ANALYSIS OF THE TRANSIENT BEHAVIOR OF RUBBING COMPONENTS

ANALYSIS OF THE TRANSIENT

BEHAVIOR OF RUBBING COMPONENTS

Mongi Ben Quezdou

R. L. Mullen

Department of Civil EngineeringCASE WESTERN RESERVE UNIVERSITY

Cleveland, Ohio 44106

Final Report NASA Contract NAG3-369Principal Investigator: R. L. Mullen

NASA Technical Off icersR. C. HendricksG. E. McDonald

( (NASA-CR-176546) A N A L Y S I S OF THE T R A N S I E N TBEHAVIOR OF R U B E I K G C C M P O N E N 1 S Final Eeport(Case Western Reserve Univ . ) 51 p

I HC A04/HF A01 CSC1 11GG3/27

N86-19464

(Judas05516

TABLE OF CONTENTS

Abstract i

List of Symbols ii

Introduction 1

Literature Review 3

Formulation of the Partial Differential

Equations 7

Finite Element Formulation 10

Problems and Solutions 16

Conclusions 30

References 31

Appendix I Shape Functions 33

Appendix II Stiffness Matrix 34

Appendix III Input Data Form 37

Publications Resulting from Grant 46

ANALYSIS OF THE TRANSIENT

BEHAVIOR OF RUBBING COMPONENTS

ABSTRACT

Finite element equations are developed for studying

deformations and temperatures resulting from frictional

heating in slid'ing system. The formulation is done for

l inear s teady s ta te motion in two d imens ions . The

equat ions include the e f f e c t of the ve loc i ty on the

moving components. This gives spurious oscillations in

their solutions by Galerkin finite element methods. A

method called "streamline upwind scheme" is used to try

to dea l with this d e f i c i e n c y . The f in i t e element

program is then used to invest igate the f r i c t i o n of

heating in gas path seal.

LIST OF SYMBOLS

C Specific heat at constant pressure

E Young's modulus

F Body force

H Hilbert spaces

k Thermal conductivityA

k Artificial conductivity

N Shape function

q Heat flux

T Absolute temperature

U Displacement

V Velocity

W Weighting function

a Thermal expansion coefficient

X Lame's constant

jj Shear modulus

p Density

£,r) Natural coordinates

v Poisson's ratio

a) Angular velocity

ii

Chapter 1

INTRODUCTION

The first law of thermodynamics expresses the energy balance

during mechanical and thermal process. In the analysis of rubbing

problem, the loss of mechanical energy (frictional energy) is

transformed in its largest percentage to thermal energy. During

high speed sliding, contact patches are formed. An analytical

treatment of stresses (or displacement) and temperature distribution

near the contact patches is necessary. A transient finite element

heat conduction analysis has shown (ref. 14) that within a very short

time after establishment of the contact zone the temperature

distribution approached a steady state relative to a stationary

observer. The length of time required to reach this quasi-steady

state is so short that it may be concluded that within the contact

patches a steady state temperature distribution occurs. Therefore

it is not necessary to do a transient temperature analysis. A finite

element formulation will be done for linear steady state. The

formulation will be given in a form that could be expanded to

inelastic, non-linear problems.

The advantage of the Finite Element Method is that it is

possible to model finite geometry of complex shapes or different

material properties. Both temperature and stress analysis could be

done by similar modeling. Indeed, a thermomechanical analysis could

be carried out using one element grid and two linked finite element

programs. The major difficulty in applying a Finite Element Method

is that the convection operators are nonsymmetric. For instance the

Galerkin Finite Element Method is successful when applied to linear

symmetric operators, but these methods usually give spurious

oscillations in their solutions when applied to convection dominated

problems. A "streamline upwind scheme" (ref. 10) is used to deal

with this problem by adding an artificial conductivity in a manner

which stabilize the solution without destroying the physics of the

problem.

In this work, a review of literature about the principal

subjects is given, followed by a formulation of the weak form for the

heat transfer equation and the thermoelasticity equation. Then a

finite element formulation is developped for both thermal and

thermoelastic equation for a two dimension solid. The resulting

finite element program, which gives the displacement and the

temperature distribution, is first compared to an analytical solution

such as a semi-infinite plane under a heat flux (ref. 1). The

program is then used to a problem of rubbing contact at high

velocities in a gas path seal.

Chapter 2

LITERATURE REVIEW

The heat transfer theory started with Fourier's law of heat

conduction:

i A dTq = -k A -j—^ dx

When the body is moving with a given velocity, a convection

term is added to the equation (ref. 13 & 17). The use of Galerkin

Finite Element Method to solve the heat problems for a moving body

give rise to spurious oscillations. These oscillations can be

removed in this case by severe mesh refinement which undermines the

practical utility of the methods (ref. 8, 9 and 10).

New schemes were developped trying to deal with this deficiency.

The first scheme appeared by Roache in ref.6 as a classical upwind

difference scheme. It has been noted that the Galerkin Finite

Element Method produces central difference type approximations to the

advection (conduction) term. In finite difference theory, the

adverse behavior of central differences in these circumstances has

long been noted. But this method was considered as inaccurate.

Heinrich proposed a new scheme (ref. 8). The Finite Element

Method is applied using weighted residual formulation with

bilinear quadrilateral element shape functions, and non-symmetric

weighting functions which are different from the shape

functions, and depend on parameters which allow the amount of

"upwinding" to be controlled. An increase in accuracy could be

obtained by varying these parameters from element to element. This

method is now known as Petro-Galerkin method. But the two

dimensionnal quadrilaterals proposed by ref. 8 distord the diffusion

(conduction) term when upwinding is applied. It seems very difficult

to find an upwinding function that does not disturb the diffusion

(conduction) operator, yet upwinds the advection (convection) term.

Hansen and Von Flotow (ref. 16) noted that it might be better to

apply upwind weighting to the advection (conduction) term only and

central to the remainder of the equation.

Another simpler technique proposed by Hushes (ref. 9) called

quadrature upwinding was based on moving the integration points in

the Galerkin Finite Element Method. But he came later with Brook

(ref. 10) to propose a new multi-dimensionnal upwind scheme. The

method was applied successfully to one dimension, and then

generalized for two dimensions. This method, called "streamline

upwind scheme", is applied to the advection-diffusion equation, and

then to Navier-Stokes equations.

The thermoelasticity equation is derived from the principles of

thermodynamics. A simple formulation of the equation is presented

in ref. 2 & 3 as:

uUitjj + (X+y) Ujfji - (3U2u) aTfi + Fi = 0

Together with the heat transfer equation, the thermoelasticity

equation (two equations in dimension two) leads to the determination

of the displacement and temperature fields in a thermomechanical

problem such as the problem of rubbing contact at high sliding

velocities.

The thermal analysis of bodies in sliding contact has attracted

the interest of many investigators because of its importance in many

situations in which friction occurs: bearing, seals, brakes,

clutches,... Various methods have been proposed, but none has proven

universally acceptable. Many surface temperature analyses have been

based on heat source methods (ref. 1), in which the solution for

temperature distribution due to a point source on a surface is used

to develop the solution for a distributed heat flux within a contact

patch on the surface of an infinite half space. The difficulties

involved with application of heat source techniques to bodies of

finite dimensions led to the development of integral transform

technique presented by Ling (ref. A). Although this method have

been successfully applied to a number of problems with different

geometries, its limitations to simple shapes and its mathematical

sophistication have kept this from being widely used by engineers.

Kennedy tried to solve the problem of rubbing contact at high

sliding velocities by using the Finite Element Method. He applied

the problem to two examples: aircraft disk brakes and gas path seals

in turbine engine (ref. 11). The first examples was presented

before in the study of transient temperature in disk brakes in

ref. 5, considered as one of the first documention that use Finite

Element Method in such problems. He retreated the second example

experimentally and analytically in ref. 1A & 15. In ref. 11,

Kennedy used one element grid and two linked finite element programs

to make a thermomechanical analysis of the contact.

Chapter 3

FORMULATION OF THE PARTIAL

DIFFERENTIAL EQUATION

3-1. Heat transfer:

Let k. . , the thermal conductivity, be constant, let p, the

density, and C , the specific heat be constant. Let q be the heat

flux. In a steady state, the heat transfer partial differential

equation for a moving body with a constant velocity V. is:

k.. T . . - p C V. T . + q = 0ij ,ij P i ti s

The first term represents the conduction heat transfer. The

second term represents the convection heat transfer. The problem

defined in the equation above could be given after applying

Galerkin's method, as:

Find T GH2 for all W G H° scuh that:

[ W kij T,ij - * P Cp V. Tf . + W q ] df> = 0

Where T is assumed to satisfy the essential boundary

conditions. An integration by parts allows us to rewrite the problem

as:

Find T e H1 for all W G H1 such that:

f [ W k. ..T . + W P C V. T . •] oft = [ WqJ0 'J *J •! P i ,1 J Jo

8

Here homogenous natural boundary conditions have been assumed

in those sections of the boundary where T is not specified.

3-2. Thermoelasticity:

The constants X and y are respectively Lame's constant and shear

modulus. They are related to Young's modulus E and Poisson's ratio

v by:

v E EX = ; y =

(1 + v) (1 - 2 v) 2 (1 + v)

The Navier's equation, with temperature changes, in terms of

displacement U is given by:

y lh .. + (X + y) U. ... - (3X + 2 y) a T);. + F4 = 0

Where a is the coeficient of the thermal expansion and F. is the body

force.

Applying the Galerkin method to the equation above within an

element results in the following formulation:

Find U e H2, T 6 H1 for all W G H° such that:

f [ W y IL .. + W (X + y) U. .± - W (3X + 2y ) a T ±

+ W Fi ] d°. = 0

After an integration by parts, we write the formulation as:

Find U G H1, T € H1 for all W e H such that:

f [ W . P U. . + W . (X + y) U. . + W (3X + 2y ) a T,i ]Jo »3 1»3 »x J»J

= I W F.Jo i

'Rdo.

This equation, called the weak form, could be obtained if the

law of conservation of energy is used.

Chapter 4

FINITE ELEMENT FORMULATION

4-1. Heat transfer:

4-1-1. Streamline upwind scheme:

The heat transfer equation has a convection operator which has

been shown to carry spurious oscillations in finite element solution

(ref. 10). The conventional approach to mitigate these oscillations

is to introduce an artificial diffusion (conduction) term in the

heat transfer equation. This method is called "streamline upwind

scheme".

The weak form becomes:

f [ W (k + k ) T + W p C V T ] dfi = f W q dQJQ »x J-J -"-J »J P i ti Jjj

A

If the artificial conductivity k. . is correctly chosen, no

oscillation will occur in the Galerkin Finite Element formulation.

In ref. 10, the following technique is presented:

A A A A

Assume k. . = k u. u .

A ^where u. = = with M u l l 2 = u. u. and u. = D C V.i ,,..,, l l ~ l l 1 1 i ^ p i

u.i

uA

k is a scalar artificial conductivity.

Assume that the coordinates are chosen such that locally xl-

direction is aligned with streamlines and the x2-direction is

perpendicular. Then the artificial conductivity matrix in this

10

11

coordinate system is:

k = k

In case of bilinear quadrilateral, in two dimension, k is chosen

as:

f. A

endr|define the location of the quadratic point and are given by

= coth a --

* 1n = coth a --

where J(2k)

are the element length (Fig. 1)

Figure 1. Typical four-node quadrilateral finite ele-ment geometry.

12

u and k are evaluated at the origin of the element in£-n

coordinates, a- could be expressed by:

PC V. h.p a 1

2 k

4-1-2. Heat transfer finite element formulation:

The weak form is know given by:

[ w§i (k. . + k. .) T> . + w p cp v. Tfl idn

If the body is divided into a number of finite elements, the

weighting function W and the temperature distribution T within a

discrete element may be approximated by:

W » W N

Where N-, is the shape (interpolation) function of the element.

W, and T~ are constant at the nodes. The majuscule subscripts I

indicate the node number.

Substituting these approximations into the weak form we get:

L [ wi »i.i (ku + v TJ NJJ + wi NI ( P CP v

TJ NJ.I ' dnit

= [ Wj Nj q dfiJfi

which may be simplified as follows:

f [NI§1 (k;n

=f N TJo -1

q

13

In the matrix notation, these equations has the following form:

[ K ] [ T ] = [ Q ]

dflwhere [ K ] = [ NT (k + k ) N. -f N. < P C V ) N. ]J -1*1 aJ *J J»J -1 P * J»J

is the stiffeness matrix.

[ T ] = Tt

[ Q ] =[ N, q dfiJo i

These equations are the finite element equations which will be

formed, assembled, and solved for the temperature distribution

through the body. Because of non-symmetry of the second term

in the stiffeness matrix [k], its presence require the use of

solution routines different from those used in most finite element

programs.

To compare the solution obtained with the solution without theA

use of the upwind scheme, we can just set k. . = 0. The same finite

element equations can also be used for both stationary and moving

components of a sliding system by setting V = 0 for elements in the

stationary body.

4-2. Thermoelasticity:

The weak form of the thermoelasticity equation was found to be:

•t w T dn

The weighting function W, the displacement distribution U, and

the temperature distribution T, within a discrete element may be

approximated by:

W = W, N,

U

where the majuscule subscripts indicate the node number and N are

the shape function defining the type of element. For some elements,

the shape functions are given in the appendix. WT, UT and T, are

constants at the nodes.

After substituting these approximations into the weak form, we

get:

+ Wj Nj (3A+ 2V ) a Tj Nj ± ] dJi =J Wj Nj Fj

which may be simplified as follows:

f [ NT (3A+ 2U) a NT .] dfi )TT = f NT FTJ 1 J,i J J i i

These equations can be written in the matrix notation as:

[ KE ] [ U ] + [ K£T ) [ T ] = [ F ]

15

where [ K^ ] = [ NT . V N . . + NT . ( * -I- V ) N . ] dft is the^ '^ ^

elastic stiffeness matrix.

[ U ] = U

[ K™ ] = [ NT (3 A + 2U ) a jo . ] dfi is the coupledJ Q J. J »1

stiffeness matrix.

[ T ] = T

[ F ] = f N F dJ 1 -1 fl

Based on the equations above and the heat transfer finite

element equations, defined in the previous section, a finite element

program has been written. This program, which will be used in a

thermomechanical analysis, will give the displacement and the

temperature distribution through the body. It should be noted that

one element grid can be used. With this program simpler problems

could be treated. For example:

-Heat transfer without thermal stresses problem (an example of

this problem is given later): Set no forces and a = 0.

-Elasticity problem without temperature change: Set no flux.

Some basic routines and a typical input data are given in the

appendix.

Chapter 5

PROBLEMS AND SOLUTIONS

5-1. Semi-infinite solid under heat friction:

5-1-1. Description of the problem:

The semi-infinite source strip is defined by: xe[-b,b];

ye[0,°°]; in the plane z = 0. The heat is applied at the

rate Q per unit time per unit area over the strip. The

surrounding media moves across it with velocity V in the

direction of the x-axis. This problem was treated by

Carslaw and Jaeger (Ref. 1) using the heat source method.

K-2b

Fig.2-Seroi-infinite solid under friction.

16

17

5-1-2. Finite element method:

The problem defined above is treated by the finite

element program. The elements used in the mesh are quadri-

lateral (linear or quadratic). The heat flux is applied

uniformly distributed on a width of 2b at the nodes. Four

meshes were used, which are:

- mesh 1: 200 elements, 231 nodes, four-node quadrilateral

element, with upwinding (Fig. 3).

- mesh 2: 200 elements, 231 nodes, four-node quadrilateral

element, without upwinding.

- mesh 3: 300 elements, 981 nodes, eight-node quadrilateral

element, with upwinding.

- mesh 4: 460 elements, 1493 nodes, eight-node quadrilateral

element, with upwinding.

5-1-3. Results:

Carslaw and Jaeger (Ref. 1) gave the temperature as the

following integral:

71 kV J x-B " "° ̂ T "• ' dU

where Ko(x) is the modified Bessel function of the second

kind of order zero and X, Z, B are dimensionless quantities

introduced as:

x.a.z.fe.B.a, with K . i-£.1*. f̂t. ZK \J \J_

P

Some values of the surface temperature of the solid

are shown in Fig. 4 for B = 10. The curve represents

18

EL

|5

_t-J_ I CL

IT?'

Q.-0

n> .J3^ '&. -FJfl.3-Linear quadrilateral mesh for a "semi-infinite solid under

heat friction".

19

I

u

2oum

§

IW3/A11JL

T3 II0) CQ

3 80 -H01 0)

01

C m-4 0)I -O

•^ -He ~»o> inu

o u•H

Si<0 0)

e A0) (N

o -a«2 '»V) o;*

20

(7iTkV/2KQ) vs. (x/b) given by Ref. 1. The results by finite

element method, for each mesh, is plotted on the same

representation.

5-1-4. Discussion:

When the quadratic elements (8 nodes) are used the curve

(TI TkV/2KQ) vs. (x/b) approaches the one given by the heat

source method (Ref. 1). When a smaller number of linear

elements (A nodes) is used the curve is less accurate but

has the same pattern. The oscillations in the solution are

minimized when the upwinding is used. But it is still

necessary to make a mesh refinement in order to get accurate

results.

5-2. Gas path seals:

5-2-1. Description of the problem:

The gas path seals are used in the turbine engines of

modern aircraft to prevent the axial flow of working fluid

(air) around rotating engine components. Reduction of the

clearances between rotating and stationary component of these

seals can decrease the consumption of the specific fuel and

increase the effeciency of the engine. However, such

reduction in clearance may result in occasional rubbing

between the rotating and the stationary seal components as

engine deflection occurs. These rubs, which occurs at very

high sliding speeds, can cause high surface temperatures,

excessive wear of the seal components, and possible damage

21

to the engine. The development of gas path seal designs

have been retarted by an incomplete understanding of the

temperatures, stresses, and deformations which occur during

high speed seal rubs (Ref. 11).

An attempt to get the temperature and deformation in this

sliding system is done using the finite element analysis.

A model has been developed to simulate the rubbing contact

which occurs in gas path seals between the rotating knife

edge and a stationary seal segment. The outer gas path seal,

assumed to have a circular section with 5 layers (Fig. 5 & 6),

is rotating at 20,000 r.p.m.. The friction force is taken as

100 lb.. The material properties are given in Table 1. On

the external surface the temperature is taken to be 70°F and

the displacement is zero. On the internal surface the flux,

considered concentrated at the lower point is taken to be:

q = N P V = 100 x (0.3) x (20,000 x 27tx A.85)

= 1.8284 x 107 BTU/min in2

where u is the friction coefficient.

5-2-2. Finite element method:

The thermomechanical program developed before is used

to the rubbing contract problem. A finite element mesh

based on quadratic quadrilateral (8 nodes) elements, shown in

Fig. 7, is used in the development of the model.

5-2-3. Results:

3-1. Temperature:

22

. Siaior

Fiq.5_Hiqh pressure lurbine ouler qas path sea!.

i

Fig.6-Cross section of outer gas path seal (units mm)

Table 1

Material properties

Layer

Int. radius (in.)

Ext. radius (in.)

Material 100 YSZ 85% YSZ 70% YSZ 40% YSZ MAR-M-15% CoCr 30% CoCr 60% CoCr 50gAl Y Al Y Al Y

Young's modulus 2.00 2.00 8.00 17.75 15.60(Ib/in2)xl06

1

A. 85

A. 91

2

A. 91

A.9A

3

A.9A

A. 97

A

A. 97

5.00

5

5.00

5.11

Poisson's ratio

Coef. of expansion(in/in°F)x!0~6

Thermal cond. .(BTU/min in°F)x!0

Density(lb/in3)

Specific heat-

0.25 .

A. 83

8.11

0.155

0.161

0.26

7.70

16.30

0.180

0.161

0.27

8.38

20.95

0.205

0.161

0.28

9.52

25.52

0.25A

0.158

0.30

12.20

A22.98

0.320

0.155o(BTU/lb"F)xlO

23

T=70°F, U=0.

u=20,000 rpm.

Fig.?- Finite element mesh for gas path seal friction problem.

25

The finite element results give the temperature at any

node. A plot of the temperature on the internal surface vs.

the angle near the contact is given in Fig. 8. A peak of

temperature occurs at the location of the friction contact.

3-2. Deformation:

The finite element results give the displacement in

x-direction and in y-direction of any node. The curve of the

deformation in x-direction vs. the angle for the interior

surface is given in Fig. 9. The displacement is positive for

the nodes at the right of the friction contact and negative

for the nodes at its left. The curve of the deformation in

y-direction vs. the angle is given in Fig. 10. This curve

presents a peak at the friction contact.

5-2-4. Discussion:

At the location of the friction contact, the temperature

and the deformation are increased rapidly. The temperature

might reach the melting point of the material. The

temperature peak was predicted by Harsher (Ref. 12) and

confirmed by Kennedy (Ref. 11). The latter conformed also

that the maximum amount of deformation occurs on the contact

surface, with magnitudes decreasing very rapidly in a

direction normal to the surface.

The oscillations that appear clearly in the x-direction

displacement (midside node) are due to the presence of the

oscillations in the temperature distribution. When the

26

scale

270

Fig.B-Internal temperature distribution under friction nearcontact.

27

£!')

Fig.9-Displa(.ement in x-direction near contact.

28

scalp;

250 270 290Fig.lO-Displacement in y-direction near contact.

upwjnding term is increased, the magnitude of temperature

and displacement decrease but the oscillations don't vanish.

Chapter 6

CONCLUSIONS

A finite element program has been developed which can

efficiently and accurately predict temperature and deformation

distribution in a thermomechanical problem such as sliding systems.

The method includes velocity effects with the use of the streamline

upwind scheme which eliminates spurious oscillations and is therefore

very useful in cases involving high speed sliding. The comparison

with some results that used heat source method shows the accuracy

of the method. A special application to gas path seal problem shows

the existence of a peak in the temperature and deformation near the

contact, indicating a possibility of melting and excessive wear

which might provoke an early thermomechanical failure. Further

studies could investigate the importance of thermal conductivity of

the moving and stationary component on decreasing surface

temperature and deformation.

30

REFERENCES

1. H.S. Carslaw & J.C. Jaeger: "Conduction of heat in solids". 2ndedition. Clarendon Press, Oxford, (1959).

2. B.A. Boley & J.H. Weiner: "Theory of thermal stresses". Wiley Inc,(1960).

3. A.D. Kovalenko: "Thermoelasticity: Basic theory and application".Translated from Russian by D.B. Macvean. Wolters-Noordhoffpublishing Groningen. Netherlands. (1969)

4. F.F. Ling: "Surface mechanics". Wiley, N.Y., (1973).

5. F.E. Kennedy & F.F. Ling: "A thermal, thermoelastic, and wearsimulation of a high-energy sliding contact problem". Jour. Lubr.Tech., Vol. 97, 497-508, (jul. 1974).

6. P.J. Roache: "Computationnal fluid dynamics". Hermosa publishers.Alburquerque, (1976).

7. I. Christie, D.F. Griffiths, A.R. Mitchell, and O.C. Zienkiewicz:"Finite element methods for second order differential equationwith significant first derivatives". Int. Jour. Num. Meth. Engng.,Vol. 10, 1389-1396, (1976).

8. J.C. Heinrich and O.C. Zienkiewicz: "Quadratic finite elementschsemes for two-dimensionnal convective-transport problems". Int.Jour. Num. Meth. Engng., Vol. 12, 1359-1365, (1978).

9. T.J.R. Hughes: "A simple scheme for developping 'upwind' finiteelement". Int. Jour. Num. Meth. Engng. Vol. 12. 1359-1365. (1978).

10.T.J.R. Hughes and A. Brooks: "A multi-dimensionnal upwind schemewith no crosswind diffusion". California institue of technology,Pasedena. CA.

11.F.E. Kennedy: "Thermomechanical phenomena in high speed rubbing".Wear, 59. 149-163, (1980).

12.W.D. Marscher: "A phenomenological model of abradable wear inhigh performance turbomachinery". Wear, 59. 191-211. (1980).

13.J.P. Holman: "Heat transfer". Chap 1 & 5. Me Craw-Hill Inc.(1980).

31

32

14.F.E. Kennedy: "Surface temperature in slinding systems. A finiteelement analysis." Jour. Lubr. Tech., Vol. 103, 90-96.(Jan. 1981).

15.F.E. Kennedy: "Single pass rub phenomena : Analysis and experi-ment". Jour. Lubr. Tech., Vol. 104, pp582. (Oct. 1982)

16.J.S. Hansen & A.H. Von Flotow:"Finite element operators: In-expensive evaluation of upwind schemes". Int. Jour. Num. Meth.Engng., Vol. 18, 77-88. (1982).

17.H.Wolf:"Heat transfer". Harper & row publishers, N.Y., (1983).

18.R.L. Mullen & R.C. Hendricks: "Finite element formulation fortransient heat treat problems". NASA. Technical Memorandum 83070,(Mar. 1983).

Appendix I

SHAPE FUNCTIONS NI

1-Linear quadrilateral element:

N = id -€) (i -n)(-1,1)

N2 = i (i (i -n)

N3 = i (i +€) (1 + n)

-

3(1,1)

2-Quadratic quadrilateral element:

d+C)(i-n

N7 = i d-62 )d+n )

N8 = i d-n2 )(!-€)

(-1,1)

5

-f3(l,l)

33

ORIGINAL PAGE ISOF POOR

Appendix II

STIFFENESS MATRIX ROUTINE

0001COO.'COC-3occ<ooosCCOtO O C 7ooc-tCOOS'00 JOco:i?*!?(•'.•KC-t'liO O i iC'017O C J £001P

OOl'O0021002:-P022?E*et".C*002?

'«Co5'l00300031oo?r00?3C034C03S

S03<C37002tOC7-V0040OC<100420043sm00440047w»0050I»OM

S°oll0054005tm*«0||

oo«:00<300t4OOti00610067006660690070

J ME CfT I f c yk E A L « E < A -

l t U X t N & r t r . l Y M

t»»MCa»»»C»» t»C»»f tC t a a aCat atC»»»»C a a a ar»>»»C0«1c»«»»C»»*»C a i ^ aCtaaaCai t tC a t a af t t»aC a a a aL»aiaC»»»»taaa iC a a a atitiica***c a a a ifa iaaCl»aiCaaaaC » a a ac»a»»c a a a a

T H 7 E E'JJ-ROtn J W i

t fc

C A L C U L A T E S

T 1 F

TUF. V A L U F U F I H f E T l f f K C t l r . A I F . J X' ' '

»<f AUIPi. * £

NAKF.

101't

l 'YI-»tF A lIIII-lS

UxJACOth^>CK'oXtsTrf-EK T Y f Et

£Ul f :ULMlNLS

»t

I-

(X) / I in »/rl". : .'}•

(llj

tl £1

A X l i . v i n M f . l t

an*•«»«f»»»»»•»aa t k» y » »»» » •• < » »• » » «»»»ii«»»»

.

5W H C I FK l t f - l *M t b L K

I'DTOI'D

I 00!IDC'F

Jf:MJNM x1 N I £ XE l t r l M

OFIt 'COKt S

* Ki 'KVEl -

1YMl 11»F

t »»»!»»«»»«»

» » « >! ««»

I f JUC-CR

LftAICfHf'IGHFf,!'

C «A1Jf

« TURKS THf. CI N I T I A L I Z E THE frMAF TK A U J J i M U L T I f l K A T I t l K

H U l T ) »

!»»»»»»*

»»>»t*«*»»»»

f Jf.lN, jut*

CMl'Tt ) tf

COHM'N /UTLl

fttt*JMIN«V.V9l t+20JHAX--JH1KFET OF ELEMEMT

f>jh l» . ' .»>j rA>. . |! E i N t i f t E ! 6 , ? > . P i J i F < E t 9 > » r £ l l T » * > • ( V)

1

£***»C

}F(!t .NE. JESfHi^'NIilNHPC 1 11'liNli?

m»»*

.»<£. NCX) CALL C6MFI < JEKtWGXJ

ZEf>0 OUT POI'T FORCE VECTOR »a»»

10 22I2* l«NnQKTF

CALL CMAT(t)TYPE.).DO 10R«0.0bXl>F-PXHEtm<f

.11-lrNb

0.0

lv«i

007300740073

5

4D r E E < j . i 2 » * pPF |E<2f I2 )» | iXpF» t 'S I«E( I2» I I ) - I ) J ( l>E»rSI ' f

35

ORIGINAL PAGE ISOF POOR QUALITY

C-C76OC-7VoorcooriOOE?OOP2

181*OOf*0067OC'SBOOF9OPSO00 VIoo?:1C-07?OOV4oo?r-

8811009*0100

010301 C<01 CiOlC-t0:0?OlOfc0109011C0111l\\'i0114

01160117one011V0120012101 ::•

If l.'.M.l't< .11. jniK; J«lKi.jAtfi»'IF <J*.CCI' ,CT. Jhf.X) JHtyr i^tOIIf tJACdh ,LE. 0.0 SlOf 4

• *MK]». •>»»•t»»»t C*LCULftU THEC

•WAJK11

Si!*012E

01310132

S133134

013501360137013601395140

0147

015101

0155

0157

Sii!01(001610162

WU-V

S O /(11<? t )3 ) ' 0 .0

t <i t ::• •

r<4,13).iTtl '2.|«&il3i'0.0V ( 6 > l S > ^ 0 . 0F<1 t |3 ) *O.C

UtSJJJ-*-*tl (1*I?)>0,0tT<r» !3 ) *0 .0P T ( 3 r I ? ) r C . P1-1 < 4 r J 3 ; « Q . O

Kit I3)«C'^6|(2,}3>«HfEt>(3t I3> '0.0K4rI3)«Df'E|(5»I3)«0.0K*.13)«0.0F<1.13)^0.0

rc3.12>«=0.0F(4>13)*0.0f < 5 1 1 3 > « l» . 0

)«F

Kl»I3)«0.0»<2>I3>*0.0

ELII?tnEE(12.11

;<2tj3)«o.iF(3>I3)cO.OF(4>13)*0.0

tiif NUE

,,„„C»*»t SET UP THE CONPUC1 ION-STATIC ELASTICITY TERM. «JTH *>*!

C.»M

810

CALL

feV'l

MPRp(fvt| t»T,«f6tNIO:T55t<ibfeKK?'* |̂illV»"H THE 1HEKKAL ITMIN «»•

CALL C«FRr(RCrv»P»KV»6f*tNlOCALL OTFRt«FrRVr8KT2.*.KDtND)?.">•! • 0

e i?BirNi>2CONTINUEIF (JE.NE.O) RETURNJE»1CONTINUEgETUMICMP

36

OOCloo&r0002

004<'007ocotC-0'.'*0010

001!001 fcCO)"*CC-lt

core-

IBOCJ4002SC'Oi'f002?00 3 Poo:*0030003101-3?003?003UCOZ'JOOJiC02 /orJtCC3*OOAO004100420043C(>4*004S004600*7-04C

04VOiC

0051t0^20052oor-40055

8016W2ooi.9

tos?iotcOC-*10042;0630640«5

*r*L9et t>nnttf!

17.VU ) i

«f-HtO-2l

i v < < > i r M t > ,!«»•. t* :

to5

40

I'd 1 Jr)l«t'U J l«-l.tt> < 1 » .1 > « C . Cuf w < i i j )-o.c

V ' )«O.C

. x» t i ) • ̂ »: ; « 6 : ; • «' < * )fc.< i ifrLit "! » e » 6 )

> t >• t < >

P < 1 t J) «• C-. C

M'«%M«tN V « ( X l '

CD 10 It

i f ( x v i iCD 10's

t '• * *:

CO 10 SC-^^•.5c•lt^•

If (f TA'0.0CO IP

(?. »XK (Ml

)-(J . /ALC1A)r .HA^-<M/2 .>* ( t^SU^O(K^YFl )»Cf Jt

CALL guru-CA'.L C1fRI<

20 eONUKUt

fLANE

Y " \ t W V

»»*»

200MPITC«<200) ALfORMAT<4X.'A'.»' i

.<'S.«FK («TYfE»

Appendix III

INPUT DATA FORMAT

Columns Format Descriptiontype v

\ 1-80 20A4 Title to be printed at beginning ofoutput.

2 1-5 15 (Number of elements in this problem.5-10 15 Number of nodes in this problem.

3 1-10 G10.4 Young's modulus for material type 1.11-20 G10.4 Yount's modulus for material type 2.

51-60 G10.4 Young's modulus for material type 6.

1-10 G10.4 Poisson's ratio for material type 1.11-20 G10.4 Poisson's ratio for material type 2.

51-60 G10.4 Poisson's ratio for material type 6.

1-10 G10.4 Expansion coefficient for materialtype 1.

11-20 G10.4 Expansion coefficient for materialtype 2.

51-60 G10.4 Expansion coefficient for materialtype 6.

1-10 G10.4 Conductivity in x-direction formaterial type 1.

11-20 G10.4 Conductivity in x-direction formaterial type 2.

51-60 G10.4 Conductivity in x-direction for

37

38

material typo 6.

1-10 G10.4 Conductivity in y-direction formaterial type 1.

11-20 G10.4 Conductivity in y-direction formaterial type 2.

51-60 G10.A Conductivity in y-direction formaterial type 6.

8 1-10 G10.4 Density for material type 1.11-20 G10.4 Density for material type 2.

51-60 G10.4 Density for material type 2.

1-10 G10.A Specific heat for material type 1.11-20 G10.4 Specific heat for material type 2.

51-60 G10.A Specific heat for material type 6.

10 1-10 G10.A Velocity in x-direction for materialtype 1.

11-20 G10.4 Velocity in x-direction for materialtype 2.

51-60 G10.4 Velocity in x-direction for materialtype 6.

11 1-10 G10.4 Velocity in y-direction for materialtype 1.

11-21 G10.4 Velocity in y-direction for materialtype 2.

51-60 G10.4 Velocity in y-direction for materialtype 6.

12 1-10 G10.4 Element height for material type 1.11-21 G10.4 Element he ight for mater ial type 2.

39

51-60 G10.4 Element heigh for material type 6.

13 1-5 15 Node number9-9 II Displacement fixity in x-direction at

this node= 0 applied traction= 1 applied displacement

10-10 II Displacement fixity in y-direction atthis node.

11-11 II Temperature fixity at this node= 0 given flux=1 no temperature

12-21 G10.4 x coordinate of node.22-31 G10.4 y coordinate of node32-41 G10.4 Force at node in x-direction42-51 G10.4 Force at node in y-direction52-61 G10.4 Flux or temperature at node

14 1-5 15 Element number6-10 15 Node number for first node on element11-15 15 Node number for second node on element

41-50 15 Node number for eigth node on element57-57 II Material type for element64-64 II Element type

Data cards can be omitted for nodes equispaced between node N.

and N. as long as the data for nodes N. and N. are included.J J

ORIGINAL PAGE fSOC POOR QUALITY

Example of input data

TH SEAL340 I

.20001*070.20001 +070. EOOOE407C . 177:£-fOBO.JifcOE4C-f O.OOOOE400

FAi.100

.2SCO 0.2600 0.2700 O.i'800 0.300C O.OOOOt + OC

. n ji»i -

3 :isJ «1!J _ e

IFlc

20

33

31-3ft

II35404;4;-

• •„ „" t :

i-

00'

COo?OC-C'ob

"

: f: ot -, o. o'.ftor i

- . ' O . i i j O L - ^ r c

,.i- « . fT-oi »i A C i* ^

* . 0 ft . 0

4.:

r«

?•» ^fK66 :-.4:-v

orooocoooooo

4.CC:

.3.4T

-

.4*0'683

l.»76

1.'0461 .05V

O . O O O G E i C OC-.OCC-C-E + CC0. OOOCE-i 00? . OOOCF400O . O O O O E 4 0 0 0 . C O O O E - I O O0 . OOOOt 40'.':•. OCOOE4CC0. O O O O E 4 C - C •>. OOC'Jf + 000 • OCOOe 400'.' • O O O f E 4 OC1

r/. OOOOE-i OCC I i&0< 0£4 C 00 .0000£ < 000.000:-E4C-0

6.'ooc.oE.4ooo.'oooolto6O.OOOCE4000.000CE+00C.Of'OOE4000.00'JOE400O.OOOOE4000.000CE40C'O.OOOOL4000.0000E400O.OOOOE4000.000CE4000.OOOOE4000.COOOE400

).OOOOE400

00C(OOC' .OvOOt40COOC.OO<(-f 400 ^.0 • 1.04500 1.9C4-00 ^.002OC 2.612OC 3.4E200 3.49350 4.047

- 4.4591.4724.7704.°?£

_____C.COOOE4000.0000E+OOO.OOOOETOO0.0090E+OCO. OOOOE4000.0000t+00o . oooor

4.770«,41.94.47?4.047

00000CO00

3.4932.fl?1.9942.002

OOC'.OOOOE400oo i.or/CO 2.00800 2.OKOC0000OC0000 4.499OC 4.7990'.' 4.9M'00 4.9"»C

3.5043.5144.0714.461

.OOOOf 4 000.OCOOE4 00•:•. C900E 4 C'OO. OOOOE4 00O.OOOOE+OOO.OOOOE400C.OOOOE4000.0000E4COO.OCOOE4000.0000E4-OCO.OOOOETOOO.OOOOE400O.OOOOE+000.OOOOt+00O.OOOOE4000.0000E400O.OOOOE+OOO.OCOOE40CO.OOOOE4000.0000E4-00O.OOOOE4000.0000E400O.OOOOE4000.0000E400O.OOOOE4000.0000E+00

O.COOOE40pO.OOOOE4000.gOOOE400

4006. 6<>66E466f>. 6000E400O.OOOOE4000.0000E400O.OCOOE4000.0000E400O.OOOOE4000.0000E40CO.OOOOE4000.0000E40CO.OOOOE4000.0000E40CO.OOOOE400C.OOOOE400C.OOOC'E4000.0000E400O.OOOOE4000.0000E400O.OOOOE40CO.OOOOE4UC

OC 4.9*0 O.OOOOE+OOO.OOOCE4000.0000OOO.OCOOE400 4.9̂ 5 0.OCOOE4000.0000

4.9704.799

4 . 4 V V4.07]3.5043.514

*'.'o6e2.014J .OC'OOE400C..OOOOE4C.Cf

O.OOOO

OOO.OO.H'£40C'00 J. ' . - 'E

2.0212-01-7

0.00 JOE4000.OOOOE4000.OOOOt400O.OOOOE4000.000CT400

00CO00OC 3.1200 3.534

5.00C4.B2B

4'.5274.096

3.'53?

. .

S .OOOOE-tOOO.OOOOt+OO.OOOOE + C-OO. OOOOE 400

O.OOOC't 4000.POOOE+000. OOOOE 4000. OOOOE 4000. OOOOt •* 000. COOOE 400.

O.OOOOE400C.OOO{-£-'00

eO 00 '.0*< 2.E<« O.C-POOf + OCO.OOC.OE + OCt: 00 4.M2 2.021 O.OOOOE.+ OOC..OOOOE4006:' 00 4.5?y' 2.02"7 C.OOOCE+OOO.OOOOE+00«3 00 4.826 1.076 0. OOCOE^f 000. OOCOE400f4 00 4.985 C.OOOOE4000.0000E+OOO.OOCOE40065 00 5.000 O.OOCOE4000.0000E+OOO.OOOOE40066 OOO.POOOE400 5.055 0.OOOOE+000.OOOOE40067 1120.0000E400 5.110 -0.OOOOE+000.OOOOE40070.066 112 1.102 4.934 -0.OOOOE+000.0000140070.069 00 2.049 4.576 O.OOOOE4000.0000E+0070 112 2.071 4.626 O.OOOOE+000.COOOE40070.071 112 2.906 4.166 O.OOOOE4000.0000E40070.072 00 3.574 2.574 O.OOOOE+000.OOOOE+0077. 112 3.613 7.613 O.OOOOE4000.0000E + 0070.074 112 4.1*6 2.90P O.OOOOE4000.0000E+0070.07L 00 4.£?« 2.04? O.OOOOE400C.OOOOL4007« 112 4.€?.t 2.071 O.OOOOE4000.000CE*p07C.O7> 11" 4.934 J.102 O.OOPCE4000.0000E4007C.07B OC 5.055 O.OOOOL4000.0000E40CO.0000140079 112 '. lit' 0.00''OE-K'00.00';Ot4000.0000£-tOC-?O.C'60 00-i.OOO O.OOOOt400C.OOOOF.4000.0000E400PI 00-S.035 O.OOOOr4000.0000E400C.OOOOHOO

tl 11J-5.110 O.OOOOE400C.OOOOE4000.0000E40070.03 00-4.E26 1.076 O.OOOOE4000.OOOOE400

E4 112-4.934 1.102 O.OOOOE4000.OOOOE+OC70.085 00-4.527 2.C27 «.OOOOE4000.0000E400

f t, OO-4.576 2.04V C.OOOOC4000.0000E4007 112-4.626 2.071 O.OOOOE4000.0000E40070.0 i

EP 00-4.09* 5.6*6 O.OOOOE4000.0000E400 1BV U2-4.iei 3.9C6 O.OOOOE4000.00COE40070.C ,90 00-3.536 3.536 0.OOOOE+OOO.OOOOF400 :91 00-3.174 3.574 O.OOOOE4000.0000E400I'.: 111-3.613 3,613 0.000014000. OOOOE + 0070.693 OC-2.84* 4.096 0.OOOOE4000.0000E40094 112-2.^OB 4.166 O.OCOOE400C-.OOOOE40070.095 00-2.027 4.527 O.OOOOE4000.0000E*00V6 00-2.OA9 4.576 O.OOOOE4000.OOOOE400 :97 112-2.071 4.fc26 O.OOOOE+OCO.OOOOE40070.0 jV6 00-1.076 4.626 O.OOOOE1OOO.OOOOE400 i99 112-1.102 4.934 O.ODOOE4000.00CCE40070.0100 00-4.970 O.OOOOE4000.<>OOOE40C-O.OOOOE400101 OC-4.9E5 6.0000E4000.0000E4000.0000E400102 00-4.79? 1.072 O.OCOOE4000.0000E400103 00-4.499 2.014 O.OOOOE4000.0000E400104 00-4.513 2,021 O.OOCOE1OOC.COCOE400105 00-4,071 2.E29 O.OOOOE4000.00COE400106 00-3.514 3.514 O.OOOOE + 000. OOOOE-t-00107 00-3.525 3.525 O.OOOCE4000.0000E400106 00-2.6?? 4.071 O.OOOOE4000.0000E40010* -00-2.014 4.499 O.OOOOE+000.OOOOE400110 00-2.021 4.513 O.OOOOE4000.0000E400111 00-1.072 4.799 0-OOOOE4000.0000E400112 00-4.940 O.OOOOE4000.0000E4000.0000E400113 00-4.955 O.OOOOE4000.0000E4000.0000E400114 00-4.770 1.065 O.OOOOE4000.0000E400}}? 99-*'*2? 2'5°2 $<$§OSE4000.0000E400116 60-4.48? 5.OOP C.OOOOE4000.0000E400117 00-4.047 2.312 XJ.OOOOE4000.OOOOE400116 -00-3.493 3.493 O.OOOOE4000,OOOOE400119 50-3.504 3.504 <.OOOOE4000.0000E400120 .00-2.$1? 4.047 3.0000E4000.0000E400121 00-2.002 4.472 O.OOOOE4000.0000E400122 00-2.OOE 4.486 «.OOOOE4000,OOOOE400123 00-1.065 4.770 O.OOOOE+000.OOOOE400124 .00-4.910 C.COCO£*OOO.OOOOE4000,OOOOE400125 00-4.925 'C.OOOOE4000.OOOOE+000.OOOCE+00126 -00-4.741 t-05V O.OOOOE4000.0000E+00127 00-4.445 1.990 O.OOOOE4000.OOOOE+00126 00-4.459 1.V96 O.OOOOE4000.0000E400129 00-4.022 2.795 O.OOOOE4000.0000E400130 00-3.472 3.472 O.OOOOE4000.0000E400131 00-3.463 3.483 . io.OOOOE4000.0000E400132 00-2.795 4.022 O.OOOOE4000.0000E400133 00-1.990 4.445 O.OOOOE4000.0000E400134 00-1.996 4.459 O.OOOOE4000.0000E400135 00-1.059 4.741 O.OOOOE4000.0000E400136 00-4.650 O.OOOOE4000.0000E4000.0000E400137 00-4.BEO -O.OOOOE4000.0000E4000.0000E400138 00-4.663 1.046 O.OOOOE4000.0000E+00139 00-4.391 J.?66 O.OOOOE+000.OOOOE+00140 00-41416 . iJ/6 6.0000E + OOO.OOOOE + oS141 00-3.973 2.760 O.OOOOE4000.0000E400142 00-3.429 3.429 O.OOOOE+000.OOOOE400

-

;.<*

11VISO

li?i.r-4

"•*

5 tC164: *•>

17Cl'-'in v>

1V2

o>-;.:;

OP-J 1'00 1 .<0000.

. .3fl

00-. i00-00-1.19200-3.200

• i.'fo'̂'4.75?

.00- 1 . 75?.oc-?.2J:>oc- 2.?«too-r.rao

337Br<?Jt:>-If 1JE"1E2

cc-:-.~r7t•'.•-•«. lit00-? .1??

6o-ilf'V'?SO- 3 . S'.t0-;.?••••00-3.t73

ob-V.il/-oo-4.ro?00-4.42*

00-4.715

4.O ••4 . 17

.-3.3U-:••./<'

-1.53:.-1 .5̂ 4-.'9£4-.eoe?

JfS

1VO1V3

1V2

00-J.212OC'-J .215

( 0-2'.267QC-7.27400-2.74400-3.16600-3.17i-

00-3'We00-3.V2000-4.23400-4.49100-4.50900-4.744

-4.B32-4.B4?-4.1?)-4.44E

..-2.221-1.I.4?

17/ V»'Vf'V'WV»fc.

i?e oo-.4310199 OC-1.215200 OO-l.i'23

?C2203

206?OV

M4

Jl*

OOO.OOOOE400-4.97000-.4310 -*•'*£•OC-1.215 -4.E4200-l.i'?3 -4.87*00-J.775 'I'̂ lt00-2I2E700-2.7*£100-3.1Bt00-3.19r,

SO-3.5S-?0-3.93200-3.

c . or oc--:-* c oc . c COC-E t0 . C-OOOt *000 . OOC VK +

221 00-2.3G1222 OC-2.777223 Cu-3.204

0 .OOOOC400C- .<VOOO£-»00O.OOOOE40000 . OOOOL-* 000

.O.OOOOi-tOOOO.OOOOE-»OOOO.OC-OOE + 000O.OOOOf.+OOOO.OOOOE40CO

-3.75*-3.8.-C-7

:!:f&-2.632

bb-4.l3^ -1*54300-4.773 -.618-ooo. oooor 400-4.9?.r-OOJ.OOOOt400-5.OCv00-.434P -4.9t'000-1.12i -4 "'

.0000E400

. OC-OOE4 00

.OOCC'f iCO

.OOCOt^OO.OOOOE400. 000014 CO.OOOOE- tOC-.OC-OOL + '.''.-

o . r-c C-OE4 C'OO . oooott ci-• • . O C O O E 4 0 0 ' . - . O O C O E - K C0 . COOOl +000 , OC C-OF40« %

0 . OOC-Or 1 0V C/. OOOC1.4C CO . O O C - O S + .'-f-O.OPCOE-K'C

-4.74P

4.2103.61?

-3.B3C-3.37?

C.COOC-t 4000.00001*000 . OOOCE4 O:-} . OOOOE4000 . Ov('9£ 4 ?'-•(• . 0500E40V

O.O0.

O.C-OOOE4000.000014000.OOP9E400C.OOCOE4000 .0 :.OC1E4000. OOOOE40ti0.{"./OC£4000.0000E4C0O.OOOOE400C>.OOi'PE40<..CC-C.CE4000.0000E4C6.OOOCE400C.OO;'C-t400,DC-OOt4000.0000E400

O.OlOOE4000.000C'E*00O.OOCOE4000.0000E400O.OOOOC4000.0000E4COC.OOOOE4000.0000t+00O.OOOOE4000.0000E400C.OOOOE4000.COOOE400O.OOOOE4000.0000E400O.OOOOE4000.000C-E400O.OOvOE4000.0000E4000.OOGOE4000.OOOOE400O.OOOOE4000.0000E+00C.OOOOE4000.0000E40CC.OC0064000.OOOOE400O.OOOCE4000.0000E400O.OOOOE4000.0000E40CO.OOOOE4000.0000E4COO.OOOOE+OOO.OOOOE40CO.COOOE4000.00COE400O.OOOOE4000.0000E40CO.OOOOE4000.00C-OE400O.OOOOE4000.0000E400O.OOOOE4000.0000E400O.OOOOE4000,OOPOE40»>O.OOOOE4000.0000E400O.OOOOE4000.0000E400O.OODOE4000.0000E400O.OOOOE4000.0000E400O.OOOOE4C-OO.OOOOE400O.OOOOE4000.0000E400O.OOOOE40C>O.OOOOE400O.OOOOE4COO.OCOOE4000.OCOOE4000.OOOOE400

OE40C-O.OOOOE400O.OOOOE4000.0000E400O.OOOOE400C.OOOOE400O.OOOOE4000.0000E400O.OOOOE4000.0000E400O.OCOC'E4COO.OOOOE4000.00?OE400&.OOOOE400

O.COOOE4000.O.OOOOC400C-O.COOCE400C.0.000014000.C.OOOOE400C.O.OOOC'E400C.O.OOOOC-iOOO.

0000E400OCOOE400OPOOE40C0000E400OOOOE + 00OC-OOE40CCOOOE40C

CO- 4.21;:,OC-4.ii.r-

OOO.OOOOE+00-1120.000CE+6C-117-.648800-1.243

112-1.2J-7112-1.82500-2.317

112-2.352112-2.839

117-3. *?£•00-4.01'

112-4.05:.112- 4.360

112-4.90700 <.8Ci

112 4.9074.5634.6134.6644.285

OC00

11200

380

00<00112 2.35200 1.785112 1.E2500 1.23000 1.243112 1.257llEo'.64Ce00 4.77300 4.536

SO 4,5500 4.26000 3.94400 3.V5fr00 3.58?0000

-.£412

-.*64K!-1.572-1.590-1.607-2.248-2.297-2.849-2.PEO-2,912-3.37?-3.4t»-3.?30-3.872-3.914-4.210-4.303

-4.615•-4.74B-4.652-4.P06-4.9*0-5.014-4.990-5.100

C.OOCCE+000.

00 3.9200 3.932

<0 3.5670 5.175

00 3.18500 2.74400 5.27400 2.28100 1.76400 1.215

80 J.21V00.6272

00 4.71500 4.48100 4.495

P.OOOOE-tOOO.OoC-OE + CCO.OOOOE-iC-OO.OOOOE + OC'70.

.OOOOE + OOO.OOOOE + 00

.OOvOE+OOO.OOOOE+0070.6.0000E+000.0030E+0070.0.OOOOE+OOO.OOOOE+000, OOOOE+OOO.OOOOE+ 0070,O.COOOE + t-OO.OC'OOE+OO?^.O.OOOOC+000.OOOOE+000.OOOOE+000.OOOOE+0070E + 0 0 . O O O + 0 0

E + OC?.OOOOt+C070t A COO . OOOOE + C>0f> . OOOOt A COO

O.OOCOC + C.OO.OOO0. OOOOt +000,

.OOPOt+OOt'.OCO0. C

JO.OOt'OE + OC. . . 50.0C-00£+C'C70.CC.OOOOE+OOO.OOOOE+000.OOOOE + 000. OC'OCE + 000.OOOOE+000.OOOOE+0070.0O.COOC'E + OOO.OOOOE + OC0«00'*0£ + COO,OC'OOETOO"0.0O.OPCOE + OOO.OOOC-E + OO

.'oc'c-OE + OCC-'oOOC-E + OC-'C-.C-.OOOOE+OOC.OOOOE+P?

O.OC'OOE+OOO.OC'OOE + t'C-?0.00.OOOOE+OOO.OOOOE+000. OOOOE + OOO, OOOC-E + 00C.OOOOE+OOO,OOOOE+0070.0C.OOOOE + OC'O. OOOOE + 000.OOOOE+000.OOOOE+0070.0

S:°o0oSr+0oS8:S80o°o!4+880.OOOOE+OOO,OOOOE+0070.00. OOOOE + OOO.OOOOE + 00O.OOOCE+000.OOOOE+0070.00.OOOOE+OOO.OOOOE+00C.OOOOE+OOO.OOOOE+000.OOOOE+OOO.OOOOE+0070.0

S.OOOOE+OOO.OOOOE+00,00&('E+OOO.OOOOe+0070.00.OOOOE+000.OOOOE+00

S.OOOOE+OOO.OOOOE+00.OOOOE+OOO.OOOOE+00O.OOOOE+OOO.OOOOE+000.OOOOE+OOO.OOOOE+000.OOOOE+OOO.OOOOE+00D.OOOOE+OOO.OOOOE+00t>. OOOOE+OOO. OOOOE+000,OOOOE+OOO.OOOOE+000.OOOOE+OOO.OOOOE+00

S.OOOOE+OOO.OOOOE+00.OOOOE+OOO.OOOOE+000.OOOOE+OOO.OOOOE+00O.OOCOEtCCe.OCOOc-fOC

. 6.OOOOE+OOO.OOOOE+00).OOOOE+OOO.OOOOE+000.OOOOE+OOO.OOOOE+00-1>.-OOOOE + OOO. OOOOE+00ii,OOOOE+OOO.OOOOE+00•>.OOOOE+OOO.OOOOE+000.OOOOE+OOO.OOOOE+00.OOOOE+OOO.OOOOE+00.OOOOE+OOC.^OOOE+00..OOOOE+OOO.OOOOE+000.OOOOE+OOO.OOOOE+000.OOOOE+OOO.OOOOE+000.OOOOE+000.OOOOE+000.OOOOE+OOO.OOOOE+000,OOOOE+OOO.OOOOE+000.OOOOE+OOO.OOOOE+00

S. OOOOE+OOO.OOOOE + 00.OOOOE+OOO.OOOOE+000. OOOOE + OOO. OOOOE-r 000.OOOOE+000.OOOOE+000.OOOOE+OOO.OOOOE+00

:<:;? c-c j.i.-.:e?--2 oo s.evr314 00 S.Vdf315 00 3«54£?16 00 3. It*31V 00 3,16*316 00 2.727219 00 2.260720 00 2.2d7321 00 1.7533.72 00 J.r06323 00 J.21i'324 000-623'325 OC 4.65?326 00 4.42*

?5s Sc 4!jsr3?* OC- 2.F4V?'».0 00 3.E72?,~.J 00 3.50:332 00 3.1183?3 CO 3.137

335 CO 2.' 232336 OC 2.2«d337 00 1.73233S 00 1.193339 CO 1.20014*

1

X456

8V101112131415161716&£021§1324

^

Si?930

M33

3637

- 3639404}42434445464748495051

S3

&

-;-.:'07-2!eo«. '

-3* 7 7 3-4. 134

-4*446-4.662-4.P17-4.F3i:

-.7964-l.Si'S

-'I-'.teo"ft ' 2%m

~J* 7ll-3. /-.'.£

-<.'3fcC--4.4C7-4.601-4.75V-4 . 766

OOO.C-15S- -4,E40611

21C13

II28

iJ37•O4548515fr5V

|?

909553103

IW3*11511*

'li12713C1323

13914214;153156

lit173136155160165170175124JE2»|5

g13

&§2331343742

41!5 •si565962657073/67967w676590

5310310610935115

25127130133

155160165170175

III1%5166mnrTOO203206

3613•>«2E

343V

5̂4633

§96267707376£267

1?60|5

%1001031C6109112m121124

111150155?60165170

\[l

165

iff194197m

i6

11

*3g

is2S283439

M485356596260

lo595100103

111

i!!124

Ho9133136139

tti14615315C163168173150155160m17517916?185

o I oooot'-f ('00 ', OOC-OL-I (• :C..OOOCE + Oi>O.OCOOE-}CO

j'.OOOOE + OOo'.OOOOE+OC'<!.OOOOF + PClO.OGl>Ct+ COO.OOOOEtOOC.OOOOE+OC

o'.OOCC't + COO*. OOOOf -tOO

0*. OOOOtfOOo'.OOOOEiOOO.OC'OOE + OOC.OOOPt + OO

0 .'C'OOOr + 000 .'OOC'OE + CC-0.000'.'f + OOO.OOOC>t*OC0 . OOOCt + '.'Cf< , OOOC-E + C'OO.OCOOE+OC-t-.COfCC + fC'0 .COC-OE + PC-0, OOf-Ct +PCC.OOOOf.+OOC.OOOCEtC-O0 . OPCOE + C'CC , OOC'jf + 00o.ooccr+ooo.cooot+oo

oloc-ooc+f oolf-cooe+ocO.'oCOOE+OOo'cOOOE-tC-O0 . OOOOt + OOC . OOOOt i OC'0. OOOOf+OOC. OOOCt +00C.CC'OOt + 0'.-O.CVOOF. + OO712J?~-J

30333fc41

V5055»646V72?I66

8104.107

HS11611?12238126

1342414014314621541591641691741371E1184187190193125199m

51C-

2?.2cOS

l!4043464524576063F̂71

?$64

f49»9£3

II102

titill114

123126129

\llis:-157162167172177160163186

If?19519B201204

27

* •»

1724

t"'7°33E

444•47"52

116166697275£1

ft>01

1̂110113

ill125

111

ffi

«3°14614915415?164169

1?̂161184m1931V6199202

49

H51015202£2932354043464VE<57

S363

96102105

i?f114

11?123126129132135136141to1511563611*4171

US157162

\̂ l177160

1B£J

111

2

13

34444555W

5

!4

\

1£

!Iiitii223•?3

i

».*-

IS15!5

1?

5S5

'5

]_ 5

' 5

' 5«•

1

1*

^|5

555

1r5

' 1

PAGE ,sPOOR QUAUTV

••6 191 20V 20i 186 20P 207 205 169ii7 194 512 20? 191 211 210 20P 19258 .112 100 212 194 113 213 211 1955V 200 216 fit' 197 217 214 214 19660 203 221 216 200 220 219 21? 20161 206 22* 2ri 203 223 221 220 20462 20V 227 224 20ft 226 21'S 223 207*3 212 230 22? 20V 229 22P Sr* 2U6-? 100 80 230 212 101 i3l 22V 113c-5 215 .'36 T3? 215 235 2s< 222 Sit6t 2::i 239 236 218 r?i: 23 '/ 235 219t> j w ~ & •" ̂ » •: 3 5 2<?3 *J^- 2^0 ^ 3 c' 2 *.'•.'6fc 22V 24T, :̂j- *-24 24^ 243 I;41 22?

70 "to e!' 216 220 *Yl 24* i47 23 17J 252 St.* 75 <5 25? 2M >E 25V72 257 25* 254 212 5f£ 256 253 25:.73 262 26* 2ty 257 263 261 I'SE 26074 267 2!-.V 264 2̂ 2 26E 266 2*3 2t575 272 274 2t5 2<7 27Z 271 26B 27076 225 233 274 5/2 222 276 273 27577 2?B 252 65 51 27* 250 *4 277"t 2S1 257 252 276 282 25L 279 2SO79 2f>4 2eT- 257 2E1 255 2dO 2£2 2E36(* 2^7 267 161' 264 ?&£ 261 2C5 2E661 ;•«•'.• 2>:: 2<7 2F7 241 2?0 2Sf 2E?P2 19/ 215 272 290 2M 275 291 292f- 29* 2Ve 51 3? 295 277 50 29384 2V? 261 27S? 2V4 298 2EO 295 296B'J 300 ?B4 2R1 297 301 283 29E 299fc6 3<--? 2E7 y*». 300 304 28c 301 3C2

§7 306 2SO 2B7 203 307 2BV 304 SOSP 179 197 290 306 196 292 3O7 5oB

€9 310 294 37 23 311 293 36 30590 313 297 294 3tO 314 296 311 B1291 316 5.00 297 313 217 299 314 31592 219 303 300 316 320 3C2 317 31E93 3:2 20* 203 319 323 305 320 32194 150 175 306 322 176 306 323 324V5 326 310 23 21 327 309 22 3259c 329 313 310 326 330 312 327 32697 332 316 313 329 333 315 330 331•?6 335 319 216 332 336 316 333 23499 3*E 322 219 Z35 339 321 33<£ 237100 148 150 322 336 149 324 339 340

3"

34

44445C

•,r.Jj*,*5

4444t.JQ

3

32*322

\22

I

«,5••

55

£,r

».

r;5ft

t.

F

55

I55r

5r.

5r,r.

555

55

355

S'

PUBLICATIONS

NASA Grant NAG3-369

R. L. Mullen and R. C. Hendricks, "A 3-Body Approach to FrictionContact Modelling", to be presented 22nd Annual Meeting ofthe Society of Engineering Sciences, Pennsylvania StateUniversity, University Park, October, 1985.

R. C. Hendricks, M. J. Braun, R. L. Wheeler III and R. L. Mullen"Two-Phase Plows With Ambient Pressure above the. Therrao-dynamic Critical Pressure", Bently Roter Dynamics ResearchSymposium of Instability in Rotating Machinery, CarsonCity NV, 1985.

R. C. Hendricks, M. J. Braun ,R. L. Mullen, R. E. Bucham andW. A. Diamond, "Analysis of Experimental Shaft Seal Datafor High-Perromance Turbomachines As for Space ShuttleMain Engines", Proceedings Workshop on Heat and MassTransfer in Rotating Systems, 1985.

M. J . 'Braun, R. L. Mullen, Andre Prekwas and R. C. Hendricks,"Finite Difference Solution for a Generalized ReynoldsEquation with Homogeneous Two-Phase Flow", ProceedingsWorkshop on Heat and Mass Transfer in Rotating Systems,ASME,1985.

_R._L._Mul.len, Andre Prekwas. M. J. Braun and R. C. Hendricks,"Finite Element and Finite Difference Methods for aReynolds Equation using a Power Law Fluid", Workshopon Heat and Mass Transfer in Rotating Systems, ASME.1985.

R.L. Mullen, R.C. Hendricks, G. McDonald, "Finite ElementAnalysis of Residual Stresses in Plasma-Sprayed Ceramics",9th Annual Conference on Composites and Advanced CeramicMaterials, 1985 (Also in press, Ceramic Engineering andScience Proceedings) 1985.

Robert L. Mullen, M.J. Braun and R.C. Hendricks "Finite ElementSolutions for the Stiffness and Damping of A Three-Dimensional Journal Bearing Using a Non-Newtonian Fluid"Proceedings of the 10th Canadian Congress of AppliedMechanics Volume 2 pp. E-23,London, Ontario, 1985.

M . J . Braun, R .L . M u l l e n , R . C . Hendr icks , Robert L . Wheeler ,"Fluid Flow and Heat Transfer in Annuli with Axial PressureGradient a Non-Isothermal Rotating Inner Cylinder and anInsulated Outer Cylinder", to be presented at the 1985 ASMENational Heat Transfer Conference, 1985.

R. L. Mullen, "Quasi Eulerian Formulation of Thermal ElasticContact Problem Involving Moving Loads". PresentedWinter Annual Meeting ASME, 1984.

47

M. J. Braun, M. L. Adams, and R. L. Mullen, "Analysis of a TwoRow Hydrostatic Journal Bearing with Variable PropertiesInertia Ef fec t s and Surface Roughness". To be publishedIsrael J. Of Technology and presented at 18th Israel Con-ference on Mechanical Engineering, June 27, 1984 Haifa .

J. Padovan, B. Chung, M. J. Braun, R. L. Mullen "An ImprovedPlastic Flow Thermomechanical Algorithm and Heat-TransferAnalysis for Plasma-Sprayed Ceramics" , 8th AnnualConference on Composites and Advance Ceramic Materials,American Ceramic Society, 1984. To be published CeramicEngineering & Science Proceedings. ;

R. L. Mullen, M. J. Braun, and R. C. Hendricks, "Finite ElementFormulation of a Reynolds Type Equation for a Power LawFluid", presented, Fifth International Symposium onFinite Element Methods in Flow Problems, Austin Texas,1984.

M. J. Braun, R. C. Hendricks, and R. L. Mullen."Studies of TwoPhase Flow in Hydrostatic Journal Bearings", 7th AnnualEnergy Sources Conference, Cavitation and Multi-PhaseFlow,FED-9 ASME, pp. 61-65 , 1984.

M. J. Braun, R. L. Mullen, and R. C. Hendricks,"A ParametricStudy of Pressure and Temperature in a Narrow Gap Between aStationary Outer Cylinder and an Inner Rotating Mis-alignedShaft", presented National Heat Transfer Conference, 1984.

M. J. Braun, R. L. Mul l en , and R. C. Hendricks , "A ThreeDimensional Numerical Method with Applications toTribological Problems", Proceedings of the 10th Leeds-LyonSymposium on Tribology, 1983.

R. L. Mullen and R. C. Hendricks, "Finite Element Formulationfor Transient Heat Treat Problems", ASME-JSME ThermalEngineering Joint Conference Proceedings, Volume 3, 1983,pp. 367-374.(also NASA Technical Memorandum 83070).