in triangular coordinates is very complicated. The
simplffit way to fi nd the fi r ite difference approximations
in triangular coordinates is to transform them directly
from rectan gular coordinates. Doing so, first the relations
between thes~ two systems . are found. Fig.5 shows the
rectangular coordinates, x and y, and triangular coordinates,
u, v, and w.
y
0
Fig.5 Triangular Coordinates
Assuming the direction u coincident with the x-axis,
and calling a and ~ the angles between v and u, and w and
u; the transformation from rectangular to triangular coor
dinates becomes:
x = u + v ·cos cl. ·+ w ~ cos f3
y = v sino(+ w sin f3 ( 2-13)
The partial derivatives of x and y with respect to u,
v, and w are therefore:
~X 1 ~X ~X xu =au= . Xv=~v =coset ; xw= ·dw =cos/3 '
10
( 2-14)
Yu =~~ = 0 Yv -s~ ==sin d. ; fl.__
Yw= a w- sin f3
A function a(x,y} may be considered a function of u, v,
and w through the intermediate functions x, y . defined by
Eqs.(2-13), and its derivatives may be computed by the rule
for the differentiation of composite functions. Thus by
E q s • ( 2-14) :
Ju - ~X xu t J-y Yu = dx -Jv - Jx xv i" }y Yv = 6x coso{ + ~y -'dw = dx Xw .... dy Yw = Jx cosf3 t dy
The operators are ··obtained as follows:
d -~ c:::Ju- dx
sincl (2-15)
sin f3
( 2-16)
In general:
d.n ~ dun (} xn
cln (co siX ddx + sinot~ )n
dvn d-y
dn ( cosf3 txt sin~?>fy )n c} wn
When n = 2, we get
- a.2 dx2
~ 2 = cos 9 'd-~ 22 ..... 2cos.vsin~L + · 2-1 d-2 _, 2 ';;!( -. "" l S 1DOI. 'y2 OV X ~x~y ~
J 2 c12 . 2 d-2 cos2~f2cos~sin~--- + S1nfo~2 ~x dxdy oy
11
( 2-17)
( 2-18a)
( 2-18b)
( 2-18c)
Substituting i:q.(2-l8a} into E~s.(2-18b) and (2-18c) and solv-
~~ . ing for · ~'fa. . y1elds
a,., . 1 r: ,x- a2 d.l. 4 J ~r -K ~in213( dV2. - cos~ d.u2.) - sin2~( dw2 - cos2f3 ~;;.t.r)
where K = 2si11olsinj3sin( f3 -o() ( 2-19)
Hence, fr9m Eq .( 2-18a) and E~. ( 2-19)
2 1 r: ( ) ~ 2 . ~ c)2 . 2 o12_-, ( ) '\1 = K rin2 f3 - c}... ~- S1n2, o1v2t S1n ol olw2J 2-20
Consider a triangular net with space intervals equal to
A, r 1A. , and r2 A_. in the directions of u, v, and w respec
tively as shown in Fig.6.
12
Fig.6· Triangular Net
By using the finite difference operators derived in
rectangular coordinates as shown in Fig.3, it is found that
~~2= :2 [~ ~!2=r,~A?~ 3!2=~[~ :r;,A.:
( 2-21)
Substi tuti:ng EE!s.{2.:~n} into· E~ .f2..::.2o)·t give,S the correspond
ing v 2 difference operator whieh is shQwn in Fig.7.
).2 2 2x:-v2 = ''- r1 r:a.
where T ~·2 G,2r}sin2(~ -ot) - rfsin2 f3 + r12sin2~ Fig.7 ~~Difference Operator
u
13
For the commonly used eouilateral triangular net, with
ol.=60o,)3=120o, r,=r2 =1, Eq.(2-20) reduces to:
( 2-22)
The corresponding~2 operator is given in Fig.8~
3 2
u
Fig.B ~2. in Equilateral Triangular Coordinates
For the ~4 operator it is found from Eq • .( 2-20) that
~4 = ~2~2= I}- [ sin2(J3-ol) ~u; -sin2~:~+sin2d. ::~ 2
K2~4=~in22(~-cl) J'!4 +sin22J3 cl~!4+sin22ct :;4
4 4 :..2sin2(,e -ot) sin2~ <h ...6-2sin2(p3-d) sin2cl ~ 2 c:lu vG' olu olw
4 -2sin2clsin2 f3 d.v~a-,2 J ( 2-23)
Using the operators in Fig.3, it can be found that
( 2-24)
d4 1 o2u2dv2 == rp ~
u
In defining the position of each point, the
symbols of Fig.9,are used.
14
( 2-24)
15
\ /v Waa-VWaa-- Yaa
I\/ \I\ UWaa- Wa --Va--UVaa
!\/ \/\!\ Ubb-Ub--Io-- Ua-Uaa--u
\1\1\1\1 UVbb-Vb--Wb-UWbb
\b(\wL\wt, Fig.9 Point Designation for ~4
Substituting· Eqs.(2-24) into Eq.(2-23), and collecting
all terms falling in the same :point, the ' ~4 operators
are found~as follows:
Io=6 [ r 14r}sin22(J3-cl) + r1sin22 J3 + rfsin22cl]
-8[r,2rfsin2(f3-d.) sin2 J3 -r,4rtsin2(~-ot) sin2ot
t r 12r}sin2clsin2 13] Ua= -4r,4rfsin22(J:3-d) + 4r12rfsin2(J3-d.) sin2 13
-4r,4rJsin2(~ -ol.) sin2ol.- 2r,2r:fsin2 d... sin2 f3
Ub=Ua
Uaa= r 14rfsin22(J3-ot)
Ubb=Uaa
Va = -4~4sin22 J3 t 4r12r1sin2(13-c:l) sin2 ,!?>
t2r14r£sin2(J3-ot) sin2 ol t 4r,2r:tsin2cl sin2 f3
Vb=Va
Vaa= rfsin22f:3
( 2-23a)
Vbb=Vaa
Wa= -4r14sin22 cl- 2r12r}:sin2(13-ol.) sin2 f3
-4r,4r:fsin2(J3-~) sin2ot + 4lj2r22sin2 ~ sin2 f3
Wb=Wa
Wbb= Waa
UVaa = -2lj2r24 sin2(J3-ot) sin2J3
UVbb=UVaa
UWaa = 2r14r¥sin2(t3-cl) sin2 o1.
UWbb=UWaa
VWaa=-2r12r;sin2 at sin2 f3
VWbb=VWaa
For the equilateral triangular coordinates the '14
operator is shown in Fig.lO.
Fig.lO V4 in Equilateral Triangular Coordinates
Rectangular coordinates is one special case of tri
angular coordinates with ~=9<f, 13=13~ r, =1, r;.=T2. Sub-
16
17
stituting these values into Eqs.(2-23a), yr4appea.rs as in
Fig.ll. It is exactly the same as the yr4 operator obtained
by using rectangular coordinates as shown in Fig.4.
Fig.ll Rectangular Coordinates as One Case of Triangular Coordinates
C. Boundary Conditions for Various Plate Supports
The application of finite difference operators, such as
\74 and "\? , along, or near, the boundary will include those
points outside the boundary. It is necessary to define the
values of the outside points in terms of the inside
points, and these relations depend upon the boundary con-
ditions.
1. Boundary Conditions in Rectangular Coordinates
a. Simple Supports
As in Fig.l2 assume one edge along the y-axis is
simply supported.
18
y
I r
I I I I I I I ,
1 11 lj 0 r zr I I I I I I I
0 X
Fig.l2 Point Designation for Boundary Conditions
The boundary conditions along the y-axis are
J (O,y)= 0 ( 2-25)
[ ~~~- + ,u ( 2-26)
Enuation (2-26) arises from the following:
Mx( o,y) =-D( $~~ + AJ ~~) = 0
Since 3;~=0, it follows ~2~=0. From Eq.(2-8), Jo=O,and
neglecting the O(h2) terms, the following equations results:
( 2-27)
b. Built-in Supports
For a plate built-in at the edge x=O, the boundary
conditions are
J(O,y)=O
(.sll.)x-o=O r} X -
( 2-28)
( 2-29)
19
Substituting Eo.(2-7) into Eq.(2-29), it ca~ be found that
d.J Jr-J~ = 0 ~X 2h
2. Boundary Conditions in Triangular Coordinates ( 2-30)
a. Simple Supports
Suppose there is a triangular plate with one edge, say the
u-axis, simply supported as shown in Fig.l3.
y
x,u
7 6
Fig.l3 Point Designation for Boundary Conditions in Triangular Coordinates
The boundary conditions are
J (x,O)= 0 ( 2-31)
( ¥-J. ) 0 ( 2-32) ~ y=o=
By Eq. ( 2-19) and Eo. ( 2-32), and noting that f!2 ·= f!2 :=0'
it is found-that
.$]. = ~ [ sin2f3;$- sin2ot f!2J == 0 ~
( 2-33)
The corresponding boundary condition to E~.(2-32) in
triangular coordinates is
( 2-34)
Applying Eq.(2-34), in turn, at points 1 and 2 of Fig.l3,
the following equations are found:
r:fsin2P( JJ + d-1 ) - r,2sin2ot( M + J6 ) = 0
r22sin2p( ;¥ + }e ) - r 12sin2ol( 'J5" + J1 ) = 0
( 2-35)
( 2-36)
20
where ~, ;J1 , and J8 are three unknowns external to the plate.
However, only two equations are available. One extra equation
can be found on the basis of the following analysis. It is 2
known that along the u-axis ~ = 0. If the subdivisions of u 2
u, v, and w are sufficiently small, the condition J 02 = 0 olu
can be applied as a limit to the line passing through points
6, 7, and 8. From Eq.(2-8) it is found that
ri2A 1 ~ = ~ ( ~6 -t- Je - 2 J1) = 0
from which the third eauation is obtained
( 2-37)
mu1 tip1ying F.']. {2-3S) by r 2 sin 2 p; subtracting from it Eq. ( 2-36)
mu1 tip1ied by r, sin2 ol , and using Ea. ( 2-37}, J1 is found in
terms of J3, J...., afld Js :
1 ( 2 s 1 s 2 1_ - sf J3 - s ~ Js ) ( S I - 52 ) 2 (}+
( 2-38)
where s, = r,2sin2 ~
~= rtsin2 f'
In the case wher;e the edge along the v-axis is simply
supported, an eauation similar to Eq.(2-38) can be found by
replacing r 1 , cl, rz , and f3 by r 2 , (f?>-cV , 1, and ( 180° -01.)
respectively. In this caseJ7 is found to be_(see :Fig.l4):
~r1= 1 2 (25;! s4 ?- s.2·T- s2 ~ ) (2-39) 0 1, ( 53 _ 54 ) 04 .,. o3 ~ os
where s 3 = r: sin2(j3-o1.)
s4 = -sin2Dl
For the plate simply supported along the w-axis, ~~is
~ - 1 ( 2s s ~ s2 ~ - s2 -~ ) 07 - ( s5 - s 6 ) 2 s 6 o+ - • o3 5 os
where s5 = -sin2,9
s 6 = -z7 sin2(t3 -cl)
Fig.I4 Point Designation for Boundary Conditions in Triangular Coordinates
( 2-40)
21
22
b. Built-in Supports
The same method as used in simply supported edges
can be employed for built-in edges; the results are as
follows:
along the u-axis,
d7 = -1
{ 2j, j2j:._- j2J- j} J3) { 2-41) { . . )2 I 5 J, -J2
where j 1=r1 COSol,
j 2= r2 cos J3
along the v-axis,
J'T= -1
( 2 j3 j4~- ·26 . 2 ) { 2-42) { . ".J2
J3 5- J.4 03 J3 -J
where j3= r.z. co s{J3-ol)
j 4 = -coscl
along the w-axis,
J7= -1
l 2 js j6~+- ·2 a ·2 } ( 2-43} ( . . )2 J5 '5- J6 03 Js-J6
where j5 =-COS f3 j 6= -r,cos (j3 -ot}
23
V ILLUSTRATIVE PROBLEMS
A. Types of Structures and the Supporting Conditions
Two types of structures are investigated. They are
triangular plates and regular polygons, as shown in Fig.l5.
(a) a
(b)
Fig.l5 Types of Structures Solved
For the- triangular plate, 18 differen~ cases have been
solved. They are~
·~· ····--~--~··
Supporting ra_ condition eX =TI"/ 2 cl =Tr/3 d..=Tr/6
Simple support 1 1 1 1 1 1 1 1 1 along three edges 2 3 2 3 2 3
---- -·- - ··-- - ·-
Built-in support 1 -t _!_ 1 l _!_ 1 __!_ __!_
along three edges 3 2 3 2 3 "-------------------·-··---------- --
For the regular polygon five cases have been solved. The
.supporting conditions . remain the same in all cases. The
number of sides (n) of the polygon are varied from 4 to 8.
With the supports as shown in Fig.l5(b), a regular polygon
can be divided into tirf• identical isosceles triangles with one
edge simply supported and the other two edges built-in.
24
B. Subdivision of Plates
The same number of subdivisions are used for all caseso
Each edge is divided into nine eoual spaces. The space
distance aloncs the u-axis is .A, along the v-axis is r,A._, and
alonq, the w-axis is r2~• The location of each poit·t is shown
in Fi~.l6.
2 3 4
Fig.l6 Subdivision of Plate and Point Designation
C. Loading Condition
Only the uniformly distributed transverse load is
considered.
D. Technique of Solution by Using Digital Computer
All numerical calculations of this thesis are done by
IBM 1620 computer machine.
1. Opera tor for \7 4
25
As shown by Eq. ( 2-23) and Eq. ( 2-23a) , the yr4 operator
is a function of d.., f3, r 1 , and r..J ,where only c.J., a nd r 1
are independent. These operators must first be calculated
by the computer a nd stored in the machine. The operators
are symbolized as P(I,J) as shown in Fi g .l7.
I,J-2 I+l ,J-2 I+2,J-2
Fi g .l7 Symbol for Operator '\74
2. Determining the Deflections
The relation between deflections and loadirg of a
plate is given by Eq.(l-7)
( 1-7)
Applying the V4 operator at each point inside the plate
yields the following equation:
( 3-1)
where P, q, and D are known.
26
The deflections along the edges are zero. The deflec
tion of those points outside the plate can be expressed in
terms of the deflections of iPside points by using Eqs.(2-38)
through (2-43). Thus only the deflections of inside points
are unknowns. Since at each point Eo.(l-7) can be applied,
the number of unknowns and the number of equatiolls are
always consistent. The solution of the set of simultaneous
equations gives the results. 28 inside points are involved
in this thesis. The solution of the set of simultaneous
equations are accomplished by using the subroutine CALL
GAUJOR which solves the equations by the Gauss-Jordan
method.
3. Determining the Moments in x and y Directions
Once the deflections are known, the moments at each
point can be found by using Eq.(l-5) ard Eq.{l-6). Expressing
27
a ~ c> x2 and cly2 in terms of ~~~'
is found that
52~2' and ~2!"2 by Eqs.(2-18a)
and (2-19), it
where u1 =1 +2: cosolcospsin{i3-or)
U2=- ~ sin2J3
U3= ...AL sin2d. K
and
My=-D(V 1~2~ + V 2 ~!2+ V 3 ~2!2 )
where v1 =.u + ~ cosclcos,ssin(p-ot}
V2 =- _1.. sin2J3 K
V3= + sin2ol
The operator for Mx is shown in Fig.18.
Fig.18 Operator £or Mx
( 3-2)
( 3-3)
u
4. Further Assumptions Made for Deflections of the Outside
Points near the Plate Corners
28
The application 6f Eqs.(3-2) and (3-3) at the corner
points, such as points (3,3), (12,3), and (3,12) in Fig.l6,
will use the deflections of points ( 2, 3) , ( 3, 2); ( 13, 2) ,
(13,3); and (2,13), (3,13). However, the boundary conditions
at the corners cannot be found, and the deflections at these
points must be assumed. In this thesis they are all simply
assumed to be zero because the deflection near the intersec
tion of supports is very small. The complexity of boundary
conditions at the corner will be discussed later.
E. Results
The maximum deflection, the maximum moments both in the
x and y directions, ard the locations are shown in Tables 1,
2, and 3.
Figures 19 through 25 show the constant deflection and
moment.
'I' able 1 Maximum Deflection, Maximum Moment, and their Locatwns for Simply Supported Plates
Shape of Plate
(cases)
( 4)
o1. a
(7)~5 oJ::30° a 2· 4·. 1 3
J ( 1)
6.50
1.31
0.41
5.73
1.04
o.30
1.40
( 8) 5 ci=30
~ 0.225
.x a
(~9~ o~,-30° '"S •. 0.059
1
+MX ( 2)
18.5
s.oo
3.20
17.7
5.80
2.61
6.50
2.30
0.968
-Mx ( 3)
6.80
1.75
0.58
7.30
2.90
1.60
2.21
1.00
0.61
+My ( 4)
18.5
9.70
5.80
15.8
8.66.
5.10
9.80
4.70
2.85
-My
( 5)
6.80
2.70
1.40
7 .so
4.67
4.68
3.90
29
13.05
8.60
Table 2 Maximum Deflection, Maximum Moment, and their
Locations for Built--in Supported Plates
( 1) + Mx ( 2)
Mx ( 3)
+ My ( 4)
My ( 5)
30
Shape of Plate (cases) A.
( 164 1fJ ( 10-3q~ ( 10-3q a2) ( 10-3q a2) ( 10-aq a2)
2.38 8.44 18.1 8.44 18.1 3 1"·4
(10) ~ ot=900 a 2.
<:). a r-----~-=--------~~----~~-----+--------+--------+--------
(11)~ 0 2 ot=90 t a·~ :4
5 0.445 2.60 5.50 4. 50 9. 20
oc. a 1-;---,-----__;_;_--------t------f-------+-------+-------- --- ----·- ---
(12~d.=900 i ·21 0.128 1.35 2.70 5.02
.. 4 d 5
2.19 8.70 14.4 8.20 17.0
0.38 2.40 3.94 3.95 8.30
<;~>~ -g-~ ~~~~--~a~-------1--------1-------T-------T------T-------
0.091 0.99 1. 31. 2.18 4.30
( 16)~-J..=30° a .2 3
•"1 4
0.515 3.00 5.75 4.70 9.70
( 17) t-o{ =30°
~ a~. a
0.056 0.537 0.796 1.714 3.14
~-----------------;-----_, _____ -T------r-------T------- -( 18) d..= 30o
~ o.o11E o.1ss 0.434 0.82 1.74
o1. a
Table 3 Maximum Deflection, Moment, and their Locatiions for Regular Polygons
31
r-----------.,..------r------.,--------.-----···-·-·· ,-·---
Shape of Plate
(cases)
J ( 1)
+ Mx
( 2) - Mx
( 3) + My
( 4) My
(5)
1-----------+-----+----+----+---·----------1 ( 19)
0.864 4.18 6.05 6.65 7.20
1-----------t------t-·----+----+---·--- ----- -·---·- --( 20)
( 22)
\ / \ I
\ I \I --- -}\-- --
&3 /2: 1\3,5 ./ 4· \.
1.848
2.849
3.744
7.13 8.41 9.10 7.80
10.5 17.1 10.9 7.85
12.4 21.5 I 11.6 7.76
~------~~~-------r------+------t----~-------- ------~ ( 23)
'\ \
r I
I
4.90 14.4 25.7 12.5 7.60
a
Simply Supported Along Three Edges
(case 1)
cJ..= 90°
a
-3 2 Mx ( 10 qa )
Deflection (lo4 9;4)
0
positive moment zero moment negative moment
Fig.l9 Deflection and Moment Pattern, Case 1
32
Simply Supported Along Three Edges
(case 4)
a
Deflection (la4 qn2>
---positive moment -·-·-zero moment -----negative moment
33
------------1 I
I
Fig.20 Deflection and Moment Pattern, Case 4
Simply Supported Along Three Edges
(case 7)
~\ a
positive moment zero moment negative moment
Deflection (1~4 ~4) D
Fig.21 Deflection and Moment Pattern, Case 7
34
a
-1
-l
Built-In Supports Along Three Edges
(case 10)
a
positive moment -·-·- zero moment -----negative moment
Fig.22 Deflection and Moment Pattern, Case 10
35
~------------------·-·····--····
Built-In Supports Along Three Edges
(case 13)
a
Deflection (lo4 ~) D
positive moment -·-.- zero momen-t ----- negative moment
Fig.23 Deflection and Moment Pattern, Case 13
36
Built-In Suppor~s Along Three Edges
(case 16)
positive moment zero moment negative moment
Deflection (165 °;4)
Fig. 24 Deflec~ion and Moment Pat~ern, Case 16
37
Regular Polygon 1\umber of Sides= 6
(case 21)
Intermediate Supports
Deflection ( 1<!4 ga4 ) D
positive moment zero moment negative moment
Fig.25 Deflection and Moment Pattern, Case 21
38
39
VI DISCUSSION
A. Comparison of Results with Published Solutions
As a check of results obtained from this research, a
few cases are compared with results found in Timoshenko (1).
The comparison is shown in Table 4. It is found that Poisson's
Ratio has a slight influence upon the magnitude of moment. A
larger value produces a larger moment. the relation is shown
in Fig.26. This curve is used to correct the solutions
presented in Table 4 to a common basis with respect to ..u. The
comparis01: shows the results obtained by using the method of
this thesis are within .± 10% of published solutions.
_M_ Mo
1.15
1.10
1.05
1.00
0.95
0.90 0.8
..u 0 = 0.15
1.0 JU
Alo
1.2 1.4
Mo =The moment when Al- 0.15
Fig.26 The Relation between Moment and Poisson's Ratio
40
Table 4 Comparison of Results with Published Solutions
Moments Mx=8.47 Mx,= 8. 30 -lo4 % I
( H)3qa2) My,= 8. 25 My 1=8.12 -1.1 % My3 = 17.85 My/= 18.3 +2.5 %
( ..u=O. 20) ( ..u=O. 20) i
( 21)
Moments Mx,=9.45 Mx,=l0.38 +9.8%
( 103qa2) My1= 11.0 My,= 9.01 -9.0 %
( ..u.=O. 20) ( ..u=O. 20)
a
( 1)
Deflection J,8.10 J/=7.52 -7.1%
a ( -3 g a~) 10 Et
a
As a self-check two observations are made:
1. The normal moments along simply supported edg es
approach zero as shown in Figi.l9 through 25.
2. As shown in Fig.27 the numerical value of the out-
side deflections along line 1 should be the same as the
inside deflections alon g line 2, since in these two cases
41
both inside and outside pointslie on the same perpendicular
line to the edge and they should have the relations of
Eqs.(2-27) and (2-30). From the results obtain ed in this
thesis, it is found that the relations of Eqs.(2-27} and
(2-30) are followed very nearly in both cases •
,--/.!i ~-
1
11 2 I ,a
- 2.0 L -
. , '~1.6 /"" . '
'-l
/
' ';2·7 / '
'
a
' '
~2-0 -- - --=-3..4- - - 1- -
(Case 1)
' ' / ) -.ql ( I
I I
I 1 1 I
I
(Case 4 )
Fi g .27 Check for Boundary Condition
42
B. Accuracy of Finite Difference Methods
There are two distinct kinds of errors involved in
fi nite difference approximations. One is the error due to
the basic approximation of replacing the continuous problem
by the discrete model. Another is an add itional error
whenever the discrete eauation s are not solved exactly.
This latter error depends upon the method employed in
solving the discrete equations.
The Gauss-Jordan method is used in this thesis. By
checking the values of the deflection at points (4,4),
(10,4), and (4,10) in case 4, it is found that the deflec
tions at these poin ts are, as they should be, very near.
Therefore the error introduced by using the subroutine
CALL GAUJOR in this thesis to solve the set of simultaneous
equations is neglig ible.
For the purpose of reducing the error due to discrete
intervals, there are two differen t viewpoints according to
wh ether t h e approach is "m a thema tic a l" or "eng i n eering".
The "mathematician" tends to look for better opproximatior
by a use of formulas containing differences of higher order,
s u ch as using O(h4) or O(h6) i n stead of O(h2) as used in
this thesis. Using the higher order accuracy incr e as e s
the complexity of finite differen ce operators as i ndicated
by following two e quations:
£ __ J,_ [~ +0(h2) dx2- 112
a2 1 ~= 12 h~
43
The "engineer" finds the best route to a close approximation
by employing sufficiently small intervals. However, the use
of small intervals increases largely the calculation work.
Since high speed calculation machines are now available, it
seems to be better to adopt smaller intervals in the approxi-
mations.
The reason for improving the accuracy by using a finer
net can be seen by the following example. Given Poisson's
equation
- f(x,y) ( 4-1)
the corresponding finite difference approximation is
Ji.,j = f(x,y). ( 4-2)
Solving for d at the central point, say it is Jo , gives
~--1 4
44
where f{xo,yo) is constant at the point "o". It is found that
the value o~ J" is the function of the ari thematic mear of the
0 values at the corners. It can be expected that a better
value of ~· can be obtained if its corner points are located
sufficiently close to it. This fact is found to be true for
all orders of difference either in rectaT'gular or triangular
coordinates.
C. Treatment of Boundary Conditions in Triangular Coordinates
As mertioned before the boundary conditions for tri
angular plates of general type are difficult to define. The
assumptions made in this thesis are one method to solve this
difficulty, and they induce a certain error along the boun
dary, especially near the corner of plates. The effect
appears as a non-zero normal momert along simply supported
edges. Fortunately, this error is small and car, be re~~lected
in most cases ..
The difficulty of boundary conditions near corners is
that the number of unknowns and eauations is not consistent.
Let
Mu• =moment perpendicular to the u-axis
Mv• =moment perpendicular to the v-axis
By resolution of stresses in X and y directior, s into the
stresses in the plane perpendicular to the v-axis and using
the relation between stress and moment, the followin~ equa
tion is obtained:
{ 4-4)
45
From Eqs. ( 3-3) ·and ( 3-4), and the follow1· I~g_ t · f . equa 1on or Mxy
M _ n < 1-u > r _} d~ J' ,l}J l xy- sin(s-ot) Lcos01. duo;:W - cosf3 Ju <fvJ
Eq.(4-4) becomes
where Ci are functions of ol, ard J3.
( 4-5)
( 4-6)
Eq.(4-6) includes eleven points as shown in Fig.28.
Fig.28 Points Near the Plate Corner
The values of J at points 2, 7, 10, and 11 are unknown. Only
two equations, namely Mu= 0 and Mv'= 0, are available at
corner 1. Therefore these four unknowns can not be deter-
mined. In this thesis the values of O.:J. and J,., are simply
assigned to be zero. Table 5 shows the value of J1<> by ex
trapolating it from values of 08 , Oq, and d m. It can be I
seen that mo:e.t J,~ do approach zero.
46
Table 5 Extrapolation ~,, from as, J9 , ar:d 'J m ---- ------- -· -
Cases* Jm J8 Jg J 10 I :
1 -3.7 -3.3 -2.0 +0.2 i
i 4 -2.5 -1.85 -0.97
I
+0.2 i -· ··--I
7 -2.09 -0.79 -0.134 : +0.2 -----
10 9.9 11.2 9.6 +4.5 l - ____ ...,. ______ --
7.33 4.88 2.1 ! o.o I 13 I i --r--- -1 16 4.45 1.90 0.36 I +0.2
...1
* Refer to Tables 1 and 2
47
VII C01CLUSI01S
From the results obtained in this thesis, the following
conclusions were observed:
1. Maximum moments of Mx and My lie on the opposite
side of the point of maximum deflectior:. My lies near the
x-axis, while Mx lies near the v-edge if r,<r2 , or near the
w-edge if r,> r2 •
2. So long as r:J. remains constar,t, the variation of r,
does not change the momert pattern of plates of the same
loading and supporting conditions.
3. If p indicates the absolute ratio of maximum posi-
tive moment to maximum negative moment, then :
" a. When ~=90, p increases as r 1 decreases.
b. When o(<90°, p decreases as r 1 decreases for simple
support, and p increases as r 1 decreases for built-in
support.
c. For a regular polygon, p of Mx decreases as the
number of sides increase, and p of My increases as the number
of sides increase.
4. For a regular polygon, the maximum negative moments
in the x and y directiors always occur at the same poirt.
5. The finite difference method is applicable in
solving triangular plates of general type by using triangular
coordinates. The boundary conditiors for various supports are
still open to further investigations. The accuracy of the
results depends upon the fineness of the net as well as the
assumptions at the boundary.
VIII APPENDIX A FLOW DIAGRAM FOR COMPUTER PROGR.Aivl
~-------------------'------------------ - -- -·-- -
----..~ I1FU1' PROBLEM FARAMETER
r--~ --------------1
ot =AO( IA)
:r ---------------, t
I t I t
CALCULATE CALCULA'rE 50 - OPERATOR ~ -
P( I) FOR V" J3.- r 1 ,r2 JA=l,3 I I I I I I I I I I I I '--
PRI:r\T P(I) ,ot.,~
1-- I1PU'l' COEFFICIEl\T MATRIX G(I,J)
I I I I I I I I
I I I I I I I
• • I I I I I I I I I I I I I I I
OF SIMULTA~~OUS EQUATIONS
PR!l\T SOLUTIO~l SUBH.OUTI:t-.E I VECTOR I CALL GAUJOR
~ TRA1SFORM SOLUTIO~ VECTOR Il\ TO DEFLECTIO~.
l 1G.ALCULATE DEFLECTIO!\ I OUTS IDE THE BOU~'DARY
l r1 CALCULATE CALCULATE OFERATOR ~ p=o.l5 Mx,M_y_ FOR Mx, My
~ PRINT !dx,My - 50 STOP
I L-------_tl L _________ __,
I I
48
APPENDIX B
COMPUTER PROGRAM
49
r-
c TRIANGULAR PLATES SOLVED BY FINITE DIFFERENCE METHOD DIMENSION P(l9),G(28,29),A0(3),RATI0(3),Z(l4,14) READ lOO,(A0(1) 1 1=1,3) R E AD 10 1 , ( R A T I 0 ( I ) , I = 1 , 3 ) DO 50 lA= 1, 3 A=AO (I A) DO 50 JA=lt3 RA=RATIO( JA) TB=ATANF(SINF(A)/(COSF(AJ-RA)) B=3.1415927+TB Rl=l./RA R2=Rl*(SINF(A)/SINF(B)) C=SINFC2.*A) D=SINF ( 2.*8) E=SINFC2.*(B-A)) DK=2.*SINF(AJ*SINF(B)*SINF(B-A) F=l./((Rl*R2)**4*DK**2) Ql=6.*((Rl*R2)**4*E**2+R2**4*D**2+Rl**4*C**2) Q2=-8.*(Rl**2*R2**4*E*D-Rl**4*R2**2*E*C+(Rl*R2)**2*C*O) P(l)=F*COl+Q2) Q3=-4.*((Rl*R2)**4*E**2-Rl**2*R2**4*E*D) Q4=-2.*(2.*Rl**4*R2**2*E*C+(Rl*R2)**2*C*O) P(2)=F*C03+Q4) P(3)=P(2) P(4)=F*((Rl*R2)**4*E**2) P(5)=P(4) Q5=-4.*{R2**4*D**2-Rl**2*R2**4*E*D) Q6=2.*(Rl**4*R2**2*E*C+2.*(Rl*R2)**2*C*O) P(6)=F*(05+Q6)
- - - - PT7) =PC 6) P(8)=F*(R2**4*D**2) P(9)=P(8) Q7=-4.*Rl**4*C**2-2.*Rl**2*R2**4*E*D Q8=-4.*(Rl**4*R2**2*E*C-(Rl*R2)**2*C*D) P ( 10 ) = F * ( Q 7 +Q 8 )
- - - PT11)=P ( lO) P(l2J=F*(Rl**4*C**2) P(l3J=PC12)
. ---- · ·----- ----
CJl 0
-Q-I
PTI4) =-F*2• *R l**2*R2**4*-E*DP(l5J=P(l4) P(l6)=F*2•*Rl**4*R2**2*E*C P(l7)=P(l6) . PC18)=-F*2•*(Rl*R2)**2*C*D PC19)=P(l8) PRINT-199 PRINT 200,RA,A,B PRINT 20l,(P(KS),KS=ltl9) Sl=Rl**2*C S2=R2**2*D S3=R2**2*E S4=-t S5•-D S6=-Rl**2*E XU=2.*51*52/(51-52)**2 XV=2.*53*54/(53-54)**2 XW=2.*55*S6/(55-56)**2 YU=-51**2/(51-52)**2 YV=-53**2/(53-54)**2 YW=-55**2/(55-56)**2 TU=-52**2/(51-521**2 TV=-54**2/(53-54)**2 TW=-56**2/(55-56)**2 Hl=(TU+YU*YV)/(1.-YU*TY) H2=CYV+TU*TV)/(l.-YU*TY) H3=CTU+YU*YW)/(l.-YU*TWJ H4=(YW+TU*TW)/(l.-YU*TWJ H5=CTV+YV*YW)/(1.-YY*TW) H6=CYW+TV*TW)/(1.-YV*TW)
----~DO 11 1=1,2 G ( I t I +5 ) =0 • G (I +5 , I ) =0 • G (1 +5 t I +7) =0 • GCI+ll,l+l3)=0. G(l+ll,l+l8)=0. G (I+ 18, I+ 16) =0. GCI+l8,1+20)=PC4) G(l+l8,1+23)=P(l4) Gfl+l9.1+191=PC11 c.n .....,
,-
GCI+l9,I+22)=P(l0) G(I+l9,I+23)=P(6) G(I+l9,I+25)=P(l8) G(l+20,I+l8)=P(5) G(I+20,I+22)=PC16) G(I+22,I+l5)=P(13) G(l+22,I+l9)=P(l1) GCI+22,1+20)=P(l7) G(I+22,1+23)=PC2) G(I+23,I+l4)=P(9) G(I+23,I+18)=P(15) G(l+23,1+19)=P(7) G(I+23,I+22)=P(3)
11 G(l+ll,l)=O.
DO 12 I =1, 3 G ( I +8 , I + 19 ) = P C 8 ) GCI+9,1+18)=P(12) G(I+l3,1+15)=P(4) GCI+l3,I+l9J=P(l4) G(l+l4,I+14)=P(l) GCI+l4,1+18)=P(10) GCI+l4,1+19)=PC6) G(I+14,I+22)=P(18) GCI+l5,I+l3)=P(5) GCI+l5,I+l8)=P(l6) G(I+l6,I+22)=0. G(I+l6,I+25)=0. G(I+l7,I+25)=0. GCI+l8,1+14)=P(ll) G(I+l8,I+l5)=PC17) G(I+l8,1+19)=PC2) G(I+l8,I+9)=PC13) G (I+ 19, I +8) =P ( 9) G(I+l9,1+13)=P(l5) G(l+l9,1+14)=P(7) G(I+l9,1+18)=P(3) GCI+22,I+l4)=P(l9) GCI+22,1+16)=0. G(I+25,I+l6)=0.
12 GCI+25,1+17)=0.
01 ('J
DO 13 1=1,4 G(l+l,I+l4J=P(8) GCI+2,I+3J=P(2)+P(l9)*TU+PC13)*XU
------~G~(l+2,I+l3)=P(l2)
-<>
G(I+7,1+9J=P(4) GC1+7,l+l4)=P(l4). b1I+8JT+8 ) = P\TI G(I+8 1 I+13J=P(10) G(I+8,I+l4)=P(6) GTr+S,I + 18 l =P ( 18) G ( I +9 , I + 7 ) = P ( 5 ) G(I+9,I+l3)=P(l6) GTI+T3_;_I +zr:: P-f13 J GCI+l3,1+8)=P(ll) G(l+l3,I+9)=P(l7) GTT+TT,T+l4) =P ( 21 G (I+ 14, I+ 1) =P ( 9) G(I+l4,1+7)=P(l5) GTT+Ilt;I +8 J =P ( 71 G(I+l4,I+l3)=P(3) G(I+l8,1+8)=P(l9)
13 G(1+18,I+ll)=O. DO 14 I=l 9 5
r- GCI,I+2J=P(4)+PC13)*TU _______ G(I,I+8)=P(l4)
G(l+l,I+l)=P(l)+P(9)*TU+P(l9)*XU+P(l3)*YU G(l+l,I+7)=P( 10) G(I+1,I+8)=P(6) G(I+l,I+13)=PC18) G(I+2,I)=P(5)+P(9)*YU G(I+2,I+7J=P(16) G(I+7,I+U=P( 11) G(I+7,I+2)=P( 17) G(I+7,I+8)=P(2) G ( I +8 , I ) = P ( 15 ) G ( I +8 , I + 1 ) = P ( 7 ) G ( I +8 , I + 7 ) = P ( 3 )
14 G(I+l3,I+l)=PC19) DO 15 I=l,6 m G(I+l,l)=P(3)+P(9)*XU±PJJ9)*YU ~
I I
----------------------- - ----- -- ---G(I+ll,I+22)=0. G(l+l2,1+22)=0. G (I +22, I+ 11) =0.
15 G(I+22,I+12)=0. DO 16 I=1,8 G (I , I+ 10) =0 •
16 G(I+lO,I )=0. 00 17 1=1,9 G(I,I+4)=0. G ( I , I +9 ) =0 • G ( I +4, I ) =0 • G ( I +5 , I + 13 ) =0 • G ( I +9 , I ) =0 •
17 G(I+13,I+5)=0. 00 18 I= 1, 10 G(I+5,I+18)=0. G ( I +6 , I + 18 ) = 0 • G (I+ 18, I +5) =0 •
18 G(l+18,1+6)=0. 00 19 1=1,15 G(I,I+3)=0.
19 G(I+3,I)=O. J=O N2=14
20 00 21 I=l 9 N2 K=14+I+J G(I,K)=O.
21 G(K,I)=O. J=J+l N2=N2-l IF(N2)22,22,20
22 G(1,l)=P(l)+P(5)*H2+P(l6)*XV+P(l2)*TV+P(9)*Hl+P(l9)*XU+P(13)*YU G(l,2)=P(2)+P(l9)*TU+P(l3)*XU G(2,3)=G(1,2) G(l,7)=0. G(l,8)=P(6)+P(l2)*XV+P(l6)*YV G(l,l2)=0. G(2,13)=0.
CJl .t... G(l,13)=0.
G(l,l4)=P(8)+P(l2)*YV - ------- - -- ----------
G(7,1J=O. G(7,7)=PC8)*YW+P(l4)*XW+P(l)+P(4)*H4+P(9)*TU+PC19)*XU+PC13)*H3 G(7,8J=O. G(7,13)=P(8)*XW+P(l0)+P(l4)*TW GC7,14)=0. G(7,18J=P(l2)+P(8)*TW G(8,1J=P(7)+P(l61*TV+P(5)*XV GC8,6J=O. GC9,7)=0. G(8,7)=0. G(8,8)=P(l)+Ptl2)*TV+P(l6)*XV+P(5)*YV GC8,13J=O. G(8,14)=P(6)+P(12)*XV+P(16)*YV G(8,19)=P(8)+P(12J*YV G ( 13, 1) =0 • GC13,7)=P(Tl)+P(14)*YW+P(4)*XW GC18,13J=G(l3,7) GC22,18J=G(13,7) GC25,22J=GC13,7) G(27,25)=GC13,7) GC13,8)=0.
---G ( 13, 13) =P ( 1) +P ( 8) *YW+P ( l4)*XW+P( 4l*TW G(18,18)=GC13,13) G(22,22J=GC13,13) G IT5;£5T=GTI3f13) G(27,27J=GC13,13) GC13,14)=0.
---GTI~, 18 )=G (7, 13) GC18,22J=Gt7,13) GC22r25)=GC7,13J
-o-G(25,27J=GC7,13) GC27,28J=G(7,13) G( 13,19)=0. G(l3t22J=G(7,18J
I G ( 18 t 25 J =G ( 7 t 18) GC22,27)=GC7,18) GJZ2_t 28 l =G C7, 18 l G(14,1)=P(9)+P(5)*TV GC19,8)=GC14,1) c.n
01 __ _ G ( 2 3 , 14 ) = G ( 14 , 1 ) ----------------------------------------
G ( 26, 19) =G ( 14, 1) G(28,23)=G(l4,1)
___ _:::_G ( 14,7) =0. G(14,8)=G(8,U G ( 19, 14) =G ( 8, 1) G(23,19)=G(8,1) G ( 26, 23) =G ( 8, 1) G(28,26)=G(8,1) G(14,12)=0. G(15,13)=0. G(l4,13)=0. G( 14,14 )=G (8 ,a) G ( 19, 19 ) =G ( 8 , 8 ) G(23,23)=G(8,8) G(26,26)=G(8,8) G(l4 9 18)=0. G(14,19)=G(8 9 14) G(19,23)=G(8,14) G ( 23, 26 ) =G ( 8, 14) G(26,28)=G(8,14) G(14,21)=0. G(15,22)=0. G(l4 9 23)=G(8,19) G ( 19, 26) =G ( 8, 19) G ( 23, 28 ) =G ( 8, 19) G(15,24)=P(8) G ( 16, 25 ) =P ( 8) G(l6,23)=P(12) G ( 17, 24) =P ( 12) G(17,19)=0. G ( 18, 20 ) =0. G(18 9 7)=P(l3)+P(4)*YW
______ G~<?~13)=G(18,7) G ( 25, 18) =G ( 18,7) G(27 1 22)=G(l8,7) Gl18,14)=0. G ( 18 , 19 ) =0. G(18,23)=0.
CJI (J)
r
G ( 19, 13) =0. G ( 19, 18 ) =0. G ( 19, 22) =0 • G(20,27)=P(8) G(21,26)=P(l2) G(21,28)=0. G(22,19)=0. G(22,23)=0. G(22,26)=0. G(22,28)=0. G(23,18)=0. G(23,22)=0. GC23,25)=P(41 G(23,27)=0. G(24,24)=P( 1)
G ( 24, 26) =P ( 10) G(24,27)=P(6) G(24,28)=P(l8) G(25,23)=P(5) G(25,26)=P(16) G ( 26, 20) =P ( 19) G ( 27, 21) =P ( 19) G ( 26, 21) =P ( 13)
------~G~(26,22)=0.
G(26,24)=P(ll) G ( 26, 25) =P ( 17) G(26,27)=PC2) G ( 27, 20) =P ( 9) G(27,23)=P(15) G( 27i-24)=Pl7-) G(27,26)=P(3) G ( 28 , 21 ) =0 • G ( 28, 221-=0. G(28,24J=P(l9) G(28,25)=P(4)*YW+P(l3) G(23,27)=P(l4) GC28,27)=P(l4)*YW+P(4)*XW+P(ll) G(28,28)=P(l2l*H5+P(8)*H6+P(l6)*XV+P(l4)*XW+P(5)*YV+P(l)+P(4)*TW
(]1
-=!
(
-~<:>
I(
(
(
00 23 1=1,28 23 G(lr29J=l./C9.**4)
CALL GAUJOR (G,28,29,28,29J PRilfl300 PRINT 301,(G(I,29J,I=lr28) J=O N4=7 N5=0
31 00 32 I=lrN4 K=I+N5
32 Zll+3,J+4)=G(K,29) J=J+l N5=N5+N4 N4=N4-l 1Fl7-J)33,33r31
33 00 34 1=3,12 ZCI,3J=O. Z(3,I)=O. IM=l5-l
34 Z(IM,I)=O. Z(4,2)=H1*Z(4,4J Z(2,4)=H2*Z(4,4) Z(l2,2)=H3*Zl10,4) ZC12,4J=H4*ZC10,4J Z(2,12J=H5*Zl4,10J Z(4,12)=H6*Z(4,10) PRINT 390,Z(4,2),Z(2,4J,Z(12,2),Z(l2,4),Z(2,12),Z(4,12) 00 35 1=1,7 Z(I+4,2J=XU*Z(I+3,4)+YU*ZCI+2,4)+TU*Z(I+4,4) 1101=10-I
------~I~l~l~I=~l~l~-~~~--------------~-----------------------------------------------------1121=12-I Zlll2Ir1+4J=XW*Z(Illi,I+3)+YW*ZCI12Ir1+2J+TW*Z(Il0I,I+4) Z { 2 , I +4 J ::: )(_y * Z ( 4 , I + 3 J + Y V * Z ( 4 , I + 4 J +TV* Z { 4 , I+ 2 )
35 PRINT 390,Z(I+4,2),Z(112I,I+4J,Z(2,1+4J U=0.15 OK=2.*SINF(AJ*SINF(B}*SINF(B-AJ AB=2.*COSF(A)*COSF(B}*SINF(B-A)/0K AUl=(U+ABJ*CRl*R2)**2 AU2=-D*R2**2/0K
en 00
AU3=C*Rl**2/0K AV1=(1.+U*AB)*(Rl*R2)**2 AV2=U*AU2 AV3=U*AU3 Z(3,2)=0. ZC2,3J=O. ZC13,3)=0.
( Zll3,2)=0. Z(2,13)=0. Zl3,13)=0.
( F1=-81./(R1*R2)**2 J=O N6=10
( 36 DO 37 I=l,N6 (
I
(
-.. --··· CHl=-2.*lAUl+AU2+AU3)*Z(l+2,J+3)+AU1*lZ(1+1,J+3)+Zl1+3,J+3)) CH2=AU2* ( Z (I +2, J+2) +ZJ I +2, J+4)) +AU3* ( Z ( 1+3_,_J±2 ).+_l (_I_+ 1 ,J+~)) CH3=-2.*lAVl+AV2+AV3)*Z(I+2,J+3)+AVl*(Z(l+l,J+3)+Zll+3,J+3)) CH4=AV2*(Z(I+2,J+2)+Z(I+2,J+4))+AV3*lZli+3,J+2)+Z(I+l,J+4))
___ _:_X.~=F l* ( CH3+CH4) YM=F1*lCHl+CH2)
37 PRINT 402,XM,YM J=J+l N6=N6-l IF(10-J)50,50,36
50 CONTINUE 100 FORMAT (3El6.8) 101 FORMAT (3E5.0) 199 FORMAT l3X,26HRATIO, ANGLES AND OPERATOR) 200 FORMAT (Fl0.4,2Fl8.8) 201 ~ORMAT (4El8.8) 300 FORMAT (3X,11HDEFLECTIONS)
---301 FORMAT l4El8.8) 390 FORMAT l3El8.8) 402 FORMAT (2El8.8)
CALL EXIT END
.--·--·-··--------
01 c.o
APPEJ.I'DIX C
RESULTS FOR CASE 1
60
r-'
RATIO, ANGLES AND OPERATOR 1.0000 1.57079630 2.35619460
.20000000E+02
.lOOOOOOOE+Ol • lOOOOOOOE +0 1 .19901067E-14 .89221226E-07
DEFLECTIONS
-.80000001E+Ol -.BOOOOOOlE+Ol
.19999996E+Ol • 20000000E+O 1 .89221226E-07
-.80000001E+Ol -.BOOOOOOlE+Ol
.19999996E+Ol .20000000E+Ol .89221226E-07
.20242445E-03 .33381948E-03 .37804526E-03 .25945306E-03 .15275805E-03 .57389168E-04 .54697347E-03 .60548945E-03 .52626951E-03 .15916197E-03 .37816662E-03 .60560210E-03 .49510377E-03 .25116974E-03 .34538260E-03 .49523715E-03 .28792038E-03 .26000100E-03 .25135044E-03 .15374999E-03 .15961518E-03
-.20242445E-03 -.20242445E-03 -.57389168E-04 .OOOOOOOOE-99 .l9629417E-04 -.l9629417E-04
-.33381949E-03 -.68485076E-04 -.33385343E-03 -.37804528E-03 -.15672071E-03 -.37816664E-03 -.34510292E-03 -.23735545E-03 -.34538263E-03 -.25945309E-03 -.26959023E-03 -.26000103E-03 -.15275807E-03 -.23755910E-03 -.15375001E-03 -.57389181E-04 -.15736726E-03 -.58888265E-04 -.51203320E-ll -.69347921E-04 -.52540821E-ll
-----~.2'1943575E-09 .14629050E-08 .18093647E-09 .12062432E-08 .21668980E-09 .14445987E-08 .24911582E-09 .l6607721E-08 .30022078~-09 .20014719E-08 .26024545E-09 .l7349697E-08 .13347754E-09 .88985034E-09 .75152243E-10 .50101493E-09 .62212029E-10 .41474686E-09 .69727832E-03 .46485224E-02 .l8095486E-09 .12063658E-08 .66l59782E-02 .66136406E-02
---.8-5267948E-02 .10833016E-Ol .80804200E-02 .13136279E-Ol .62611274E-02 .13919233E-Ol
.lOOOOOOOE+Ol
.lOOOOOOOE+Ol
.l9901067E-14
.89221226E-07
.34510289E-03
.33385342E-03
.35647146E-03
.63648592E-Q3
.52654722E-03
.35690917E-03 • 5 8888 25 2E-04
en f-1
.~6782405E-02 .13412906E-Ol .86078669E-03 .11717070E-Ol
-.16818001E-02 .88355911E-02 ------~.38164289E-02 .48500127E-02
.90525556E-09 .75653723E-09
.10837859E-Ol .85232891E-02 ~14400410E-Ol-- .14394994E-Ol
( .1~'43452E-Ol .17585762E-Ol .99166628E-02 .18299441E-Ol .46866134E-02 .16722197E-Ol
-.10783237E-02 .12947343E-Ol -¢-
-.61379503E-02 .69441311E-02 .16262489E-Q8 .28466025E-09 I
----~·~13145954E-Ol .80762583E-02 .17593538E-Ol .13540473E-Ol .l6046325E-Ol .l6044895E-Ol .10445314E-Ol .15503435E-Ol .23583901E-02 .ll903228E-Ol
i ' -.67003203E-02 .51861161E-02 .23659519E-08 -.62486716E-10 .13940719E-Ol .62556497E-02 .18311036E-Ol .99188056E-02 .15503592E-Ol .10452150E-Ol .75098484E-02 .75190269E-02
-.38719503E-02 .87769592E-03 .23261314E-08 -.32795674E-09 .13464083E-Ol .36720010E-02 .16735837E-Ol .47001056E-02 .11899812E-Ol .23759323E-02 .86553341E-03 -.38576502E-02 .15109930E-08 -.48371284E-09 .11840288E-Ol .B7426736E-03
-----'---'
.12946111E-Ol -.10413986E-02
.51715616E-02 -.66656187E-02
.96966457E-09 -.39761941E-09
.91028186E-02 -.14828689E-02
.68800524E-02 -.61151281E-02 ----~·38302256E-09 -.21988334E-09
.51401863E-02 -.23373885E-02
.OOOOOOOOE-99 .OOOOOOOOE-99
Q')
£'V
63
APPE~TIIX D
\7 4 OPERATOR
0;= 1.0000 ci-::1.57079630 !3=2.35619460 0.20000004E+02 -0.80000018E+Ol -0.80000018E+Ol 0.10000002E+Ol -0.80000018E+Ol -0.80000018E+Ol O.l0000002E+Ol 0.20000004E+Ol 0.20000004E+Ol O.OOOOOOOOE-00 0.20000004E+Ol 0.20000004E+Ol O.OOOOOOOOE-00 O.OOOOOOOOE-00 O.OOOOOOOOE-00
1/r,= 2.0000 ol=l.57079630 !3=2.67794520 O.l3400002E+03 -0.20000001E+02 -0.20000001E+02 O.lOOOOOOlE+Ol -0.80000020E+02 -0.80000020E+02 0.16000004E+02 0.80000013E+Ol 0.80000013E+01 O.OOOOOOOOE-00 0.80000013E+Ol 0.80000013E+Ol O.OOOOOOOOE-00 O.OOOOOOOOE-00 O.OOOOOOOOE-00
?1;= 3.0000 Dl=1.57079630 !3=2.81984220 0.56400031E+03 -0.40000012E+02 -0.40000012E+02 0.10000002E+01 -0.36000020E+03 -0.36000020E+03 0.81000048E+02 0.18000006E+02 0.18000006E+02 O.OOOOOOOOE-00 0.18000006E+02 0.18000006E+02 O.OOOOOOOOE-00 O.OOOOOOOOE-00 O.OOOOOOOOE-00
Vr;= 1.0000 d,= 1.04719750 ,8=2.09439520 0.18666672E+02 -0.44444445E+Ol -0.44444445E+01 0.44444441E+OO -0.44444467E+Ol -Q.44444467E+01 0.44444472E+OO -0.44444461E+01 -0.44444461E+Ol 0.44444462E+OO 0.88888914E+OO 0.88888914E+OO 0.88888904E+OO 0.88888935E+OO 0.88888935E+00
J.i= 2.0000 oi-= 1.04719750 ,8=2.61799410 0.14933335E+03 O.l0666680E+02 O.l0666680E+02 0.48000048E-12 -0.85333358E+02 -0.85333358E+02 0.16000009E+02 -0.28444443E+02 ~0.28444443E+02
0.17777771E+01 -0.55425669E-05 -0.55425669E-05 -O.l8475214E-05 O.l0666666E+02 0.10666666E+02
'/rt = 3 • 0 0 0 0 01. = 1 • 0 4 7 1 9 7 50 f3 =2 • 8 0 8 11 9 6 0 0.72266697E+03 0.70222275E+02 0.70222275E+02 0.44444585E+OO -0.45600024E+03 -0.45600024E+03 O.l0000007E+03, -O.l0400001E+03 -O.l0400001E+03 0.39999981E+01 -0.13333357E+02 -0.13333357E+02
--~~_-0.26666701E+Ol 0.40000007E+02 0.40000007E+02 q= 1.0000 (._~=.52359878 {?=1.83259580 0.10744621E+03 -0.60103006E+Ol -0.60103006E+01
O.l0000002E+Ol O.l0000002E+Ol o.ooooooooE-oo O.OOOOOOOOE-00
O.lOOOOOOlE+Ol 0.16000004E+02 O.OOOOOOOOE-00 o.ooooooooE-oo
0.10000002E+Ol 0.81000048E+02 O.OOOOOOOOE-00 O.OOOOOOOOE-00
0.44444441E+OO 0.44444472E+OO 0.44444462E+OO 0.88888904E+OO
0.48000048E-12 0.16000009E+02 O.l7777771E+01
-0.18475214E-05
0.44444585E+OO 0.10000007E+03 0.39999981E+01
-0.26666701E+Ol
0.28718669E+OO (J)
~
0.28718669E+OO -0.60103145E+Ol -0.60103145E+Ol 0.28718729E+OO ___ " ___ 0 .28lla_7 29E +00 -0.62276860 E+O 2 -0.62276 860E+02 0 .ll"-L9-L9.L-99.L-9..L!6..uE~+,_,0~2~-------
0.ll999996E+02 0.57437397E+OO 0.57437397E+OO 0.37128098E+Ol 0.37128098E+Ol 0.37128137E+Ol 0.37128137E+Ol
/'t1=2.oooo a~.=.52359878 P-=2.72630950 ' o.96l2311BE+03 o.27881008E+03 o.27BBl008E+03 o.a5743837E+Ol
0.85743837E+Ol -0.51491322E+03 -0.51491322E+03 0.82297543E+02 0.82297543E+~2 -0.41538432E+03 -0.41538432E+03 0.47999977E+02 0.47999977E+02 -0.53128171E+02 -0.53128171E+02 -0.40574386E+02
" -0.40574386E+02 O.l2570251E+03 O.l2570251E+03 1'tt_:::_3.()000 ct. =.52359878 .8=2.91144000
0.51156957E+04 O.l2892924E+04 O.l2892924E+04 0.40861660E+02 -0.31656029E+04 -0.31656029E+04 0.65575487E+03 -O.l5581530E+04 -O.l5581530E+04 O.l0799987E+03 -0.32738495E+03 -0.32738495E+03
-0.13286164E+03 0.53224596E+03 0.53224596E+03
0.40861660E+02 0.65575487E+03 O.l0799987E+03
-0.13286164E+03
m CJl
66
IX BIBLIOGRAPHY
(1) S. Timoshenko and S. Woinowsky-Krieger {1959), Theory
of Plates and Shells, Second Edition,
McGraw-Hill Book Company, Ir.c., New York.
(2) R. V. Southwell (1940), Relaxation Methods in Engineer
ing Science, First Edition, Oxford Uni
versity Press.
(3) R. v. Southwell (1946), Relaxation Methods in Theore
tical Physics, First Edition, Oxford
University Press.
(4) L. E. Grinter (1949), Numerical Method of Analysis in
Engineering, The Macmillan Company,
New York.
(5) L. Fox (1962), Numerical Solution of Ordinary and Partial
Differential Equations, Addison-Wesley
Publishing Company, Inc.
(6) M.G. Salvadori {1952), 1umerical Methods in Engineering,
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6·7
VITA
Hao-Yang Huang was born in Taiwan, China on May 24, 1938.
He attended high school at Provincial Hsinchu High School. He
received his Bachelor of Science degree in Civil Engineering
in June of 1961 from the National Taiwan University, Taipei,
Taiwan, China. In February 1964 he entered the graduate school
of the University of Missouri at Rolla, Rolla, Missouri, for
further studies in Structural Engineering leading to the de
gree of Master of Science in Civil Engineering.