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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,

Prentice Hall, Inglewood-Cliffs, N. J.

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.