Deflections of a Ring Due to Normal Loads Using Energy Method and Stiffness Matrix Method

Rochester Institute of Technology

RIT Scholar Works

Theses

Thesis/Dissertation Collections

1970

Deflections of a Ring Due to Normal Loads Using

Energy Method and Stiffness Matrix Method

A.J. Rabinowitz

Follow this and additional works at:

http://scholarworks.rit.edu/theses

Recommended Citation

DEFLECTIONS OF A RING DUE TO NORMAL

LOADS USING ENERGY METHOD AND STIFFNESS MATRIX METHOD

Approved

by:

by

A.

J.

Rabinowitz

A Thesis Submitted

in

Partial Fulfillment

of the

Requirements for the Degree of

MASTER OF SCIENCE

in

Mechanical Engineering

Prof.

Name Illegible

(Thesis Advisor)

Prof.

William Halbleib

Prof.

Name Illegible

Prof.

Name Illegible

(Department Head)

DEPAR~ENT

OF MECHANICAL ENGINEERING

COLLEGE OF APPLIED SCIENCE

ROCHESTER INSTITUTE OF TECHNOLOGY

ROCHESTER, NEW YORK

ABSTRACT

Deflections

and reactions of a

ring

of 1.855" mean dia

meter, fixed at one _{point, simply} supported at another point and loaded _normally at an _arbitrary point were determined

using the energy method and the stiffness matrix method.

These results were verified _{experimentally} and

it

was found that the values of displacements and reactions obtained

by

the two theoretical methods were within

*4.5%

and 0.57o res

pectively of experimental results. When effect of transverse shear was

included

in

the computations done _using the _energy method

it

increased

the values of deflections

by

less than

1.5% thus _confirming that the effect of transverse shear

is

negligible on deflections of a _{normally loaded} ring.

Finally,

the

investigation

established that either the _energy method

or the stiffness matrix _method, can be _successfully used to

predict deflections and reactions of an _{arbitrarily normally}

TABLE OF CONTENTS

Page

List of

Tables

iv

Nomenclature v

I. Introduction 1

II. Literature

Survey

**

III. Statement of the Problem 5

IV.

Theory

6

V.

Energy

Method 12

VI. Stiffness Matrix Method 23

VII. Application of the Two Methods to a Specific Ring.. 36

VIII. Experimental Verification

*>U

IX. Discussion <. - **6

X. Conclusion 50

XI.

Bibliography

51

XII. Appendix A 53

XIII. Appendix B

Table I

Table II

Table III

Table IV

Table V

Table VI

Table VII

LIST OF TABLES

Page

Deflection and Forces

by

Energy

Method

-Transverse Shear Included

39

Deflection and Forces

by Energy

Method

-Transverse

Shear Neglected 39

Deflection at Nodes - Stiffness Matrix

Method

41

Deflections - Experimental Method

45

Comparison of Deflections - Matrix Method

vs Experimental Method

46

Comparison

of Results Obtained

by

Matrix,

Energy

and Experimental Methods

47

J = _{polar moment of}

inertia

kj = _stiffness

coefficient, force

in

the

i

direction due to a unit displacement

in

the

j

direction

k-f

a = _{stiffness coefficient associated with constraint}

iJ

energy

s

k- = _{stiffness coefficient associated with strain}

iJ

energy

ficl

= _stiffness

matrix, n x n matrix of stiffness coefficients

1 = _{length of beam}

Ml^2^3

= _moments

N,S,X,Y,Z,

=

locatiors

_on

ring C,L

P,F^,F2,QS

= _{normal loacb on}

ring

qi

= _generalized

coordinates,

displacements,

i

=

1,

2,

. .n

fq\

- displacement

vector, n xl column matrix of J

generalized displacements

q^

= displacement

in

the

i

direction of node a of

an element

q^

= displacement

in

the

i

direction of node b of an element

n x 1 column matrix of structural

displacements,

displacements at one end of an element with

respect to structural orthogonal axes

n x 1 column matrix of elemental

displacements,

displacements at one end of an element with

respect to an orthogonal set of axes applicable

to the particular element

tt

force vector, n x 1 column matrix of generalized

forces

a

Qj

= _force

in

_the

i

_{direction of node} a of an element

Q^

= _force

in

_the

i

_{direction of node b} of an element

to

n x 1 column matrix of

forces

and moments applied at one end of an element with respect

to the coordinate system for the structure

n x 1 column matrix of

forces

and moments applied at one end of an element with respect

to the coordinate system of the element

R = _{mean radius of curvature of circular beam or}

ring

[Rs-e]

rotation matrix which transforms a structural

system

into

the elemental coordinate system

s = _{length along curved beam}

T1'T2T3

= _Torques

U =

energy

Us

= _strain

energy

Uc

= _constraint

energy

UB

= _strain

energy due to

bending

Urp

= _strain

energy due to torsion

Ups

= _strain

energy due to transverse shear

u^

= _displacement

in

_the

i

_direction

Jut = displacement _vector, n x 1 column matrix of

displacements

V;l,V2,V3

=

vertical _{shearing forces}

W = _work

wl"w7

=

x,y,z = _{position variables}

in

_{a three dimensional}

rectangular coordinate reference system

*l~xll =

simplifying

coefficients

GLj&j

lft

UJ

-angular locations on _ring

^iq

= _{variable angular location on}

ring

Os

, os

= _{deflections at location S on}

ring

J

-

deflection

A.

i

= _{Lagrangian multiplier, reaction force}

in

_the

i

direction

Q$ Qs

= _angular

displacements,

_{slope at location S on}

'

ring

-&$. = angular rotation: of beam due to

bending

moment

&&

= _{angular rotation of beam due to}

twisting

torque

T

=

shearing stress

IS

= _{Poisson's ratio}

^

=

INTRODUCTION

A

ring

is universally

defined as a curved

bar

or beam

whose cross-sectional area dimensions are small

in

comparison

with

its

radius of curvature.

Virtually

every

industry

involved with the design of

mechanical components of structures

is

faced with the need

to analyze the stresses and strains

in

rings subjected to

various combinations of loads. In submarines, for example,

the hull

is

fabricated from a number of rings. These rings

must -be capable of _withstanding radial forces due to shock

and water pressure. The rings also have to withstand normal

forces

due to thermal expansions. Other examples of _ring

applications are to be found

in

frameworks of aircraft and

missiles,

bearings,

radar, etc.

Further applications are found

in

the construction

industry

where rings are used as _supporting foundations for

reinforced concrete water tanks. Rings are also useful

in

the design of high temperature _piping such as used

in

the

engine room and reactor room of atomic submarines.

With such wide applications

it

becomes

imperative

to

know,

in

detail,

the state of stress and strain

in

_ring

structures.

Three methods can be used to

determine

displacements

energy

principles,

starting

with Castigliano's second theorem

(principle

of

least

work) to find the external and

internal

redundants,

and then employs either Castigliano's

first

theorem,

or principle of virtual displacements to find the

displacements

of the ring.

A second method employs _stress, strain, and equilibrium

relationships to

develop

a differential equation. The

equation

is

then solved

in

conjunction with the appropriate

boundary

c ond

it ions

.

A third _method, combines the principles and techniques

of both aforementioned methods

into

a matrix oriented approach

which

is

_very useful

in

_solving complex structures that are

difficult to

handle-by

exact analytical approaches. This

method, which

is

_particularly adaptable to computer calcula

tions,

is

_commonly referred to as the stiffness matrix

approach, or direct stiffness _method, or finite element

method. It

involves

replacement of the continuous structure

by

one composed of finite elements. The stiffness matrix

approach

is

_{relatively new;}

in

fact

it

has _{only become}

increasingly

popular

in

the last 20 years

during

the advent

of the "computer age". Its _popularity

is

particularly high

in

the aviation and space

industries

where

it

is

used to

analyze stresses and strains

in

numerous structures such as

It

is

the objective of this thesis to investigate the

deflections

in

a _normally loaded _ring

by

the classical _energy

method and the modern stiffness matrix method and

further,

LITERATURE

SURVEY

Several authors

have

dealt

with the problem of _normally

loaded

rings.

Volterra,

for _example, used methods of harmonic

analysis to

determine deflections

of circular beams _resting

on an elastic

foundation

and loaded normally. He obtained

his results

in

a closed form under the assumption that the

foundation

reacts

following

the classical Winkler-Zimmerman

hypothesis.

Rodriguez2

analyzed _normally loaded rings

by

manipulating

the classical equilibrium equations.

Patelf

Kamel and

Reddy,

have analyzed _normally loaded curved beams

but have not used their results to analyze rings specifically.

Levy

used a combination of Castigliano's theorems and

classical techniques of solid

body

mechanics to derive dis

placement coefficients for circular rings loaded normally.

His analysis

is

limited to rings which are supported at

opposite ends of a diameter.

The above authors neglected deformations produced

by

transverse _shearing and/or limited their analysis to sym

metrical loading. The analysis presented

herein,

includes

effects of transverse shear, and

deals

with the use of two

approaches mentioned

in

the

introduction in

solving

symmet-rical as well as unsymmetrical loadings.

STATEMENT

OF THE PROBLEM

We

seek the

displacements

at various points

in

a

normally

loaded

ring

which

is

constrained at several points

along

its

circumference

by

fixed

and/or simple supports. The points

of application of external forces are _{completely arbitrary}

-We also wish to

determine

the effect of transverse shear on

the magnitude of

displacements.

The results obtained

by

_using an _energy method will be

compared with those obtained

by

_using a stiffness matrix

approach and with those obtained

by

experimental methods.

The theoretical basis for the solution of the problem

THEORY

When

a conservative structure

is

subjected to

forces

which are applied

gradually

so that at _any

instant in

the

loading history

the structure

is

in

static _equilibrium, the

kinetic

_energy of the structure will be zero. All the work

W

done

by

the

forces

acting

through certain displacements

will be stored

in

the structure

in

the form of strain _energy

U.

This

_energy

depends

on the final configuration of the

structure and not on the load-path

history,

or the manner

7 8 9

in

which the final configuration

is

reached

',0'*'

If the

force

-displacement

relationship

for the structure under

consideration

is

linear,

the work done

by

force

Fi

when

it

causes a final

displacement

of

Ui

is

given

by

(*i

(I

2

W

=\F

dui

=

V

kiiUidui

=

h

kiiui

=

h

F^i

(1)

Jo

Jo

where

kii

equals the stiffness coefficient of the _structure,

or the force at location i due to a unit _displacement

|

at location i. Wh<=>n the struc+tire is ?<?td

on

by

_a, number of

forces

Fi

through a number of

displacements,

the total strain _energy can be represented

by

U =

k?

Fi

ut

(2)

i

which

in

matrix form

is

U =

h

VuV

t<

Ff,

(3)

Since

$V]

=

[k]{u]

=

ki;j

Uj

U M

*fu]

t

[KJ(U]

=

*

i

j

which can be written

in

the

following

form

kLL

kL2

k13,

k _u

(4)

V =

h

J^i

u2

u3..Ui..uJ

'li

k21

k22

k23

kln"|

k2i"k2n

k31

k32

k33--k3i--k3n

,

kil

ki2

ki3

kii

kin

r

knl

kn2

kn3

kni

kn'n

(4a)

If we take a partial

derivative

of the strain _{energy U} with

respect to

displacement

u^, the only nonzero terms

in

the

multiplication of the right side of equation

(4a)

will be

terms

containing

ui.

Also,

^since

kij

= k

au

=

:

Qui

dv

aui

kij

uj

or

= Fj

(5)

(6)

Equation

(6)

is

a statement of Castigliano's

first

theorem.

It can also be expressed as follows:

Oil

-Qi

(7)

3Qi

where

qi

are generalized coordinates and

Q^

are generalized

forces.

For an element which

is

loaded _externally and which

Equation

(4)

for

strain _energy can be expressed

in

quadratic form as

s s s

us

=

%

i

_j

kij

qi

qj

where

kij

= _k

.^

(8)

This form

is

positive definite since strain _energy can assume

only positive values

for

_{any deflections,}

kij

represents

the stiffness coefficient for the _element, that

is,

the

generalized

force

at a point _acting

in

the

i

direction due

to a unit

displacement

_acting

in

the

j

direction,

qi

represents generalized coordinates, that

is,

three

transla-tional and three rotational displacements from equilibrium

position for each node

in

the structure.

The constraint _energy can be expressed through the

Lagrangian multiplier

in

a quadratic form as

uc

=

^?fcij-a-iqj.

<

The constraints _satisfy the equation

j

cij

^j

=

<9a>

which

linearly

involves

generalized coordinates _qi and re

presents a constraint condition. The Lagrangian multiplier

^i

has the physical significance that

it

provides the

generalized forces of reaction

in

the

i

direction on a

constrained node. The coefficient

Cij

represents the

The

introduction

of the constraint _energy renders

equation

(7)

as an

independent

set. One realizes that

if

the strain

energy

alone were used

in

equation

(7),

the par

tial

derivatives

would result

in

a dependent set of equations

because

all of the _qi would not be

independent

quantities.

A comparison of strain and constraint _energy equations show

that

they

both

have the same form

if

the Lagrangian multi

plier

^i

is

considered as additional unknown generalized

coordinate and _which,

therefore,

can be called q^.

Thus,

the total potential _energy

in

_the structure can be expressed

as

U =

Uc

+

Us

=

^Tj

k?j

*i

<lj

+

*

ij

kij

<U

<*j

=

%^-pkij

qi

qj

do)

where

kij

now represents stiffness as well as constraint

coefficients, and the

qi

represent displacements as well as

reactions.

Differentiating

equation

(10)

with respect to _q^,

crtu

, the above equation

Replacing

kij

by

gr-: and

kji

by

4

oy,

'

z

i

3

^^(M^K-U^

i

K*<^

Or,

restated,

or

(11)

(12)

;kiw-{o)

In _above,

J^Kj

is

a matrix of stiffness coefficients for the

structure,

j

qv

is

a vector of generalized coordinates of _any

point

in

the structure with respect to the structural co ordinate _system, andjQ\

is

a vector of applied generalized

force

components of which are three orthogonal forces and

three orthogonal moments applied to each point

in

the

structure or three _mutually perpendicular rotations

initially

applied to a constrained point with respect to the structural

coordinate system.

Equation

(2)

can also be written

in

matrix form as U =

%{f}

t\^\

(13)

and since

Juj

=

[aJ\Fj

where

aij

is

the

flexibility

coefficient or the deflection

in

the

i direction due

to a unit force

in

the

j

direction.

If we take a partial derivative of the strain _energy

as represented

by

equation

(14),

with respect to _any force

Fi

(with

all

Fi

considered to be

independent

and with

aij

=

aji) we _obtain,

3U

=

J>

aiiF-5

<15>

3Fi

j

3

or

du

=

u

(16)

aFi

which

is

an expression for Castigliano's second theorem.

We now proceed to _apply the above general

theory

to

ENERGY METHOD

Consider

a

force

P

acting

on a

ring

at angle

~ff

and

in

a

direction

normal to plane X-Y

(see

Figure 1).

i.OCI/5 Of roRCE _p

Figure 1

The _ring

is

fixed at point S and

is

_simply supported at

point N. The angle

X

is

a variable angle. Displacements

in

the horizontal plane are considered to be small quantities

of a high order and are therefore neglected.

We wish to find the vertical displacement of the _ring

at angle

f

when

do

lies between

"2*

and

&

.

Consider a free

body

diagram for the section to the

right side of fixed support S. The resultant beam

is

a

curved cantilever. We substitute a vertical force

F^

for

the portion of the

ring

removed at S are represented

by F2

and H2.

The

curved cantilever and applied forces are shown

in

Figure

2.

VIEW A-A VIEH/

B-g

Ytt* OC

Figure 2

From equation

(16),

the vertical deflection

c*s

at point S

equals

r

c)us

<*s=0i7

<17)

where

Fs

= _{the vertical force}

acting at point S.

Us

= _{the strain}

energy transferred to the beam.

=

UB

+

Ut

+

UTS

-(18)

Ug

= _strain

energy due to

bending

(Figure

3)

.

Vr

= _strain

energy due to torsion

(Figure 4).

Uts

= _strain

energy due to transverse shear

(Figure 5)

Figure

3

U.

r-jmce 9ffc

2EX

(19)

Strain _{energy due} to torsional shear

is

given

by

Figure

4

Vr=

**

s \ wee

0^.

=

j^

Strain energy due

to transverse shear

is

given

by

vr4=pn

Flfel/RS C

- Tr

dx

cf m u/

From equation

(21a)

, the shearing strain energy per unit

volume can be expressed as

P

=

T3'

2G

Since _shearing stress

7"

at _any point y. when a vertical

force

i3

(2/o)

V

is

acting

is

given

by

r

=

zz

ff

-'")

(21b)

the _{energy for} the elemental volume b

dy

dx

becomes

Att.-^C-f-**)^*

-sz.

^-1X1^^-*^^

'3.

(21c)

To obtain the deflection at point S due to force

FL

and

P

acting

alone

(neglecting

F2

and

H2)

it

is

first necessary

to assume an

imaginary

force

Qs

acting at S. The deflection

at S will then be equal to the partial derivative with

respect to

Qs

of the total _energy

in

the beam

(bending,

torsional and transverse shear) contributed

by

F^,

P and the

imaginary

force

Qs.

Thus,

from equations

(17),

(18),

(19),

(20)

and

(21e),

the deflection at S due to force P and

FL

. _{.,_} 10,11

can be written ' as

J0

BX

OC?s

)fi

EX

OQ,

4S

J^X

DQS

Jo

J&

0%

Jfi

T&

QQs

]fZ&QQ

(22)

where

M-l

=

Qs

R sintfo

M2

=

Qg

R sin

2&

-F-l

R sin

(3*D

-B)

M3

=

Qs

R sin

7&

-FL

R sin

(A-B)

+ _PR sin

(2-tf)

T, =

Q

R

(1-cos

~#o

)

Is *

T2

=

Qs

R

(1-cos

TCo

)

-FL

R

(1-cos

(

7C0

-B)

)

To =

Q

_R

(1-cos

"tf )

-F-,

R

(1-Cos

(

< -B)

)

+

3 s L

Vi

=

Qs

V2

=

Qs

-FL

V3

=

Qs

-FL

+ P

Substituting

Mj_, M2, M3, T]_, T2, T3, V]_,

V2,

V3

into

equation

(22),

replacing

ds

by

R

d"2fo

and

integrating,

the _resulting

deflection

after

setting

Qs

to zero

is

given

by

O _s = "

Fl

(Dl

+

D2

+

D3

+

D4)

+ p

(D5

+

D6

+

D7>

(23)

where

q.

-J?L

(-0CO4

₀

-+^L*.0 +

(Xoe0p-c**p/^cXc^CL-~J*<~p*<'ot\

C*-1*}

2EX

V

D?.=

-5^

(&

c<x.^ cj>iz?^a~.uc*x.cx. _

<&j^ot^^?~ifiotri?

-*a^t?)

^"ZS1)

*

-

*%

Czi)

Pa

= -^-

/*

-***

c*a2f*

* _<of

^

2f-^^ a. +

o/codar1

^{cm(<m/

0

JT?

I

2

^

*

-

^1^-C30)

Using

the same procedure,

it

is

found that the

deflection

at S due to

F^

and

H2

_acting alone

is

given

by

Ss

=

F2

(D8

+

D9

+

D10)

"

H2

(DH

+

Dg

=

-^

(

4

- *~

ol c+*.

cd\

(32)

fc

--|L

(g

-Z^o~oc+

*-^*~~^

(33)

D,o

=

&$

to

0.2.

- -(- C** *-

^2<--*,)

(36)

Furthermore,

the slope at S due to

F^

and P _acting alone

is

given

by

^s

=

Fl

C13

+

Diz

)

+ _P

(D15

+

D16

)

(37)

where

D/3

- ELK

/

sfrv* J3 + oo*. ^jay^U. - _{oLm6~ &} -sA6~-]3si+<* ooAeL

J

(38:)

Alsoy

the slope at S due to

F2

and

H2

_acting alone

is

given

by

0S

=

F2

(D17

+

D18)

-H*

(D19

+

D20)

(41)

0,7

in.

&*)

D"

*

{^oL.i^fL^

&3)

P,|

=

iff

(*

***"*-

c**-')

(i'i)

Da

- -Ar

^-^A^

(45)

We also know that

Substituting

equations

(23),

(31),

(37),

(41)

into

equa

tions

(46),

(47),

and _using the equilibrium equation

(48),

the constraints can be found to be

F =

wl

- p

<w2)

(W

W3

+

Wi

F2

= ₁

-FL

(50)

Fi

(W4).

+ _P

(W5)

(51)

H2

=

Wi

=

Di

+

D2

+

D3

+

D^

(52)

W2

=

D5

+

D6

+

D7

(53)

W3

=

D8

+

D9

+

D10

(54)

W4

=

D13

+

D14

(55)

W5

=

D15

+

DL6

(56)

W6

=

Di7

+

DL8

(57)

W7

=

D19

+

D20

(58)

Once the constraints

F^,

F2

and

H2

are

known,

the deflection

at S

is

given

by

Def =

Xi

+

X2

+

X3

-Xz;

.

X5

-X6

+

X7

+

X8

+

X9

+

Xiq

-XL1

(59)

where,

Xx

=

55

/

* M

(*

**"

cx-cxo*)

<**~>UJjt&J'oL

(60)

X2

=

o

ly"

j

Cv^T? C&<LUJ

(oL

4A~. ot

C&4.CC)

_-^4-*/* TPc-tXi. uj^i^o*.

^i*iio _{004. !?}sia~P-oI -h s4*~.Tfs&u+uj

(oL+s&OfCt

_CA*n.ot\

fr{\

Cos-2/* c-cra.UJ

uj

*4a~. uJ

ca.cc/j

+ ^i, 3^to.<c/^*.A UJ

-4-^yt~* co cva. &*_^v^^u*

~-^a*A*z?sO*Aw

(uj-^*44~.uj

c&*uj}\

Xo = -43^-*

I

*- *9^

^

-^c**&.

*j>u* ujc&a. ol rcU~._oL

4- CAtttAj

(

Gi--*- _^u3v.

GC

CASra^eti)

-**&<*,. UJ

^^.^oC^- uj

vp2j

-y C~<rz.UJ^&A*0. _UJ ^OrA^-.UJ CaotA. UJ -t

6&^*v.

UJ ^o&U/

^t*-*^i*.u

i<irO4<^ a.

__ ^3~6~.UJ

(s&au^Uj)

"

L.

.

, . xn

(63)

fe0*

**?

**+~> u*

-+-&~J*f

(

UO +sU*UJ CaXX.(jS)\

X5

=

Y3&"i *'OL

~" ^ _C4^6*^ -**^*<- + * c-c*lc

^v

fee/ - 2. CajoJ*>OA~>eL

-* C01.J9 <^luj^ott. 4ti _oC

CJX.eC)

u. ^i^ou

c*x.-p

^OaUa^cc

(j"</)

-+

-a^cvyS .4*/**UJ

(

OC ^i-*3- o< c-os.<X- _+*&u%-]3 o**.<o*d***~o<.

C4^<4^fu(wf^v>tO G*ti.<o) r4J~ UJ

Cjro.p

*4*~?-UJ

(i^O.)

-4*~ ,4C*UJ [OS _-$>UUJ

Ca^A-Uj)

__ ^u. 13 Cd-_cO

^-c*,4<*-2. ^X~.

p

cog.uj

I

-+ Coq^c^ ou

f

OC

f^U.OCtOtf.ot')

-*

.^c*,

W&04

V^J*?~0<-

(&$)

. -+ -***

"3*

-4^ <V

(

Ot .4^ C C**.oC *-^-*7Tc&UUJ^U^Oi

X8

=

P^..

,- 2.CO

2,

Ctf3.60'4-C* 0 - 3L XH-~ C*> <r4X.tU

- _{3,U*lcJl4A*<A/}

. 4- t-a.

?*

C*X.UJ

[lO

+

4ju*C*JGA>4-Uj)

4- _^aa^UJC^X.^^ojL?'

UJ

/)

4- .^to* *Z*

?*- **->

(

UJ

*A*J*. UJ Cat*. UJ H-^^m "3^ C_-cti UJ .?*--* *>

x =

H^.K

r_2Co^.OC-

C->-.-w/ sOA-^OC *4aC~ uj

(ct-sfruo(<uxet).

, v

^Coq.*o - _{00c to} ^a-**."3"*** Jd^*^ ** /c*>-^<X. to

Xio

-

jg.

(Fx~ F.

H-Py

*-,)

(68)

^y^~ _{oljcoMjQ ^6^-tX.}

_j. ^n^^ty

(<X--*&6.

otcj~a.cc)

Co-dp

C*raUj[ujsi*.OiJ

C4XUJ)

rl _^&a7*%J3 C-4X*CaJ^tAJ^UJ \4")

-*A^UJC^p^X~?*UJ-r4*6-*ja_^~s

Uj(oU+^6CAj<^O0Ov)

1

In the

foregoing

expressions UJ represents a specific value

of

4Q

Next,

we consider the

theory

for stiffness matrix

STIFFNESS

MATRIX METHOD

In order to solve for displacements using the general

theory

stated

in

section IV with values of applied forces as

known

quantities,

it

would be _necessary to use a stiffness

matrix for the structure. Since

it

is

very difficult to

obtain such a stiffness _matrix, one prefers to obtain a

stiffness matrix for each element

in

the structure analyt

ically

and to

incorporate

these stiffness matrices for the

elements

into

one overall matrix

JKJ

which can be used

in

equation

(12).

Thus for each _element, the

following

equation

can be written 12,13

M

{^

=

(^

(70)

where

jKeJ

is

a matrix of stiffness coefficients for an

element, _-Tq^}

is

a vector of 6 deflections at each end of

an element with respect to an orthogonal set of axes appli

cable to the particular element, and

f

Q^

is

a vector of 6

components _consisting of three orthogonal forces and three

orthogonal moments applied at each end of an element with

respect to the coordinate system for the element.

The assumptions used

in

defining

\KA

are:

1)

Deflections are small.

2)

The physical properties of the elements

Since equation

(70)

is

in

terms of the element coordi

nate _system,

its

terms must be transformed

into

the struc

tural coordinate system

in

order to

be

consistent with

equation

(12)

.

Using

a rotation matrix which transforms

one coordinate system

into

_another, the

following

equations

can be written:

[qe"J

=

[Rs-e][qs]

(7i)

{Qe'i

=

[Rs-e][Qsl

(72)

where

[Rs_eJ

is

a rotational matrix which transforms a

structural coordinate system

into

the element coordinate

system.

Substituting

equations

(71)

and

(72)

in

equation

(70)

will give:

[Ke]

[Rs-3

fcs}

=

[Rs-e]faJ\

(73)

Multiplying

both sides of equation

(73) by

the

inverse

of

|_Rs-eJ

whih also happens to be the transpose of

jRs_e\

will

give:

[Rs-e]

*

[Ke][*.-elki

=

[*s-e]

*

Mto

or

W(qi

={Qi

(7<o

where

[Kg]

=

[ksj

is

a matrix of either stiffness or constraint cooeff

icients

for

an element and will be referred to as an element stiff

ness matrix.

Since equation

(74)

now contains terms with respect to

the structural coordinate _system,

it

can be substituted

into

equation

(12),

resulting

in

a set of simultaneous linear

equations. The[K]matrix of equation

(12)

contains thefKs"l

matrices for all the elements and

is

called the structural

matrix.

Solution of equation *(

12)

for the displacements

fq^

can be obtained

by

_{employing any} convenient method like

matrix

inversion,

Gauss Jordon method etc. The solution

may be

formally

written as

H

-H

"l

M

C75)

After the unknown displacements are _obtained, the end

reactions with respect to the structural coordinate system

of each element can be determined

by

_substituting the

appropriate displacements

into

equation

(74).

The above

theory is

applicable to _any type of _structure,

i.e.,

it

is

general

in

nature. For a specific _structure, one

first has to decide on the shape of elements

into

which the

structure will be broken up. In case of a _ring,

it

can be

readily broken up

into

small curved beams or

into

small

the structure than the straight elements.

However,

the

stiffness matrix of a curved beam

is

_considerably more com

plex than that of a rectangular

beam.

With

_sufficiently

large

number of

finite

_elements, a _ring made up of straight

rectangular

beams

should _closely approach the actual con

figuration. Hence for the specific problem discussed

in

this

thesis

it

was decided to _{break up} the _ring

into

straight

STIFFNESS MATRIX FOR RECTANGULAR BEAMS

Let us consider the rectangular beam shown

in

Figure 6,

Y

4

NODfc

tfODE

b

Figure 6

Nodes a and b are located at the cross-sectional centroids

of the left and right faces of the beam respectively. Dis

placements of node a and node b

in

relation to a reference

a b

set of orthogonal axes are represented

by

q^

and

q^

where

i

varies from 1 to

6

.

qi

qf

and

q|

refer to the deflections of node a

in

3. 3. A

the _{x, y,} and z directions respectively, while _q^,

q5

and

q6

refer to the slopes or angular rotations of node a about the

x, y and z axes respectively.

Similarly,

_{q^, q2,}

q3

represent

res-pectively;

and _q^,

q5

and q,- represent angular rotations of

node b about the x, y and z axes

respectively-Loads

are represented

by

Qi

where

i

varies from

1

to

6.

AAA

Loads

Qi, Q2,

Q3,

_{for example,} represent forces _acting on

node a

in

the _{x, y} and z

directions

_{respectively;} and loads

Q Q Q

Q4

Q5 Q6

represent moments _acting at node a about the x,

y and z axes, respectively.

To

develop

the stiffness matrix for the

beam,

let us

first

consider axial

loads

_acting on nodes a and b. The

force

displacement

relationships are:

Ql

= _AE

(qi

-qj)

(76)

L

<?4

= kG"

(q

-q?)

(77)

Ql

=

_AE_

(qf

-q)

(78)

j

<?4

=

-JSG_

(q?

-q?)

(79)

L

b a

(qi

-qi) represents the axial displacement of node b

in

relation to node _a, or compressive strain of the beam.

b a

(qi*

-q/+) represents the angular rotation of node b

in

relation to node _a, or torsional twist of the beam about

its

longitudinal axis.

(qf

"

qi) represents the axial

displacement

of node a

in

relation to node

b,

or the tensile strain of the beam.

a b

(qz*

longitudinal

axis.

Now application of loads at node b such that the beam

is

displaced

in

the X-Y plane (Figure

7)

leads to the follow

ing

force-displacement

relationships:

NOOE&

Figure 7

>

ql3

******

L%*+

m

AO

qJlx

ZZ

(80)

where

b

q2

is

the

initial

vertical deflection of node

b,

which would be equal to the vertical deflection of node a.

Lqg

is

the

initial

vertical

deflection

of node

b

con

tributed

by

the slope at node a. It

is

a product

of the

length

of the beam and the slope at node a.

Q2L

/3EI

is

the vertical deflection of node b due to

bending

of. the beam. That

is,

the vertical deflection at

node b due to a vertical

force

Q2

applied at node

b while node a

is

fixed.

QgL

/2EI

is

the vertical "deflection of node b due to a moment

Qg

applied at node b while node a

is

fixed.

Q2FL/AG

is

the vertical deflection of the beam due to trans

verse _shear, which will be discussed later.

Furthermore

tf

= +

<?^

₊

^-

(81)

where

q^

is

the total slope at node b after application of loads

Q2

and

Q*?.

q^

is

the

initial

slope at node b which

is

equal to the slope at node a.

b 2

Q2L

/2EI

is

the _slope, at node b due to application of load

Q2

while node a

is

fixed.

QgL/EI

is

the slope at node b due to application of load b

Equations

(80)

and

(81)

_{may be} written

in

the

following

matrix

forms

:

4

att

OV

-6S2i

L

or

-6Sg

L

JlSxx

4

i-C-'C

?f-(82)

J

0^-!*Szi2L _

6Szi

tf-/aSz.fti

6Szgf

,4

*

L L

(83)

(84)

In the same manner as _above,

by

considering

the same

configuration but with

loads

on node a we find that:

GSziV

o

Q*

=

6S2ii%t

4.

U

Szxtt

-

i^jii

.4.

2Szsf*

(85)

(86)

For

analyzing

deflection

in

the x-z _plane, we can sub

#

-#=*

&-*?

q;-c

which _essentially

implies

replacing

_y

by

z and z

by

-y-From

efc.

(83^

Q^l-^gtf

+

6Stf'#

-.

J2^2ii

^

jSaLgll,

(87)

From

e&.

(w)

From

eo, (s\

From c^(6^

Equations

(76),

(77),

(78),

(79),

(83),

(84),

(85),

(86),

(87),

(88),

(89),

(90),

have twelve unknown

displace-a a a a a a b b b b b b

ments: qf,

q2

, q3, q*+,

qs

, qg> qi

q2

> q3 34' 35' 36*

These equations can be _conveniently expressed

in

the follow

ing

matrix form where the coeff

icent

matrix _{becomes the}

St

-*

itSzi

J*

6Szi -IZSzi

SSzi

L iZSiji -6S9.

L

-'2Sjj L8-L

Sts

_-its

^

9Z

L

2S^

65zi L

4SZI

-bSzi L

2SZ3

-St

-st

->2Sz) L8- _L. i>Sz* L4 -6Szj L

65tf,

6Sa

L

-$M

Sts

-6Syi L

2S33

6S31

</S9jL

Szi ~T""

2SZ3

-6Sz L **Sz3l a

31

323 a

33

a

34

a

35

a

"16

4

b

32

5

b

35

<&

~a

Qi

'2

ta

<tf

qS

l

n

Q?

q5

"S

<?

Q^

Q6b In above,

st-

-ae;

s =ec

L ts

V(z)J

_ EI

(zz)

C^;

Ci

= ₁

L

r^syCz)

c v _

3EI

, s

fy(z)

Ssy(z)

= _yy(zz)

Co

=

L+s

t ^ n

1"2S

, n

sy(z);

C3

=

sy(z)

The above matrix equation

is

in

effect equation

(70)

and

it

pertains to one element

in

the structure

being

analyzed,

Each element

in

the structure will have a similar matrix

equation. All of these equations can be converted

into

forms

represented

by

equation

(74)

which

in

turn can be consolidated

into

one giant matrix equation represented

by

equation

(12).

The resultant stiffness matrix

[ksj

of equation

(74),

for

example, can be considered as one element of the general

stiffness matrix

[k]

in

equation

(12)

_;

likewise,

the displace

ment matrix and applied

force

matrix of equation

(74)

for one

element can be considered as one element each of the general

displacement

matrix and applied force matrix of equation

(12)

respectively.

Thus,

in

a structure which

is

broken

into

n number of

beams,

the stiffness matrix of the general equation

(12)

expressing relationships of forces and displacement would be

an n x n matrix with each element

in

this matrix

being

a

12

x 12 matrix. The displacement matrix and force matrix

of equation

(12)

would be an n x 1 matrix and n x 1 matrix

respectively, with each element

in

these matrices

being

a

12 x 1 matrix.

To obtain

displacements,

equation

(12)

can _{be manip}

ulated to provide:

ft}

M

-1

ftl

such as the Gauss Jordon method to give displacements at _any

APPLICATION

OF THE TWO METHODS TO A SPECIFIC RING

We now proceed to _apply the two methods of

obtaining

displacements

viz. the

Energy

Method and

Stiffness

Matrix

Method to a specific numerical example

involving

ring

structure.

Consider a

ring

which

is

fixed at point S

(Figure

9)

,

simply supported at point

L,

_normally loaded at point C.

The normal load P equals 1

lb.

The physical properties and

dimensions

of the

ring

are taken as

follows:

D0

= _{Outer Diameter} = _1.945

in.

D^

= _{Inner Diameter} = _1.765

in.

R = _{Mean Radius} = .927

in.

6 2

E = _{Modulus of}

Elasticity

= _{30 x 10 lbs/in}

b = _Thickness = .090

in.

n = _{No. of Nodes or Elements Into Which Structure}

is

Broken = ₃₆

*xx

= _{Moment of Inertia About the Element's x-x}

-6 4

.Axis = 5.47 x 10

in.

yy

= _{Moment of Inertia About the Element's}

y-y

-6 4

Axis = _{5.47 x 10}

in.

Izz

= _{Polar Moment of Inertia of Element} =

10.94 x

10'& in.4

f = _{Lateral Shear Shape Factor} = _1.177

A _ring such as the one described above has been used as

a

supporting

structure for a _sefaty and

arming

device

in

_proximity

fuzes

used

in

anti-aircraft _projectiles, where deflections of

supporting rings are critical.

Figure 9

The lateral shear shape factor f

is

a dimensionless

quantity, dependent on the shape of the cross-section, which

is

introduced

to account for the fact that the strain, as

determined

by

dividing

the average shear stress on a cross

-section

by

its

shear _modulus,

is

not _{uniformly distributed}

over the cross-section,.

The lateral shear shape factor can be calculated from

the formula f = ^-2 +

11V

wherel/ = _{the Poisson's ratio.}

10 +

(i+y)

results.

Cowper's

results agree

fairly

_closely with other

authors who

have

also questioned Timoshenko's shear factor

and derived their own. Cowper derived his formula

by

inte

grating

the equations of three dimensional _elasticity theory.

The torsional shear shape factor e

is

a

factor

in

the

well known equation for angle of twist

[eq.

(20)J.

e

Tl

eG

where for circular _{cross-section,} e

is

the polar moment of

inertia

J. For other cross-sectional _areas, this factor will

change. For rectangular _{cross-sections,} for example:

4 ab3

16

-3.36

b

=

.1406a4,

if

a =b

I2a*

]

Deflection at point X on the _ring shown

in

Figure 9 was

obtained through the _energy method

by

_substituting the above

constants

into

equation

(22)

and _{solving for} <j^. . To facil

itate

calculation, the problem was programmed and run on an

1800 MPX operating system. The language used was Basic

Fortran.

Running

time for the program was nineteen seconds.

A _copy of the program

is

contained

in

the appendix.

Deflection and forces found

by

this _method, are pre

Table I

Deflection and Forces

(Energy

Method

-transverse shear

included)

Def.

(in.)

at X

Fi

(lb.)

F?

(lb.)

H2

(in.

-lb.)

.0048279 .76019 .23980 .49055

The results when the terms that contribute to transverse

shear are removed from equation

(29)

are given

in

Table II.

Table

II"

Deflection and Forces

(Energy

Method - transverse

shear neglected)

Def.

(in.)

at X

Ft

(lb.)

F?

(lb.)

H9

(in.

-lb.)

.0048202 .76008 .23991 .49047

To obtain a deflection at a similar point

by

means of the

stiffness matrix _method, the mean circumference of the _ring

is

divided

into

36 equal straight beams and the nodal points

Figure 10

The origin of the coordinate system for the _ring

is

established at node 36 with the _x-y plane

being

coplaner

with the plane of the ring.

The

(x,y)

coordinates of

jth

node

in

relation to the

ring structure coordinate system _may be expressed as:

Node 36

is

considered as a ground node. All displace ments at this node are therefore zero. At node 13 which

is

simply supported

only^

the deflection

in

the Z direction

is

zero. A

force

of one pound

is

applied at node 22.

To solve for

deflections

by

the stiffness matrix _ap

proach, an

existing

computer program called the

Strap

3

program

is

used. This _program, which was developed at

Eastman Kodak

Company,

is

based on the mathematical form ulation discussed earlier- _The program consists of _many

subroutines, or _modules, linked together

by

a _monitoring system. These modules are written

in

Fortran IV for an IBM

360 computer. Deflections of the _ring

(Figure 4)

obtained

using

the above program are shown

in

the

following

table:

i

Table III

Deflections at Nodes - Stiffness Matrix Method

(Due

to 1 lb. Force at Node

22)

Node Number Vertical Deflections

(in.)

1 -.0000385

2 -.0001496

3

-.0003218

z+ -.0005348

5 -.0007661

6 -.0009907

Node Number Vertical Deflections

(in.)

8 -.0012850

9 -.0012999

10 -.0011942

H -.0009483

12 -.0005536

13 .0000000

14 .0006903

15 .0014904

16 .0023595

17 .0032430

18 .0040942

19 .0048545

20 .0054827

21 .0059401

22 .0061826

23 .0061993

24 .0060071

25 .0056154

26 .0050737

27 .0044175

28 .0036939

29 .0036939

Node

Number Vertical Deflections

(in.)

31 .0015607

32 .0009930

33 .0005485

34 .0002342

35

.0000548

36

.0000000

The

constraints were found to be:

Fi

(lb.)

F2

(lb.)"

H2

(in.

-lb.)

.760989 .239010 .492000

Running

time for the above program was 11 seconds.

Excerpts of the program are contained

in

the appendix. The entire _program, which comprises over

150.

pages of subprograms

EXPERIMENTAL VERIFICATION

A test was conducted to _verify

deflections

due

to normal

loads

on the

ring

analyzed

by

the two methods. Normal

loads

were applied

by

a Chatillon push-pull _meter, model DPP-30 at

3

locations.

The experimental _set-up was as shown

in

Figure

11.

HEAVY

OUTY VISE

SIMPLE- -SUPPORT

APPL\*0 FORCE

Figure 11

The fixed point of the

ring

was held _rigidly

by

a

heavy

duty

vise. Points

X,

Y and Z and the simple support point were located

by

a toolmaker's microscope

(Gaertner)

. The

simple support consisted of a diameter pin.

Deflections were read at points

X,

Y and Z upon appli

dial

indicator.

Results of the experimental test were as follows:

Table IV

Deflections - Experimental Method

Experimented Run Number

1

2

3

4

5

Average"

Normal Deflection/lb.

(in.

/lb.)

Pt,

.0046

Pt. Y

.0013

Pt. Z

.0040 .0010 -.0010

.0042 .0010 -.0012

.0046 .0013 -.0011

.0050 .0015 -.0012

.0050 .0016 -.0014

DISCUSSION

Deflections

and Reactions

The

average

deflection

at Pt.

X,

(Figure 11)

which

is

identical

in

location

to node 27

in

the

finite

matrix _scheme,

was found _{experimentally} to be .0046". Pt. Y

in

the experi

mental test was similar

in

location to a position between

node

31

and node

32

in

the stiffness matrix analysis. And

Pt. Z

in

the experimental test was similar

in

location to

node 9

in

the stiffness matrix analysis. A comparison of

the results for both approaches

is

shown

in

the

following

table:

Table V

Comparison of Deflections - Matrix Method

vs Experimental Method

Matrix Analysis Experimental Test*

Node 9 -.0012999 -.0012

Node 27 .0044175 .0046

Node

31^

.0012768 .0013

* _{Average value of 5 trials.}

The deflections obtained

by

the stiffness matrix approach i

and the experimental method are thus seen to be within

4%.

The deflection and constraints found at location X on

a _ring

(Figure

11)

by

the three different methods discussed

Table VI

Comparison of Results Obtained

by

Matrix,

Energy

& Experimental Methods

Method Def.

(in.)

Fi (lb.)

F2

(lb.)

H2

(in-lb)

Experimental .0046

Stiffness Matrix .0044 .760989 .239010 .492000

Energy

.0048 .76019 .23980 .49055

The above table

indicates

that the deflections obtained

by

the _energy method and the stiffness matrix method are within

4^%

of the results obtained from the experimental method while

those obtained

by

the two theoretical methods are within

9%

of each other. Such close agreement of the results

justifies

the

idealization

of the problem. It

is

of interest to note that the _energy method tends to give higher values of deflec

tion while the stiffness matrix method gives values lower thn

those obtained

by

the experimental method. Some variation

in

results

is

to be expected due to lack of precise values of

the physical properties of the _ring such as Young's modulus

of

elasticity,

Poisson's ratio etc. It

is

also expected

that stiffness matrix method should yield values lower than

those obtained

by

the _energy method due to the fact that the

former method uses a

foreshortened

circumference _resulting

from a _ring made _up of straight beams

instead

of curved beams.

curved

beams

in

the stiffness matrix method while

keeping

the

number of nodes

fixed

or

by

_employing greater number of nodes

say

100,

while

retaining

the straight beam approximation.

The values of reactions and moments _acting on the _ring

obtained

by

the two analytical methods are

in

excellent

agreement, with

deviation

of less than one-half

precent,

which

i

can be attributed to the round-off and the truncation errors

in

computation.

Effects of Transverse Shear

. The

following

table contains a comparison of the

deflections

and constraints at point X as determined

by

the

energy method when transverse shear

is

included

and when

transverse shear

is

neglected.

Table VII

Effects of Transverse Shear of Deflection

(Energy

Method)

Def.

(in.)

Fi

(lb.)

F2

(lb.)

H2

(in.

-lb.)

With Shear .0048279 .76019 .23980 .49055

Without .0048202 .76008 .23991 .49047

Shear

The above table _clearly shows that the effects of trans

verse shear on deflection are less than one and a half per

cent and therefore

indeed

negligible. This agrees with the

observation

by Seely

and Smith that except for _relatively

short,

deep

beams,

the deflection caused

by

transverse

Since

the effects of transverse shear are so negligible

incorporation

of terms

accounting

for

it

into

the stiffness

matrix program

becomes

merely a matter of academic

interest.

Comparison

of the

Energy

Method and the Stiffness Matrix Method

Equations

in

chapter V for the _energy method were

formulated

for

a specific

loading

configuration.

Any

change

in

the

loading

scheme would necessitate _{reformulating} the

governing

equations.

Also,

the equations

in

the thesis were

formulated

to predict deflection at a specific point on the

ring^ If one

is

interested

in

_{predicting deflection} at _any

arbitrary point on the _ring a formidable amount of work

would

indeed

be required. On the other

hand,

equations for

the stiffness matrix method are fewer and less

involved

and

yet permit

incorporation

of _any combination of

loading

scheme.

It has also the advantage of _{predicting deflections} at all

the nodes almost simultaneously. It does have the disadvan

tage that

it

requires solution of 12 x n number of equations

where n

is

the number of nodes _chosen, thus _requiring signif-i

icant

amount of computer

time,

_especially

if

one needs high

CONCLUSION

The results of this

investigation

can be summerized as

follows

:

(a)

The

energy

method and the stiffness matrix method as

applied to a

normally

_loaded

ring give values of

deflections

which are within

4%%

of those obtained

by

the experimental method.

(b)

The values of reactions and moments obtained

by

the

two theoretical methods are within 0.5%.

(c)

-Effects of transverse shear on deflection of a _normally

loaded _ring

is

negligible, less than 1^%.

(d)

The _energy method

is

suitable when computer time

is

at

premium and one

is

interested

in

a specific

loading

configuration and

in

values of deflections and reactions

at a specific point on the ring.

(e)

The stiffness matrix method

is

a versatile one capable

BIBLIOGRAPHY

1.

Enrico

Volterra

2.

D.

A.

Rodrigues

3.

P. D. Patel

"Deflections of Circular Beams

Resting

on Elastic Foundations Obtained

by

Methods of Harmonic

Analysis".

Journal of Applied Mechanics. Vol.

19,

March,

1962

"Three-Dimensional

Bending

of a

Ring

on an Elastic Foundation".

Journal of Applied Mechanics.

Vol.

28,

September,

1961.

"Analysis of Continuous Bent

Beams Loaded Out of Plane

by

the

Six Moment Equations". Thesis

(Ph.D.

)

, Oklahoma State University,

1962.

4.

M. E. Kamel

5. M. N.

Reddy

6.

R.

Levy

7.

C. Lanczos

"Influence Areas for Continuous Beams

in

Space". Thesis

(Ph.D)

,

Oklahoma State 1963.

"Analysis of

Laterally

Loaded

Continuous Curved Beams".

Journal of the Structural

Division,

Proceedings of the A.S.C.E.

February

, 19S7.

"Displacements of Circular Rings

with Normal Loads". Journal of the

Structural

Division,

Proceedings of

the A.S.C.E. ,

February,

1962.

"The Variational Principles of

Mechanics". The

University

of

Toronto

Press,

Toronto,

Ontario. 1964.

8. J. Gere and

W. Weaver

"Analysis of Framed Structures"

D. Van Nostrand Co. , Inc.

Princeton,

N.

T.

T5F3

Moshe F. Rubinstein "Matrix Computer Analysis of

Structures",

Prentice-Hall_, Inc. Englewood

Cliffs,

N. J.

10.

S.

Timoshenko

11.

Seely

and

Smith

12.

J. Ashman

13. Y.

C.

Fung

14. G. R. Cowper

15.

John S. Archer

16.

S.

Timoshenko and

J. N. Goodier

17. Enrico Volterra

18.

Francis B.

Hildebrand

19. R. J. Melosh

20.

Eric Ressoner

Strength of Materials. Part

I,

3rd Edition. D. Van Norstrand

Company,

1955.

Advanced Mechanics of Materials.

John

Wiley

&

Sons.,

Inc. N. Y.

N. Y. 1966.

Structural Analysis Users Manual

1969.

Foundations of Solid Mechanics.

Prentice-Hall,

Inc. Englewood

Cliffs,

N. J. 1965

"The Shear Coefficient

in

Timoshenko's Beam Theory".

Journal of Applied

Mechanics,

Vol. 33. 15ZG~.

"Consistent Matrix Formulations

for Structural Analysis

Using

Finite-Element Techniques", Vol.

3,

October,

1965.

AIAAJ

Theory

of

Elasticity

, McGraw

Hill Book

Co.,

Inc.,

New

York,

New York. 1951

"Bending

of a Circular Beam

Resting

of an Elastic Foundation",

Journal of Applied

Mechanics,

March,

1952.

"Methods of Applied

Mathematics".

Prentice-Hall,

Inc. , Englewood

Cliffs,

N. J.

1^5.

"Basis for Derivation of Matrices for the Direct Stiffness Method". AJAA Vol.

1,

July,

1963.

"The Effect of Transverse Shear

Deformation on the

Bending

of

Elastic Plates". Journal of

COMPUTER

PBOGMM-ENFffftY

METHOD

FOR CC673

0CS(CARD,1443 _PRINTER) 1ST ALL

DEFLECTIONS OF NORMALLY LOADED RINGS

1 A=6.283 EL=3.840 B=2.268 W=4.712 P=l. R=.927 E=30.E+06 AIX=5.47E-06 AJ=10.94E-06 G=11.5E+06 H=.090 C1=(P*R**3)/(2.*E*AIX)

Dl=_{Cl*(COS(EL)*COS( B)*( A-EL-( SI N ( A)*COS( A) )+( COS ( EL )*S I N( EL ) ) ) )} 1+C1*_{( S I N (E L )*SI N( B)}*_{( A-E L)-SI N( B)*SI N(A )}*_{( S I N (}A)*COS ( EL )-COS( A)

2*S IN (EL ) )-SI N ( E L) *COS( B)*S I N( A)*S\E N( A )+SIN( EL ) *COS(B)*SIN(EL)*

"

3SIMIEL ) )

C2=(P*R**3) /( AJ*G)

D2=C2*( A-EL+SINI B ) *COS(A)-COS(B )

"*S

I N( A )+_{S I N ( EL )}-COS( A)-COS( EL )*

ISUM( A ) +( A/ 2. )*S I N ( B) *S IN( EL)+( A/ 2. ) *COS ( B)*COS ( EL )+SI N( A)*SI N ( A) /(

2 2.0 )*( SIN(B)*COS( EL)+COS( B)*SIN( EL) )+{ SIN { A )*COS ( A ) ) /2.* _(COS{B)*

3C0S(EL)-SIN(B)*SIN( EL) )-( EL/2.*SIN( B)*SIN(EL) )-SIN ( B)*COS( EL )+

"4C0S(B)*SIN(EL)-C0S(

B ) *SI N (EL ) /2.-EL*C0S( B ) *COS (.EL) /2 . )

D3=_{{ R**3/ ( 2}

.#E*AIX ) )*( A+S I M ( A)*COS ( A)*SI N ( B)*S I N ( B) -SI N(A)*COS( A)*

1C0S( B)*C0S ( B )-SI N ( B)*S I N ( B) *SI N( B) *COS( B)- _B+

S I N( B ) *C0 S( B) *_{( CO S ( B)}*

2C0S(B)-2.*SIN( A)*SIN(A) )+2.*(SlN( B) )**3*C0S(B) ) C6=R**3/(2.*AJ*G)

D5=C6#₍ 2.*A+( CO S ( B ) )*# 2*( A+S I N( A)*COS ( A ) )+( S I N( B ) )**2*(A-S I M ( A )*

lCOS(A) )+2.*C0S( B ) *SI N ( B) *( SI N( A) ) **2-4.*C0S ( B)*SIN ( A ) +4.*SIN ( B )* 2C0S( A)-2.*B-(C0S(B) )**2*( B+S I N( B) *COS( B) )-( SIN(B) )*#2*₍ B-SI N( B)* 3C0S(B) )-2.*C0S( B)*( SIN(B) )**3+4.*COS( B)*SIN( B)-4.*SIN ( B) *COS ( B) )

C3=(P*R*H**2*( A-EL) ) /( 10.*G*AIX ) .

C4= _{(r*h**2*( A-B) ) /( 10.*G*AIX)}

D8=_{( R**3 ) / ( 2}

.*E*AI X )*( S I N( B)-B*COS( B)+A*COS (B)-COS( B ) *S I N ( A ) *COS(A

1)

1-SIN(B)*SIN(A)*SIN( A) )

D9= _{(R*'*3-) / ( 2}. *E*A I X )*( A*COS ( EL)-COS( E L)*SI N(A)*COS ( A)-SI N(A)*S I N

1(A)*

IS IN (EL )-EL*COS( EL)+SIN(ED)

D10=(Rl!*3) /( AJ*G)*( SIN( B)-B-(#COS( B) )

/2.-1( SIM(B)*COS( B)*C0S(B))/2.-(SIN(B)*SIN(B)*SIN(B))/2.+SIN( B)*COS(B)

2-C0S( B)*SIN(B)+A-SIN(A)*COS( B)+COS( A)*SI N( B)-SI N ( A )+(A*COS( B) )/2.

3+(SIN(A)*COS(A)*COS(B))/2.+(SIN(A)*SIIM(A)*SIN(B))/2.)

D11=(R**3) /(2.*AJ#G)*( 2.*A-2.#SIN( A)*COS(EL)+C0S( A) *SIN ( EL)*2.-2.

1*_{SI N ( A)}+A*COS(EL )+SI N(A) *COS( A)*COS( EL )+_{( S IN}( A) )*#2*S I N( EL )-2.

2*EL+2.*SIN (EL) -EL*COS ( EL )-SIN( EL )*(COS( EL ) )**2-(S IN ( EL ) )**3)

""'

D12=(H#*2*R*(EL-B) ) /( 10.*G*AIXJ

013=(H**2*R*( A-EL) ) /( 10.*G*AIX)

D14=(-R**2)/(2.*E*AIX )*{ B*SI N(B)+COS( B)*S IN(A)*S I N(A)-A*SIN(B)

-1SIM(B)*SIN( A)*COS( A) )

D15= (R**2)/(2.*E*AIX)*( COS

CEL)*SIN(A)*SIN(A)-A*SIN<

ED-SIN(ED*

PAGE (

D16=(R**2)/(2.*AJ*G)*(2.*COS(EL)-2.*COS(A)-COS(EL)*SIN(A)*SIN(A)+

SIN (EL

)*(EL-A)+SIN(

EL)*SIN(A )*COS ( A) ) <