Q G VONKARMANCENTER - NASA · @ q g_ vonkarmancenter thermal strain analysis of advanced manned spacecraft heat shields final report to national aeronautics and space administration

@ Q

G_

VONKARMANCENTER

THERMAL STRAIN ANALYSIS OF

ADVANCED MANNED SPACECRAFT HEAT SHIELDS

Final Report

To

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

MANNED SPACECRAFT CENTER

HOUSTON, TEXAS

Contract NAS 9-1986

Report No. 5654-0Z FS / October 1964 / Copy No. 4

N65-30721(ACCESSION NUMBER)

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https://ntrs.nasa.gov/search.jsp?R=19650021120 2019-12-31T12:36:17+00:00Z

CONTRACT FULFII/MENT STAT_ENT

This is the final report submitted in fulfillment of the National

Aeronautics and Space _Administration Contract No. NAS_9-1986. This

report covers the period from I September 1963 to 31 August 1964.

Finite Element Analysis

Prepared by

Eo Lo Wilson

Senior Research Engineer

Finite Difference Formulation (Appendix G)

Prepared by

A, Zukerman

Head, Advanced Programs Department

Approved by:

W. T. Cox, Program Manager

Reviewed by:

Technical Specialist

Report No. 5654-02 FS

I

I

I

I

I

IiI

ABSTRACT

Numerical methods and computer programs are presented for the

analysis of heat shields. The finite element technique is used to

determine stresses and displacements developed in composite axi-

symmetric solids of arbitrary geometry subjected to axisynnetrie

thermal or mechanical loads. This technique is then applied to the

development of an automated ccmputer program for the _nalysis of

axisymmetric heat shields subjected to axisymmetric thermal and pres-

sure loadings. Finallyj the numerical technique is extended to the

analysis of heat shields subjected to non-axisymmetric thermal

loading o

Several examples are presented to illustrate the application

of the method and to demonstrate its validity. FCRTRAN II card

listings and descriptions of the use of the above programs are given

in the appendices. _ _ _ _0 f

Report No. 5654-02 FS Page ii

INTRODUCTION

PART I:

PART II:

TABLE OF CONTENTS

METHOD OF ANALYSIS

A. INTRODUCTIONi

B. EQU_ _UATIONS FOR AN ARBITRARY

FINITE EI_E_T

C@

De

_UILIBRIt_ EQUATIONS FCR A SYST_ (F

FINITE ELemENTS

SOLUTION OF EQU_IUM _UATIONS

E. EL_4_T STRESSES

GENIAL CCMPUT_ PROGRAR FOR THE ANALYSIS OF

ARB_Y AXISDRE_RIC STRUCTURES

A, INT_TION

B. ST_ OF TRIANGULAR

C. E_UILIBRI_M EQUATIONS FOR CCMPLETESTRUC_mZ

D. DETERMINATION OF DISPLAC_ENTS ANDSTRESSES

E. C(RPUT_ R_OGRAM

F. _MPLE

G. DISCUSSION

Page No.

I

3

3

3

8

I0

12

13

13

15

21

22

24

29

29

Report No. 5654-02 FS Page Iii

PART III:

PART IV:

APPENDIX A:

APPENDIX B:

APPENDIX C:

APPENDIX D:

APPENDIX E :

APPENDIX F:

APPENDIX G:

TABLE OF CONTENTS (cont'd)

AUTOMATED PROGRAM FOR AXISYMMETPIC HEAT

SHIELDS SUBJECTED TO AXISYMMETRIC LOADS

A. INTRODUCTION

B. MESH GENERATION

C. DETERMINATION OF DISPLACEMENTS AND

STRESSES

D. COMPUTER PROGRAM

E. EXAMPLES

F. DISCUSSION

AUTOMATED PROGRAM FOR AXISYMMETRIC HEAT

SHIELDS SUBJECTED TO NON-AXISYMMETRIC LOADS

A.

B.

C,

D.

E.

INTRODUCTION

THEORY FOR TEE ANALYSIS OF ANAXISYMMETRIC

BODY SUBJECTED TO NON-AXISYNMETRIC LOADS

COMPUTER PROGRAM

EXAMPLES

DISCUSSION

SOLUTION OF EQUILIBRIUMEQUATIONS

MATRIX FORMULATION OF THE LEAST SQUARES

CURVE-FIT PROCEDURE

MATHEMATICAL MODEL FOR SANDWICH SHELL

PROGRAM LISTING-ARBITRARYAXISYMMETRIC

PROGRAM LISTING-AXISYMMETRIC HEAT SHIELDS

PROGRAM LISTING-NON-AXISYMMETRIC HEAT SHIELDS

SUMMARY OF EFFORTS TO SOLVE THE THERMAL STRAIN

PROBLEM BY THE OVER-RELAXATION AND DIRECT

INTEGRATION OF FINITE-DIFFERENCE FORMULATION

Page No.

32

32

33

35

36

40

43

45

45

45

55

59

65

AI

B1

C1

J_J__

E1

F1

G1

Report No. 5654-02 FS Page iv

LIST OF SYMBOLS

aj,bj,ak,b k = Element Dimensions

E = Modulus of Elasticity

u = Displacement in the r Direction

v = Displacement in the 9 Direction

w = Displacement in the z Direction

- Thermal Coefficient of Expansion

B " Over-Relaxation Factor

V = Shear Strain

- Strain in the Radial, Circumferential and Longitudinal •erj9 ;z Direction

v = Poisson's Ratio

= Stress in the Radial, Circumferential and Longitudinal_r_8,z Direction

- Shearing Stress

[ IT . Matrix Transpose

[a] - Displacement Transformation Matrix

[C] - Matrix of Elastic Coefficients

_ - Displacement Transformation Matrix - Harmonic n

_] = Matrix of Element Corner Forces

[u] = Matrix of Element Corner Displacements

[k] = Element Stiffness Matrix

r_l = 1_nr1.1Pn_+. Di.placementsL_ ........

i

[R]- - Nodal Point Loads

_K_ = Stiffness Matrix for Complete Structure

Report No. 5654-02 FS Page v

INTRODUCTION

The purpose of this investigation is the development of methods

of analysis and digital computer programs to aid in establishing the

structural integrity of manned spacecraft _heat shields. The results of

the analysis which are presented in this report indicate only the cap-

abilities of the computer programs and do not necessarily represent the

behavior of a specific heat shield. The final evaluation of the structural

capability of a heat shield must be based on a certain amount of engineer-

ing Judgement, in connection with the use of the computer programs.

In this investigation the finite element method is used to de-

termine stresses and displacements developed in solids of revolution.

First, a numerical procedure and a digital computer program are develop-

ed for the analysis of composite axisymmetric solids of arbitrary

geometry subjected to axisymmetric thermal or mechanical loads. Second,

this program is specialized to the analysis of axisymmetric heat shields.

Finally, the same numerical technique is extended to the analysis of

heat shields subjected to non-axlsymmetric thermal loading. A descriptioni

of the method of analysis and the use of the above computer programs is

presented. In addition, FCRTRAN II listings of the above programs are

incorporated in this report.

Report No. 5654m02 FS Page i

During the initial phases of this contract, finite difference

techniques were used to solve the governing differential equations for

displacements of the system. However, considerable difficulty was en-

countered in the solution of the resulting set of linear equations. An

iterative approach, coupled with over-relaxation techniques, resulted in

inadequately convergent displacements. The direct solution technique

gave a matrix for the set of simultaneous equations which was ill-

conditioned. An additional difficulty of the finite difference technique

was encountered in satisfying the boundary conditions at the edge of the

heat shield. The finite element approach proved more practical and more

versatile; therefore, the finite difference method was discontinued. A

complete description of this initial investigation is given in Appendix

G.

Report No. 5654-02 FS Page 2

PART I_ METHOD OF ANALYSIS

A. INTRODUCTION

The "finite element method" is a general methQd of structural

analysis in which a continuous structure is replaced by a finite number

of elements interconnected at a finite number of nodal points -- (such an

idealization is inherent in the conventional analysis of frsmes and trusses).

In this investigation the finite element method is applied to the determina-

tion of stresses and displacements developed in axisymmetric elastic

structures of arbitrary geametry and material properties which are sub-

Jected to thermal and mechanical loads.

An assemblage of different types of axisymmetric elements is used

to represent the continuous structure. Approximations are made on the

displaceme_s within each element of the system. Based on these ap-

praximations, equilibrium equations are developed for all elements.

From "direct stiffness techniquesU, the equilibrium equations p in terms

of unknown nodal point displacements, are developed at each nodal point.

A solution of this set of equations constitutes a solution to the finite

element system.

B. _UILIBRIUM EQUATIONS FOR AN ARBITRARY FINITE EL_4ENT

I. Strain-Displacement Relationship

The first step in the determination of the stiffness (corner

forces in terms of corner displacements and temperature changes), of a


finite element is to assumea solution for the displacement field within

the element. It is desirable that this assumed displacement field

satisfies compatibility between other elements in the system. Based on

this solution for the displacements within the eleaent, it is possible

to develop an expression for the strains at any point within the element

in terms of the nodal points (corner) displacements. This expression in

matrix form is

[,] - [_3[-]

where [,][-][-]

(i.i)

i_ a column matrix of the M components of strain

is a column matrix of the N nodal point displacements

is an M x N strain-displacement transformation matrix -

this matrix may be a function of space

Stress-Strain Relationshi p

For an elastic material, the stresses at any point within

the element are expressed in terms of the corresponding strains by the

elastic stress-strain relationship. Or, in matrix form

[_]-[_] [.] ÷[,] c_.,_


where _] s a colmnnmatrix of the M components of stress

c] is a column matrix of the M components of strain

_] s a column matrix of the M components of thermal stress

[C] is an M_M matrix of material property coefficients

The size (M) of these matrices will depend on the type of element

being considered. The coefficients of matrices [0] and IT] will depend

[o]on material properties. Since is completely arbitrary 9 anisotropic

materials can be handled. Also_ each element in the system may have

different properties; therefore_ ccmposite structures are readily represent-

ed by the finite element idealization.

3. Internal Work

The internal work, or strain energy, which is associated with

an infinitesimal volume element dV within the finite element is Eiven by

IdNI " _ (el_l + ¢2 _2 "'"" + eM _M ) dV

or in matrix form

T

dNI = ,_[e] [o'] dV

The substitution of Equation (1.2) into Equation (1.3) yields

(1.3)


Equation (I.I) may be written in transposed form as

[]' [u]- (1.5)

After Equations

internal work is given by

dWl= _ a C dV + I [u [I" dV2

(I.I) and (1.5) are substituted into Equation (1.4), the

(1.6)

The total strain energy stored within the element is found by

integrating Equation _1.6) over the Volume of the finite element. Or

wi2

@ E_-terrml WorkI I,

The work supplied externally at the nodal points of the

finite element is given by

w_ = _ uI sI + u2 s2 .....,.....+ v_

or in matrix form

w_-_ s

where[u_T

(1.8)

is a row matrix of the N nodal point displacements

is a Column matrix of the N corresponding nodal point forces


5. Energy Balance

The external work, Equation (1.8), is equated to the internal

work, Equation (1.7), yielding

(1.9)

where the element stiffness matrix

dV (1.zo)

and the .thermal load matrix

Equation (1.9).represents an energy balance (scalar equation) for a single

nodal point displacement pattern. If the final displacements [U_ are

assumed to be composed of N separate displacement patte_-ns_[_ij ] J - 1,...N,

[]ing sets of f=ces, B_ S " l, .... _, Equation(!.9) m_ be .ritten as

[<[,]. [<[,] [o]'[,.]T m

To eliminate the term [_] the displacement patterns must be

Teselected in such a manner as to assure an inverse of [_] An acceptable

matrix is a diagonal matrix of the'final displacement, or

r-_ T r.q r,,___qLuJ " L=J " L _J (!;l j)


Equation (1.12) is now premultiplied by _J" I yielding

r "-i

where [lJ is a diagonal unit matrix.

Since only linear systems are consideredj the N displacement patterns

may be superimposed. Or

Is] - [_][_]• [,.] (_._

Since

J'l,...N

Equation (i.15) expresses nodal point forces in terms of nodal point

displacements and temperature changes Within the element.

C@ _U._ BQUATIONS FOR A S¥ST_ OF I_-NITE _L_TS

The first s_ep in the procedure is to express all el_nent forces in

terms of sxternal nodal point displacements far each element in the

system. This is accomplished by expanding Equation (I.14) in terms of

the N possible nodal point displacements; this will yield H matrix equations

of the form

[_] . [_][,]+ [_] , +_,....where M is the total number of elements in %he system.

(1.16)


The matrix [km] is termed the complete stiffness of element m and involves

only terms which are associated with the displacements of the connecting

nodal points. Consequently, the majority of the coefficients of this

matrix equation are zero. The matrix [r] contains all possible nodal point

displacements of the complete finite element system. The matrix ISm] is a

column matrix containing the forces acting on element m in the direction

of the nodal point displacements [r]. The thermal load matrix [Lm] and the

element stiffness matrix [km_ are given by Equations (I.Ii) and (I.IO);

however, the order (size) of these matrices has now been expanded to

correspond with the total number Of nodal point displacements.

In order to satisfy equilibrium of all nodal points, the sum of the

internal element fOrCes must be equal to the external nodal point loads,

Or

m=Ij.. _M

of Equation (1.16) into Equation (1.17)yields

[,]. .[e][,]. Zm=lj., _M re=I,...M

is the eKternal_y applied nodal point loads. The substitution

(1.18)

Report No. 5654 o2 FS Page 9

or rewritten in the following form:

M "[_I__] _ll_where

[.]-[P]- _ [,."]m=lj,, ,M

(1.20)

m-i 9,.,M

Equation (i.19)_ which is an equilibrium relationship between external

loads and internal foroes_ represents a syste_ of N linear equations in

terms of N unknown displacements.

D. SOLUTION OF _UILIBRIUM _QUATIONS

Equation (1.19) represents the relationship between all nodal point

forces and all nodal point displacements. Mixed boundary conditions are

considered by rewriting Equation (1.19) in the following partitioned form:

m !

I

_a I_

(1.22)

Report No. 5654-02 FS Page i0

where E_2

Ira]

= the specified nodal point forces

= the unknown nodal point forces

= the unknown nodal point displacements

= the specified nodal point displacements

Equation (1.22) may be expressed in terms of two separate equations, or

[,_]. [_,][_j • [_] [_] c_Equation (1.23) is rewritten in the following reduced form:

(l.25)

where the modified load vector, JR2 is given by

In Part II of this report_ the Gauss-Seidel iterative technique

is used to solve _uation (1.25) for the unknown nodal point displacements

Ira]. direct solution which is used in theAppendix A gives approach&

automated computer programs for the thermal stress analysis of heat shields,

Parts I_Z and IV of this report.


E. EL_ENT STRESSES

After the nodal point displacements have been determined, the strains

within an_ elsment in the system are evaluated by the direct application

of Equation (i.I). The corresponding stresses are calculated from the

stress-strain relationship, Equation (1.2).


PART II GENERAL C(]MPUT_R PROGRAM FOR THE ANALYSIS

OF .l_trrtt_Y AXISgNN_IC STRUCTURESi ii ii

A. INTRODUCTION

The stress analysis of an axisymmetric structure of arbitrary shape,

subjected to thermal and mechanical loads is of considerable practical

interest. Although the governing differential equations have been known

for many years, closed form solutions have been obtained for only a

limited number of structures. Thus, the investigator must often rely on

experimental or nt_aerical procedures to solve this probl_,.

Experimental methods, such as Photoelasticity, have proven to be

versatile too_s in t_e analysis of many axis_etric structures. However,

for structures composed of several different materials or structures with

thermal leading, this approach is limitedo

The finite _difference method, Which involves the replacement of

the derivatives in the differential equations and boundary conditions with

difference equations _ has been the most popular of the numerical techniques.

However, for structures of composite materials and of arbitrary geometryj

the procedure is difficult to app_v.

In this section the finite element method is used to determine the

stresses and displacements developed within arbitraryj elastic solids of

revolution subjected to thermal or mechanlcal axisy_aetric loads. The


GeACTUAL RING

I II I

b. TRIANGULAR ELEMENT APPROXIMATION

FIG. 2.1 THE FINITE ELEMENT IDEALIZATION

Report No, 5654,-02 FS Page 14

finite element approach replaces the continuous structure with a system

of triangular rings interconnected at a finite number of nodal points

(Joints). Loads acting on the structure are replaced by statically equi-

valent concentrated forces acting at the nodal points of the finite

element system. Figure 2.1 illustrates a finite element idealization of a

typical axisymmetric solid.

B. STreSS OF

i@ Strain-Displacement Relationship

Cont_uity between elements of the system is maintained by

requiring that within each element relines initially straight remain

straight in their displaced position". This linear displacement field,

which is illustrated in Figure 2.2, is defined in terms of u(r,z) and

L 'FID. 2.2 ASSUMED DISPLAC_ENT PATTerN

Report No. 5654_02 FS Page 15

w(r,z) by equations of the following form:

u(r_z) = CI + C2 r + C3 z (2.1a)

_r_z) - C4 + C5 r + C6 Z (2.1b)

If Equations (2.1a) and (2.1b) are evaluated at the three corners i, J, k

of the triangle_ the following see of equations is obtaineds

ui

U

i

0

ri zi 0 0

0 0 I

I rj zj 0

0 o o 1

z rk "k o

o 0 o I

b

ri zi

o o

rj zj

0 0

C1

C 2

C3

C_

C5

C6

(2.2)

By solving the system of Equations (2.2) for the constants Ci,....,C6,

they are expressed in terms of corner displacements. The strains i_ the

rz-plane are obtained from the assumed displacement field by considering

the basic definition of strain.


_U

;r " _ = C2 (2.3a)

= _w = C6 (2.3b)

_rz = _'z_u+ _w = C3 + C5(2.3¢)

At any point within the element the tangential strain

=e-(r'=) . u(r,z)r

The average tangential strain is found by averaging the strains at the

Vertices of the triangle,

(2.3d)

After el_i_a_ing the constants c between Zquations (2.2) and (2.3),n

the average eleme_ strains are expressed in terms of corner displacements

by the following matrix equation:

Report No. 565_-02 FS Page 17

#

¢r

cz

cg

_rz

II i

bj% o bk o %

0 ak-a j 0 -ak 0

X 0 _ 0

0

aj

0

ak-a j bj_o k -ak _ aj -bj

ui

w i

uj

wJI

or in symbolic form

[.] - [,,lE-Iwhere

aj = rj - ri

ak = rk-r i

bj = zj - zi

x = aj_-_bj

The geometry of a typical triangle is illustrated in Figure 2.3

Report No. 5654=02 FS Page 18

k

rk

rj J

ri • i

TFD_. 2°3 _ENT DIM_SI_

2, Stress-Strain Relationshipm.

One important advantage of the finite element approach is

that structures with anisotropic materials can be treated. In generalj

the stress-Strain relationship is of the form

%

_Z

_9

_rs

cn c_ cI_ c_ "r

C21 C22 C23 C24 Cz

c31 c32 c33 cj4 eo

c41 c_ c_ c_4 Vrz

_Z

_9

_rz. •

(2.5a)


P _

.h_e L'J is the_tr_ of thez_al stresses for a given temperature C_eo

For example, the stress-strain relationship for an isotropic material is

given by

_r

z

! ee

E

(1÷_,)0.-2v)

-1-,_

1)

0

v 0

l-v u 0

l-v 0

o o I-2_2

F ¢ - Tr

Z

+

ce

7rz _

(2.6)

'1"

where 7 = AT (2.7 )

3. Element Stiffness

i

The stiffness of a typical triangular ring, which is an

expressioh for corner forces in terms of corner displacements_ is given

by Eql. (I_I0) as

Ek]-I [a]T[°][a]dV

And the thermal load matrix is given by Eq. (i.ii) as

EL_ = _I Ea_ T E_ dV

Since the coefficients in matrices Ca_ and ECI are assumed not to be

a function of space the above equations reduce to

Report No. 5654--02 FS Page 20

(2.8)

(2.9)

If a one-radian segment is considered, an approximate expression for

the volume of a triangular ring segment is

V=_A

m

where r is the average radius given by

r = (r i + rj * rk)/3

and A is the cross-sectional area given by

A - (ajbk - akbj)/2

From Eq. ii,15) the six corner forces acting at the vertices of a

one-radian triangular segment is given in terms of the six corner

displacenents and the temperature change within the element by

[.].

(2 .ii)

(2.12)

(2.13)

C. EQUIW,IBRIUM EQUATIONS FOR COMPLETE STRUCTURE

The equilibrium of the complete system of triangular rings, which

is aft expression for nodal point loads in terms of nodal point

displacements, is given by the following matrix equation:

.[R] .= [K] [r]

Report No. 565_-o2 FS Page 21

F7The stiffness matrix LK_ and the load matrix LR ] are determined

by

"direct stiffness" techniques as indicated in the previous section,

(1.20) and (1.21). In addition to the thermsl loads, the [R]Eqs.

matrix is composed of concentrated external forces acting at the nodal

points of the system. Hence, pressures acting on the boundary of a

segment of the structure are replaced by statically equivalent forces

acting at the nodal points.

Mixed boundary conditions are considered by a simple transformation

of Eq. (2.14); Eqs. (1.22) to (1.26) give the details of this

modification.

D@ DETE_IINATION OF DISPLAC_NT AND STRESSES

Equation (2.14) is solved for the unknown nodal point displacements

by the applicati_ of the well-known Gauss-Seidel iterative procedure.

This involves the repeated calculation of new displacements from the

equation

where n

.." Kniri "_ ani i

i=l, •.n-I i=n+l, •.n

(2.15)

is the number of the unknown and s is the cycle of iteration.

The,only modification of the procedure introduced in this analysis

is the simultaneous application of Equation (2.15) to both components


of displacements at each nodal point.

vectors with r and z components.

Therefore, rn and Rn become

The rate of convergence of the Gauss-Seidel procedure can be

greatly increased by the use of an over-relaxation factor. This factor

.'(s)is appliedby first calculating the change in displacement _rn of

nodal point n and then determining the new displacement from the

following equation-

r(S) _(s-l) SAt(s)n m _n +

where B is the over-relaxation factor.

(2.16}

The solution of an over-relaxation factor, which gives the best

convergence, depends on the characteristics of the particular problem.

However, experience b_s indicated that for most structures, the op%iimm

ever.relaxation factor is between 1.8 and 1.95.

Since only the non-zero terms in Equation (2.14) are developed and

stored by the computer program, a solution of several hundred eqna%Iona

is possible, thereby making it possible to solve large finite element

systems.

For each element the average strains are calculated directly from'

the nodal _oint displacements by the application of Equation (204). The


average element stresses are then determined from the stress-strain

relationship for the element, Equation (2.5). In addition, at each

nodal point, stresses are computedby averaging the stresses in all

elements attached to the point.

E. COMPUTERPROGRAM

The complete analysis of an axisymmetric solid by the finite element

method involves three separate phases, First, the structure must be

idealimed by a system of triangular rings. Second, this system iS sOlved

for displacements and stresses from given nodal point forces. Third, the

displacements and stresses are presented graphically for further evaluation

and utilization.

The Selection of the system of finite elements for a particular

problem is complet@ly arbitrary; therefore, axisymmetric structures,

composed _f many _terac_g components, of practically any shape may

be han_ed, By nu_ering all elements and nodal points, in a convenient

manner, the system can be defined in the form of three numerical arrays -

nodal point arrayj element array and boundary point array. The nodal

point array contains the coordinates and the loads or displacements that

are associated with each nodal point of the system. The element array

contains, for each element in the system, the location of the element

(the ÷.hre_ nodal noint numbers defining the corners of the element and

other possible parameters which are associated with the element (i.e.j

Report NO. 5654-02 FS Page 24

elastic constants, density and temperature changes). The boundary array

indicates the type of restraint thst exists at boundary nodal points.

These three arrays, along with somebasic control information,

constitute the numerical input for the digital computer program. The

program itself performs three major tasks in the analysis of the finite

el_nt System of triangular rings. First, the equilibrium equations for

th_ System are formedfrom the basic numerical description of the system.s

Second_ this set Qf equations is solved for the nodal point displacements.

Third_ the internal stresses are determined from these displacements.

I. Input Information

To define the system of finite elements, all nodal points

and elements are numbered as illustrated in Figure 2.4. Based on this

n_be#tng System, the following sequenc e of punched cards constitutes

the input to the computer program.

a. (72H)

Columns 2 to 72 of this card contain information to

be printed with results

CONTm CARD (614,2 12.5)

Columns 1 - 4 Number of elements

5- 8 Number of nodal _points

9 - 12 Number of restrained boundary _oints

13 16 Cycle interval ...... _-_ ^_ ""_'°_o'_,'o,_forces

Report No. 565h-O2 _ Page 25

Z

R- ORDI NATE

N

R_-LOAD

Z- LOAD

NODAL POINT NUMBER

2I

ELEMENT NUMBER

FIG. 2.4 NUMBERING SYSTEMAND NODAL POINTS

FOR ELEMENTS


C@

d@

17-20 Cycle interval for print of results

21-24 Maximum number of c/Ices problem may

run

25-36 Convergence limit for unbalanced forces

37-48 Over-relaxation factor

ELEMENT ARRAY - i card per element (414, 4E12.4,

F8.4)

Colu.m 1-4

5-8

9-12

13-16

17-28

29-40

-Sz

53-64

65-72

Element Number

Nodal point number i

Nodal point number J _ in counter-J

clockwise orderNodal point number k

Modulus Of elasticity E

Density of element 0

Poisson 's. ratio

Coefficient of thermal expansion

Temperature change within element. AT

NOmL POINT ARRAY - 1 card per nodal point (114,::'

Column 1-4

5-12

13-20

21-38

29-36

37-48

4F8.1,2m.2.8) -

Nodal point number

R-ordinate

Z-ordinate

R-loadTotal force acting

Z-load _ radian segment.

R-displacement

on a one

49-60 Z-displa cement

On free nodal points, the displacements are initial guesses for the

iterative solution. On restrained nodal points, the input displacements

at6 %,_ 3p_'ified fgDa_ displacements of the nodal point.


e. BOUNDARYPOINTARRAY- I card per restrained nodal

point (214,

Columns 1-4 Nodal point numbers

5-8 0 if point is fixed in both directions

1 if point is fixed in the R-direction

2 if point is free to move along a line

of slope S

9-16 Slope S (type 2 points only)

output o ation

The following information is generated and printed by th_

2.

co_q_t er program:

a.

b'

Ced.

Input DataNOdal Point Di-splacement

Average Element StressesAverage Nodal Point stresses

3 • Timing

For the IBM 7094 the computational time required by the

program is approximately 0.004 x n x m seconds, where n equals the

number of nodal points and m equals the number of cycles of iteration.

Depending on the desired degree of convergence, it may be necessary to

extend the iteration process.

4. Pro a m List ,

A card listing of the FORTRAN II source deck for the general

axisy_ric program!s included _ Appendix D of this rApnrt. This

:program is compiled for a maximum size of 550 elements or 340 nodal points.


F, EXAHPLE

An infinite cylinder subjected to steady state temperature

distribution, for which an exact solution is known, is selected as a

means of verifying the finite element analysis. A finite element

idealization of a section of the cylinder is shown in Figure 2.5a. The

temperature distribution which is assumed constant within each element,

is plotted in Figure 2,5b. The hoop stresses are compared with the exact

solution in Figure 2.5c. Considering the coarse mesh, agreement with the

exact solution is very good except at the two boundary points. This

discrepancy is due to _he fact that nodal point stresses are calculated

by averaging the stresses in the attached elements. Therefore, the

boundary nodal point stress reflects the average stress in the elements

near the boundary.

In general, good boundary stresses are obtained by plotting the

interior stresses and extrapolating to the boundary. This type of

engineering judgement is always necessary in evaluating results from a

finite element analysis.

G. DISCUSSION

This section demonstrates the application of the finite element

technique to the stress analysis of structures of revolution. The approach

reduces the stress anaiysi_ to o- __._-_-_pro__-_1_A..... In order to use the

Report No. 565_-02 YS Page 29

6 7 8 9 I0 II 12* ' ' RADIUS i

a. FINITE ELEMENT IDEALIZATION- INFINITE CYLINDER

i- =°"

I-"

Radius

b. TEMPERATURE DISTRIBUTION

----- EXACT SOLUTION

Rodiu'-s0"r'

c. HOOP STRESSE_,

FIG. 2.5 ANALYSIS OF INFINITE CYLINDER


II

II

program, it is only necessary to select an element idealization of the

structure and to supply the computer program with data that numerically

define the system of elements. Therefore, the program may be used as a

tool in design since changes in materials and geometry of the structures

may involve only minor changes in the input data.

In addition, the program may be extended to include the effects

of anisotroplc materials. For this casej the input to the program must

be expanded to include the general elastic constants defined by Eq. (2.5).

For the analysis of a specific type of structure, this program can

be further automated by incorporating a mesh generator and the calcuiation

of temperature-dependent material properties, In the next section of

this report the program is specialized to the thermal stress analysis

of axisymmetric heat shields for manned spacecraft.

Report No. 5654r02 FS Page 31

PART III AUTOMATED PROGRAM FOR AXISYMMETRIC HEAT

SHIELDS SUBJECTED TO AXlSYMMETRIC LOADS

A. INTRODUCTION

The general computer program for the analysis of arbitrary

axisymmetric structure, as given in the previous section, can be applied

to the thermal stress analysis of heat shields. However, the use of this

program for such a complex structure involves a large amount of detail

work to select the finite element idealization and to prepare the computer

input. In addition, the convergence of the Gauss-Seidel iteration procedure

is slow for this type of structure and a solution may require an excessive

amount of computer time.

By restricting the general computer program to the analysis of heat

shields and by automating the input, a considerably more efficient program

can be developed. The geometry of the heat shield is supplied to the

computer program in the form of R and Z coordinates and ablator thickness

at various points along the bond line. The required triangular mesh and

the temperature at the grid points are generated within the program.

Material properties at various temperatures are supplied in tabular form

and the program automatically develops analytical expressions for the

material properties by least square techniques. The flanges of the sand-

wich shell are idealized by special conical shell elements and the honey-

comb core material is treated as a separate material. This approach


eliminates the need for the establishing of a pseudo-thickness for the

composite sandwich shell. The solution of the equilibrium equations,

which was previously obtained by an iterative approach is accomplished

by a direct solution procedure. Because of their significance, stresses

within the sandwich plates and at the bond line are included in the

computer outputs

B. MESH G_ERATION

A typical finite element idealization of the cross-section of a

heat shield is shown in Figure 3.1. The basic element in this system is

a quadrilateral ring, which in turn is composed of two triangular rings

(Part II). In this particular case, the sandwich shell is represented by

the first two rows of elements and the ablator is idealized by four rows

of elements; there are 30points in themeridional direction. The specific

mesh configuration is a variable which is supplied to the computer program.

In general, the geometry of the shell is given by the R-Z coordinates

of the points at the bond line between the sandwich shell and the ablator.

The points on lines perpendicular to the bond line inside the variable

thickness ablator and inside the constant thickness sandwich shell are

generated automatically within the program. Thin shellcone elements are

used to represent the face plates of the sandwich shell.

From a given temperature distribution at the bond line, the grid point

temperatures are assumed td be constant within the shell and are assumed

to vary parabolicallywithinthe ablator.


Report No. 5654.-02FS Page 34

II

I

|,

C. DETERMINATION OF DISPLACEMENTS AND STRESSES

Based on temperature dependent material properties, the equilibrium

relationship for each quadrilateral ring is developed and then combined to

form the equilibrium equations of the complete system of rings. Similarly

the stiffness properties of the face plates of the sandwich shell are

incorporated into the equilibrium of the system. The axisymmetric

behavior of a typical conical element is a special case of the non-

axisymmetric behavior which is given in Appendix C. The unknowns in

this set of equations are the vertical and radial displacements at each

grid point in the system. The satisfying of possible displacement boundary

conditions requires that these equations be modified as indicated by

Equation (1.25). Because of the physical characteristics of the heat

shield, the resulting set of equations is in band form. Appendix A

indicates the necessary modification to restrict the standard Gaussian

elimination procedure to the solution of symmetrical band systems. This

approach results in a definite increase in capacity and speed over the

iterative technique and it eliminates the problem of convergence.

After the equilibrium equations are solved for the unknown grid

point displacements, the average stresses within each triangular ring

are calculated as indicated in Part II of this report. Based on the

stresses in the two triangular rings, average quadrilateral stresses are

calculated for each quadrilateral ring in the system.


D. COMPI_F_PROGRAM

The first step in the stress analysis of an axisymmetric heat

shield is to select points at the bond line at regular intervals along

the meridian of the shield. A quadrilateral mesh is automatically

developed by the program from the R and Z coordinates. The material

properties vs temperature for the ablator and bond are supplied to

the computer in tabular form and the computer program automatically

determines analytical expressions for the properties by a least square

procedure (see Appendix B for details of method). Finally, the grid

points which are to be restrained and the external loads which act at

grid points are specified.

I. Input Information

The following sequence of punched cards numerically defines

the heat shield to be analyzed.

a.

b@

FIRST CARD - (72H)

Columns i to 72 of this card contains information

to be printed with results

SECOND CARD - (615, 2F10.2)

Columns I - 5 Number of points along the shield - NMAX

6 - iO Number of points thru the thickness-MMAX

I_ _ !_ T.n,,'.m+.Innn_"h_nd llnA - MB

16 - 20 Number of material property cards - NP


I

I

r

b

r

L

I

I

I

I

I

CQ

dt

ea

21 - 25 Number of points with radial and

axial loads - NL

26 - 30 Number of additional boundary

conditions - NB

31 - 40 Surface temperature of ablator

41 - 50 Zero stress temperature

THIRD CARD - Properties of Sandwich Core (4FI0_2)

Oolumns i - iO Modulus of elasticity

11 - 20 Poisson's ratio

21 - 30 Coefficient of thermal expansion

31 - 40 Thickness of core

FOI_TH CARD - Properties of Sandwich Face Plates

(_12.2)

Coltmms i - i0 Modulus of elasticity

Ii - 20 Poisson's ratio


31 - 40 Thickness of single face plate

GEOMETRY CARDS - (hFlO,2)

One card per point along shield, in order from axis

of symmetry to edge (NMAX cards).

Columns I - IO R-ordinate at bond line

Ii - 20 Z-ordinate at bond line

21 - 30 Temperature at bond line

31 - 40 Normal thickness of ablator


f. MATERIALPROPERTYCARDS - (4FIO. 2)

One card for each temperature (NP cards)

Columns i - IO Temperature

Ii _ 20 Modulus of elasticity of ablativemateris i

21 - 30 Modulus of elasticity of bond material

31 - 410 Coefficient of thermal expansion forablator and bond materials

g. LOAD CARDS - (2]5, 2FI0.2)

One card for each point which is loaded externally

(NL cards). N and M specify the grid location of the point.

Columns I - 5 N(Meridional direction)

6 - IO M (Thickness direction)

II - 20 Radial Load,

21 30 Axial LoadTotal load acting on

one radian segment

h. BOUNDARY OONDITION CARDS - (315)

One card for each point which is restrained(NB cards).

N and M specify the grid location of the point.

Columns 1 - 5 N (Meridional direction)

6 _ i0 M (Thickness direction)

II - 15 Boundary Code

Code = I point fixed in R-direction

Code = 2 point fixed in Z-direction

Code = 3 point fixed in both the Rand Z-directions

Report No. 5654.-02 FS Page 38

@ Output Information

The following information is generated and printed by the

computer program:

a@

b.

C@

d.

ee

f.

g.

Input data

Least squares evaluation of the temperature dependent

material property data

Coordinates and temperatures of all grid points

R and Z displacement at all grid points

Average stresses in quadrilateral rings

Stresses in sandwich face plates

Stresses in bond layer

3. T ing

The computer time required by this program for an axi-

symmetric analysis of a heat shield is approximately

time --A + B'(NMAX)'(MMAX) 2 (seconds)

The constants A and B depend on the specific computer system

which is employed. For the IBM 7094 A=20 and B=O.02, and the time required

for a 30 x 7 mesh is 50 seconds.

4. Program Listing

A card listing of the FORTRAN II source deck for the

automated computer program for the sxisymmetric stress analysis of heat

shields is Riven in Appendix E. The program is compiled for a maximum


grid size of 40 points in the meridional direction and i0 points through

the thickness. Material properties can be specified by a maximum of

50 cards.

E. EXAMPLES

Several axisymmetric analyses of a heat shield were conducted to

evaluate the significance of the various structural parameters. A

typical finite element idealization of the heat shield is shown in

Figure 3.1.

I. Effect of Mesh Size

The first example was selected to illustrate the effect

of mesh size on the accuracy of the displacements and stresses developed

within the heat shield. For a structure fixed at the edge, typical re-

sults of two analyses with different meshes are shown in Figure 3.2. This

example illustrates that two layers of elements in the sandwich shell are

adequate for the purposes of predicting stresses. It is of interest to

note that the stress distribution varies linearly within the sandwich

shell, thereby confirming the assumption made in thin shell theory. The

displacements for these two analyses differed by less than one percent.

2. Effect of Ablator Thickness on Stress Distribution

Figure 3.3 shows typical results of three analyses of heat

shields with different ablator thicknesses. In general, the magnitude of


I - 200

I-150

emm

Q.

I

elq)I,..

4=-

(t)

- I00

Q.00

-50

0

• Four-Layers in ShellFour Layers in Ablotor

x Two Layers in Shell

Six Layers in Ablotor

m

Sandwich Shellr

f Bond

< Ablator _ [

Line

Fig.3.2 Effect of Mesh Size on Stress

Report No, 5654-02 FS

Distribution

Page

I -200

I i=,i'_II o_

i To-5o

O Normal Thickness

OnS,-Half Normal Thickness

m One-Fourth Normal Thickness

m

_s

Sandwich Shell _! : Ab,lator_,

Fig. 3.3

Report No.

Effect of Thickness of Ablator on Stress Distribution

5654-02 FS Page 42

the nmximum stresses in the ablator were in good agreement. This example

illustrates that the thickness of the ablator is not an important

structural parameter at the temperature of re-entry.

3. Effect of Boundary Conditions on the Behavior of the

Heat Shield

The support condition which is imposed on the heat shield

is an extremely important parameter. Figure 3.4 illustrates the

deflected position of the bond line for two different support conditions.

The resulting stresses differ significantly. Therefore, it is important

that the boundary condition which is imposed on the finite element

system be a realistic approximation of the physical support condition

which exists in the actual heat shield.

F. DISCUSSION

The automated computer program presented in this section reduces

the analysis of an arbitrary heat shield subjected to axisymmetric

thermal or mechanical loads to a simple procedure. The program auto-

matically generates the finite element grid, evaluates temperature-

dependent material properties, solves the equilibrium equations for

the grid point displacements and calculates stresses within elements,

sandwich shell face plates and at the bond layer. Arbitrary1_oundary

conditions can be imposed since any oi _he grid pointu _._aybe restrc_ncd

in either the R or Z direct ions_

Report No. 5654_O2 FS Page 43

//

////_

|

m

._

@u

0

ID

I1,,,,,0

UIP

I1,1

°_


PART IV: AUTOMATED PROGRAM FOR AXISYMMETRIC HEAT SHIELDS

SUBJECTED TO NON-AXlSYMMETRIC LOADS

A. INTRODUCTION

In general, the heat shield of a manned spacecraft is composed of a

constant thickness sandwich shell and an ablator which varies in thickness

in both the meridional and circumferential directions. The temperature

distribution experienced by the heat shield is also non-axisymmetric.

Because the ablator, at high temperatures, is not a major structural

element, contributing to the overall behavior of the heat shield, an

approximation of its properties in the circumferential direction is

justified. The approximation, that it is axisymmetric, reduces the stress

analysis of a non-axisymmetric heat shield to the stress analysis of an

axisymmetric structure subjected to non-axisymmetric thermal loads. This

involves the expansion of the temperature distribution and the final

displacements of the system in Fourier series. By making use of the

orthogonality properties of the harmonic functions the three-dimensional

analysis is divided intoa series of uncoupled two-dimensional analyses.

B. THEORY FOR THE ANALYSIS OF ANAXISYMMETRIC BODY SUBJECTED

TO NON_XISD_TRIC LOADS

£ theory is presented for the analysis of solids of revolution

subjected to non-axisymmeSrlc loads which arc s_-_-_etric,bout a plane


containing the axis of revolution. Figure 4.1, a view of a plane per-

pendicular to the axis of revolution, shows the trace of the plane of

symmetry. Anisotropic material properties, which are constant along

any circumferential line, are included in this formulation.

The structure is idealized as a series of rings with triangular

cross-sections; the rings are interconnected at their nodal circles,

i.e., at thecircles containing the vertices of the triangles,

(Figure 4.2). Loads acting on the structure are replaced by statically

equivalent concentrated forces acting along the nodal circles.

I. Strain-Displacement Relationship

By noting the ax_sym_ofA _-_.__._-'---.,,,'_,_'_,_'.._ __.__+_.... _a_al

properties of the body and the plane of symmetry for deformations, the

displacements in r, 0, z coordinates may be written in the following form:

Ur =ZUrn(r,z) cos n@ (4.1a)

ue =_Uzn(r,z) sin ne (4.1b)

uz =_Ugn(r,z) cos n9 (4.1o)

Within each ring element the r and z variation of the Fourier

coefficients of the displacements are assumed to be linear, i.e.,

Report No_ 5654-02 FS Page 46

S Troce of Plone ofDeformotion Symmetry

Z

Solid of Revolution

Fig. 4.1 Cylindrical CoordinateEmbedded in a Solid of

System

Revolution

1Repot(, No. 5654-02 FS Page 47

Z

rk , Zk)

(ritZl} " Ring Element

Nodal Circle

Fig. 4.2 Cross Section of o Ring Element


Urn _ kln + k2n r + k3nZ

Uen ._k4n * k5n r + k6nZ

Uzn _ k7n + k8n r * k9nZ

(4,2a)

(4.2b)

(4.2c)

Now expressing the constants k in terms of the corner values

of the Fourier coefficients of the displacements, i.e., in terms of

Uirn, i i uj UJn j k k kUSn' Uzn' rn' Uzn , Urn J Ugn and Uzn

%n

kDn k5n kSn

k3n k6n kgn

Urn ue n Uzn

uj _ ujrn n zn

uk k k irn U@n Uzn

(4.3a)

with

rjz k - zjrk rkz i - riz k riz j - rjz i

I Zj - zk zk - zi zi - zj

rk - rj ri - rk rj - ri

(4.3b)

and D = rj(zk - zi) + ri(z j - Zk) + rk(z i - zj) (4.3c)

Combining Fquation (h.l) with the strain-displacement

relationships, the following expressions for the strains are found:

bu¢ = r )__2-= __ m cos ne (4.4a)

Report No 5654-02 FS Page 49

'e =_ e_-+-r=r Cen cos ne (4.4b)

'z = _ = ¢zn cos rm' (4.4c)

_Ur _% _7re = ( _T" + r_- " ) = 7ren sin n8 (4.4d)

au.. aur7rz = (r_ ÷ _T ) =Z >rzn cos ne (4.4e)

_ue _uzV0Z = (_- + I _ Veznr BT ) = sin nO (_._)

_ere

_U

= _- (4.5a)crn _r

1Con = _ (nUen + Ur) (4.5b)

_Uzn _' = -- (4.5c)zn _z

_Uen Uen nurn (4.5d)

7ren= -_r r r

BUzn BUrn

Yrzn - _r * -_ (4.5e)

8USn Uzn

Yezn = _--_-- n T (_.5_)

Within a given ring the approximate values of strain for the harmonic

n are calculated by combining Equations (4.2), (4.3) and (4.5). The

Urnhoop strain is assumed to be constant within the r_ng and -- iu

r

approximated by


i j ukU U

r__n + rn +_Z_ '

3_ 3_ 3rwith 1 (ri + rj + rk)

Thus, for the harmonic n the six components of strain within the element

are given in terms of the nine corner displacements by the following

matrix equation:

The strain-displacement transformation matrix (for convenience it is

written in its transposed form)ms defined on the next page, Eq. (4.6b).

2. Stress-Strain Relationship

The stress-strain relationship for the harmonic n is

written in the following symbollic form:

[oj: [o][,j÷ ['jr (4.7a)

For an isotropic material this becomes

(rrn

%n

_rzn

or.

L uzn

B _ B o

0 0 0 I_

0 0 0 0

0

0 0

0 0

0 0

0 0

t* 0

0 0 0 0--4

Crn

eUn

gzn

CrOn

trzn

Tn

T n

T

+ n_ (4'7b)0

0

0


D

Zk-Z i

D

0

0

.

O

0

1m

1i

3_

1

3r

1

l'lm

3_

n

0

O

0

0

0

0

0

D

ri-rk

D

D

nm m

m

3r

n

3r

nN m

3r

3r

0

0

0

D

ri-r k

D

0

0

0

D

Zk-Z i

D

0

0

D

D

ri-r k

D

X1

nm

3r

nm i

3_


- (l-u) E (4.8a)where _ = (l+u)(l-2u)

u E (4.8b)

(4°8c)

E_7n= -_-j_ Tn (4.8d)

Tn is the Fourier coefficient for the expansion of the average temperature

change within the ring

T =_ Tn cos ne (4.9)

3= Equilibrium Equation for Harmonic .n

By recognizing the othogonality of the harmonic functions

the same procedure which was used in Part I may be used to develop the

equilibrium equations for an element subjected to harmonic loading.

Therefore, Equation (1.15) is rewritten as

. e.e(4.z2)_Ln] = f EGJETj dV

Within a ring the matrices _G_ andIC_ are not a function of space;

therefore, Equations (4.11) and (4.12) reduce to

Report No, 5654-02 FS Page 53

(4.13)

(4.14)

Equilibrium of the over-where the volume V is given by Equation (2_i0).

all structure requires that the sum of the nodal circle forces for all

rings with a common nodal circle must equal the applied nodal force.

This results in an equation of the following form for each harmonic:

where the displacement vector Ern_ contains all the displacement

amplitudes urn , uen and Uzn for all nodal circles in the system.

r •

The equilibrium of the face plates is incorporated by a

similar procedure. Appendix C gives the details of this development.

4. Determination of Displacements and Stresses

The number of harmonics required to represent the three-

dimensional temperature distribution indicates the number of two-

dimensional problems which must be solved. For each harmonic Equation

(4.15) must be solved for the unknown displacement amplitudes. The

corresponding strain amplitudes are calculated for each finite element

by Equation _(4.6) and then stress amplitudes are found by the application

of Eauation (4,7). The final displacements of the system for any angle

are calculated from Equation (4.1). The final stresses are determined

from the stress amplitudes by the followinff equations:


Or(r'z'e) = _" Srn cos ne

_z (r'z'e) =_ _zn cos ne

%_r,,.,e)- L %n sinn_

Crz (r'z'e)= _ _rzn cos n_

_e (r'z,e_=_ _n sinne

Cez(r,z'e)" _ _ezn sin ne

(4.16C )

(4.16d)

(4.z6t)

C. COMPUTER PROGRAM

The use of the non-axis}_mctric heat sbSe]d program _s similer to

the axisymmetric program (Part III). The only additiona]input reouired

is the three-dimensional temperature distribution. The com_mter program

automatically develops the necessary Fourier coefficients for the

temperature aistrJbution and sums the series of two-dimensional analyses

to oro_Jce the final disnlacements and stresses in the system,

i. Input Information

The following seouence of punched cards numerically defines

the heat shield to be analyzed:

a@ FIRST C_RD- (7_)

Columns I to 72 of this caz-d contaLns _-_._.,,_*_._..+_._

be pr_ntedw_.th results

Report I_o. 5654-02 FS Page 55

t

l

I

t

l

t

t

I

be

Co

d.

e@

SECOND CARD - (615, 2FIO.2)

Columns i - 5 Number of _oints along meridianof shield - NI_X

6 - i0 Number of points through thickness -MMAX

II - 15 Location of bond line - MB

16 - 20 Number of material property cards - NP

21 - 25 Number of harmonic to be used in

analysis - NL

26 - 30 Number of boundary condition cards - NB

31 - 40 Surface temperature of ablator

_I - 50 Temperature of zero stress

THIRD CARD - Properties of Sandwich Core (_FIO.2)

Col_m_ns ! - I0 Modulus of elasticity

II - 20 Poisson' s ratio


31 - _0 Thickness of core

FOURTH CARD - Properties of Sandwich Fact Plates (_w/O.2)

Columns 1 - I0 Modulus of elasticity

II - 20 Poisson's ratio


31 - _0 Thickness of single face plate

GEOMETRY CARDS - (kF!0.2)

One card per point along shield in order from axis of

symmetry to edge (NI_X cards).


Columns i - i0 R-ordinate at bond line

Ii - 20 Z-ordinate at bond line

21 - 30 Temperature at bond line

31 - 40 Normal thickness of ablator

The temperature information is used by the program

to determine the axisymmetric temperature-dependent material properties.

f. MATERIAL PROPERTY CARDS - (4FI0.2)

One card for each temperature (NP cards)

Columns i - i0 Temperature

Ii - 20 Modulus of elasticity of ablativematerial

21 - 30 Modulus of elasticity of bond material

31 - _ Coefficient of thermal e_ansion for

ablative and bond materials

g. THREE-DIM_SIONAL T_PE3AT_RE DISTRIBUTION CARDS

A table of bond line temperature values at iO degree

increments along 9 circumferential lines is punched in the following form:

Ist. card - (9F8.O)

R-ordinates of 9 circumferential points onbond line

2nd card - (9F8.0)

Z-ordinates of 9 circumferential points on bondline

3rd t9 21st card - (9F8.0)

One card for each IO degree increment (O to 180°).

Each card contains the 9 temperatures which correspond to the above R and

Z-ordinates.


h. BOUNmRYCONDITIONCARDS- (315)

Onecard per restrained nodal circle (NB cards)

Column I - 5 N}mesh point N, M6 I0 M

ii - 15 = i restrained in R-direction2 restrained in 9-direction3 restrained in Z-direction

_ restralned in R and 9-directionsrestrained in R and Z-directions6 restrained in 9 and Z-directions

7 restrained in R, 8 and Z directions

i. PRINT AN_E CARDS - (]95.0)

Column I - 5 angle 8

For earth "print angle card" a complete set of

displacement an@ stresses are printed for angle 8.

2 • Output Information

The following information is generated and printed by the

computer program:

a@

bo

C@

d.

e°

Input data

Leash squares evaluation of the temperature-dependent

material property data

Coordinates and temperature of all grid points

Two-dimensional Fourier temperature coefficients

For each print angle

(I) R, Z, and e displacement at all grid points

(2) Average stresses in quadrilatoral rings(_ _+_°==_ _n _andwich face plates


3• Timin_

The computer time required by this program for the non-

axisymmetric analysis of a heat shield is approximately

time = A + B "(NMAX)'(MMAI)2.NL (seconds)

For the IBM 7092 computer A=20 and B=O.05, and the time required for a

30 x 7 mesh with 4 harmonics is approximately 5 minutes.

4. 5 sti.

A card listing of the FORTRAN II source deck for the

computer program for the non-axisymmetric analysis of heat shields is

given in Appendix Fo The program is compiled for a maximum grid size

of 30 points in the meridional direction and 8 points through the thick-

ness. A maximum of i0 harmonics may be considered. Material properties

can be specified by a maximum of 50 cards.

Standard input tape 5 and output tape 6 are used by the

program. Tape 20 is used for temporary storage within the program; it

may be necessary to change this tape unit to conform with local computer

center policy.

D• EXAMPLES

Two analyses of axisF_,etric heat shields subjected to non-axis_etric

temperature distribution were conducted to illustrate the application of the


program. In both cases, the sxis ymmetric finite element meshwas similar

to the meshgiven by Figure 3ol. For the purpose of reference the

station layout along the bond line is shownin Figure 4.3. The three-

dimensional bond layer temperature and ablator thickness distribution

for angles e - 0, 90°, 180°are plotted in Figure 4.4.

In Analysis A the axisymmetric properties of the ablator are

assumedto be equal to the properties of the actual ablator at e - O.

In Analysis B the ablator properties at e = 180° are used. For both

analyses the surface temperature of the ablator is I000 F and the temperature

at the bond surface is given by Figure 4.5. Station 20 (Figure 4.3) is

restrained at the inside surface of the shield to simulate the effect of

an intermediate support ring.

The computer output for a non-axisymmetric analysis contains

displacements and stresses at manypoints in the heat shield; however,

only the typical results are presented. For Analysis A the deflected

shape of the bond line at three sections is plotted in Figure 4.6a. The

non-axisymmetric behavior is significant. The displacements from

Analysis B are shownin Figure 4.6b; they are essentially the sameas

those found by Analysis A. This again indicates that the ablator's thick-

ness and property variations in the circumferential direction are not of

major importance at these temperatures. Hoop stresses at station 15 are


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plotted in Figure 4.7 for two values of e. In both analyses the stresses

within the sandwich shell are in fair agreement since the material

properties do not change_t these temperatures. However, within the

ablator, where the material properties are strongly temperature-dependent,

the stresses differ significantly. Of course, the particular solution is

most reasonable if the assumed material properties correspond with those

at the _osition of the desired stress. Hence, for 6 - O, Analysis A is

considered the best approximation and similarly for e - 180 °, Analysis B

is the most reasonable.

E• DISCUSSION

In this section a method and the resulting computer program are

presented for the analysis of axisymmetric heat shields subjected to non-

axisymmetric thermal loads. The program may be used to analyze an

approximate solution to a non-axisymmetric heat shield if a number of

solutions are obtained and then are judicially evaluated. At the section

where the assumed axisymmetric ablator prope_ies (+...... +.... and

thickness) correspond to the local properties, the stresses will be a

good approximation of the actual state of stress in the non-axisymmetric

heat shield. Therefore, for each angle 0 for which stresses are desired

a separate structure must be evaluated.


I00

!

I

!

I I

! !

!sd- sseJtS do0H

I00m

I

I0


It should be pointed out that the computer program can be

extended to include non=axisymmetric pressure loading_ displacement

boundary condiditons and the effects of elastic supports° However_

this additional investigation was beyond the scope of the present effort°

F. COLD SOAK CONDITION

A necessary approximation of the method of analysis which is pre_

sented in this report is that the non_axisymmetric ablator of the heat

shield is approximated by an axisymmetric ablatoro Since the stiffness of

the ablatcr is reduced at high temperatures_ this approximation is justified

at the temperature of re-entryo However_ at low temperatures the stiffness

of the ablator, as compared to the stiffness of the sub-structure_ is signi-

fic ant °

Figure 4°8 illustrates typical stresses developed within two

different heat shields when subjected to a uniform reduction in temperature

(185°F to -250°F)o The ablator thicknesses used in the analyses correspond

to the thicknesses at sections 0° and 180 ° (Fig° 4°4)° The resulting

stresses are comparable which indicates that for an axisy_metric heat shield,

the thickness of the ablator does not affect the stress distribution signi-

ficantlyo It is also reasonable to expect that the actual behavior of the

nen-axisymmetric heat shield will be bracketed by these results.

Report Noo 5654_02 FS Page 67

,

II I000

i

J 800

II ._ _oo

i :I _ _oo

g

I o _oo

I

i o

_ o Section at 180 °

• Section at 0 °

- Su,,d.,ch ,.,,,,,,°i"n! Ab!otorIv I

Fig. 4.8 Typical Results of Cold Soak Analysis

Report No. 5654-02 FS PsFe 68

APPENDIX A

SOLUTION OF E_UILIBRIUM _UATIONS

The equilibrium equations for a System of finite elements may be

written in the following fo_mx

AlIX I +AI2 X2 +AI3 X 3 ................... ÷ AINX N • BI

A21 XI + A22 X2 + A23 X3 ................. ÷ A2NX N - B2

A3! XI ÷ A32 X2 ÷ A33 X3 ............. _.... ÷ A3NX N - B3

mm m emem _ _ mem mm _ euem a, emmmemm e_emem

ANI XI + AN2 X2 + AN3 X3 - . + ANN XN = BN

or symbolically

where [A] = the stiffness matrix

X] - the unknown displacements

[B] = the applied loads

(Ala)

(AIc)

(-)

(Al)

Report No. 565k-o2 FS Page AI

Gaussian Elimination,,,i

The first step in the solution of the above set of equations is

to solve Equation (Ala) for X]_ Or

xI . B1/An . (AI#All)X24AI#AI,)x3 ..............(AIgA_)XN (A2)

If Equation (A2) is substituted into Equations (Alb, o, ....j N) a modified

set of N-I equations is determined.

A221 X2 + A231 X3 ............. + AINX N = BII (A3a)

A_2 X2 + A_3 X3 -- ...... ----+ AINx N - BI (A3b)

|mm, mmmmemmmm mm m -- ..ma,.em ! mm m(mmmmmmmmmmmmm|mmm| m m ----m--_e-- m m --mm

I ....• xN"X + AN3 -'

whereAI AIj/AIIiJ " Aij " Ail

B.I - B i BI/AIIl " Ail

i, J " 2, ...., N (A_)

i = 2, ...., N (A4b)

A similar procedure is used to eliminate X2 from Equation (A3), etc.

A general algorithm for the elimination of Xn may be written as

Report No. 5654-02 FS Page A2

I

I

I

I

i

I

l

[

[

i

I

i

[

_n-l..n-l_ XjXn " (_n /Ann''_. "nJ"nn.''An"i/An'l_u n + ip ,ooo_ N (AS)

. .n-i .n-I (An-I/An-l_Aij Aij "Ain " nJ "nn"

ip J _ n + Il ,o.o_ N (_)

_I _n-1,.n-l•n . B'I-A (_n /Ann ; i- n+ i_ ...., NB i

Equations AS, A6 and _7_Y be rewritten in compact forma

(A7)

A_n. An-1_- A_ I c._ i, J d n + I, ...., N

i " n 4" lj eeoej N

where_n-1 ,.n-I

Dn_ _ _1_ /Arm

•n-lt. n-I

Cnj " Anj /Ann

After the above procedure is applied N-I times the original set of

equations is reduced tothe .following single equation.

AI_IxN = _i

(Ag)

(_o)

Report No, 5654-02 FSPage A3

which is solved directly for XN

In terms of the previous notation, this is

The remaining unknowns are determined in reverse order by the repeated

application of Equation (AS).

Simplification for Band Matrices

For mar_ finite element systems it is possible to place the stiff-

ness matrix in a "band# form which resUlts in the concentratic_ of the

elements of the stiffness matrix along the rain dlagonal. Therefore_ the

following simplifications in the general algorithm (Equations ASp A9 and

A16) are poasible:.

T-!

Xn - Dn-__ CnJXj _ - n+19 ...., n÷,M-1

n .n-I .n-I O j i, J = n + I_ n +M-IAiJ = Aij - Ain ....

n . Bn-1 A..-1Bi i - in Dn i = n + i, ...._ n +M - I

CA14)

where M is the band width of the matrix.

Report No. 5654-02 FS Page ,_

I

I

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I

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I

I

I

i

I

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I

I

I

I

The number of numerical operations can further be reduced by

recognizing that the reduced matrix at any stage of procedure is

sy_uetric.

equati on:

Accordingly, Equation (AI3) may be replaced by the following

_n-i .n-i i - n + i_ ,,,,s n +M -1

_ - Ai -Ain cn I (nS)S S S S - i, ....,n+M-

since

n . AnAji iJ

The number of numerical operations required for the solution of a

band matrix is proportional to NM2 as compared to N3 which is required

for the solution of a full matrix. Also, the computer storage required

by the band matrix procedure is _ as compared to _ required by a set

of N arbitrary equations.

This technique has been used in the automated axisysuetric proEr_

for the analysis of a typical heat shield idealized by a iO x 40 mesh of

quadrilateral el_ts. A solution to 800 8i_ultanQous equations was

necessary, which required less than two minutes of computing time on

the IEM 7094e

Report No. 5654-02 FS Page A5

APPENDIX B

MATRIX FORMULATION OF THE LEAST SQUARE CURVE-FIT PROCEDURE

Consider the problem of selecting the "best" polynomial of the

-cI+c_ +c_2+c,x_,....,c.__'Ito_,_°.__°_o__form YAI

set of data points:

n XI _ YI J X2| Y2| ..... ,.............. J

If the above po_omial is evaluated at points XI

following form are found:

Cl +02 Xm +C2 _ + ............. CN_-I = Ym

or in matrix form

r.lCI

C

w

m

i

CN.

Y1

Y2

mmm

M

N-iI XI X2 ..... XI

I X2 X) ----- X2N'I

mmmum lilil

gmm_Imo_ m m qmmmm_ mm4m i_mmi i i NQ i

• ......

%o XM, M equation of the

m n i| ****_ M

(Bla)

E

Report No. 5654-02 FS Page BI

or symbolically

where

where

A] : M N matrixa x

Y] a Mxl matrlx

[.?'If Equation (BI) 'is premultipiled by _ a set of N linear equations.

in N unknowns is created, Consequentlyj

[0].- [0][A]'E.]

[D] = [A]T[_ - aNxl matrix

Equation (B2) can now be solved directly for the unknoml coefficients [0] @

This procedure is numerically equivalent to the standard least

square procedure. Howeverp it is presented here in .a form which is readi_

programmed for the digital computer The technique is not restricted to

po_Tnomlals.

Figure B1 illustrates the application of the method in the evalua-

,4^.._v..^ev.+ho.......,1,,tic modulus for temperature dependent materials, A

fourth order polynomial was used6

Report No. 5654-02 FS Page B2

I

II

Ii

I

II

II

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r

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qU0

3E

600

500

400

300 -

20O

I00 -

00

• Data Points

Fit

I I I200 400 600 800 IO00

Temperature - degrees farenheit..

Fig. BI Example of Least Square Curve Fit

Report No. 5654-02 FS Page B3

I

APPENDIX C - MATHemATICAL MODEL OF SANDWICH SHELL

IThe sandwich shell substructure is composed of a honeycomb core

I material and two steel face plates. The orthotropic core material is

I readily represented by solid triangular rings as indicated in Part IV

of this report. However, for the description of the behavior of the

I face plates, a shell type element is used. The appropriate theory is

i given in this appendix.

It is assumed that the face plates are idealized by series of

I truncated cone elements which are connected at mesh points of the finite

I element system. The cross section of a typical truncated cone elementis shown in Figure C1.

Z _ _ Nodal circle

\ j rs,Z jK_\ _L_ of Thickness h

Truncated Cone Element

I ivlfri,zi _(,

FIGURE 01 - CROSS SECTION OF TRUNCATED CONE ELDEST

I

Report No. 5654-02 FS Page CI

The displacements of the system in the r 9 @9z coordinate system

are written in the following forms

ur " _ _rn (rgz)cos

u " ) Cr,z) cos n@Z ¢_ Uzn

ue = _ _n (r,z)Binne

where Rrn (r#z), uzn. (rpz) and _n(r,z) are the two-dimensional displacement

functions (Fourier coefficients) associated with the harmonic n.

Within each truncated cone element, the two-dimensional displacement

functions are assumed to vary linearly. The displacement at some point

t within the element is given in terms of the nodal circle displacements

(ci)

(c2)

(c3)

Acc_dingly9 the displacement in the t-dlrection is


ut(%e) - ur(%0) ace ® .,-um(t,e)e_ ® (c4)

&where cos w • 7

sin (. • b

j The inplane strains within the truncated cone elemen% are

! % "_-" _'%" ooe .o (c5)|

, _ue

(C7)

By cc_bi_ing Ec[uatio_s (el)throuEh (C7)_ the inpla.e strains

for harmonic n are expressed _n terms of nodal circle displacements by

, the following matrix equations:

%,

_tn

m

. a b a b ¸

o

m

0

_-t n(_-t) t ntr--l" o r-"I _ o r-'/

i

iU_vl

!

I '

_n

(08)

Report No. 5654-02 FS Page C3

If Equation (08) is evaluated at the center of the element, the average

element strains _re given by

. ,.,. ._ a _ -_

n "6n;'., = o _ o --.-.

t •- r,a ...._._b __ ...:._. ._ :z _

where _ = (ri + rj)/2

or Equation. (C9a)written insymbolic form

r, I . [_][o] (o,,,,)Ln..l

(Cga)

From Hooke,s law, the _stresses within the element are given by

r-

|I v 0

I0 0 I-__L -_-

m

¢%n

¢@n

n m

I"1° i

T2

mo


l+uwhere _I " _2 " -----'2 Fnt Tn

I-_

I Equation (lOa) expressed in symbolic form is

I [,j- Fo_[,j.,[_] (lOb)

From Equation (I.15)_ the nodal circle forces in terms of the nodal

circle displacement for the harmonic n are given by

where

[,J, [,j [,j. [_,.[KJ-,_[oj'[o][oj'E_J-- Eo._[,]

For the truncated cone element

A-h -_

(c_)

These forces are included into the overall equilibrium of the

system for the harmonic n by the same technique used for the system

of triangular rings. When n equals zero, these equations reduce to

the axisymmetric case.


APPENDIX D

LISTIIqI3- ARBITRARY AX_ETRIC STRUCTURESi, i W , i i , ,

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APPENDIX E

PROGP_M LISTING - AX!SYMMETR!C HEAT SHIELDS


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Report No. 5654-02 FS Page F20

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APPENDIX G

SUMMARY OF EFFORTS TO SOLVE THE THERMAL STRAIN PROBLEM

BY THE OVER-RELAXATION AND DIRECT INTEGRATION

OF FINITE-DIFFERENCE FORMULATION

I. ACCOMPLISHMENTS

During the initial phases of this contract, finite-difference techniques were

employed in an effort to solve the governing differential equations for this prob-

lem: The _]iowing work was accomplished:

A. Derivation of basic equations suitable to the given geometry

B. Derivation of the finite-differences model of the equations

C. Development of a thin-shell model for the thin layers (bond, face-

plates) and its finite-differences equivalent

D. Attempt to establish proper boundary conditions for the finite-dlfference

model

E. Programing of the finite-difference model(s) and attempts to solve the

equations by (i) the over-relaxation approach (in several versions) and (2) direct

matrix, inversions.

F. Investigation of alternative methods and recommendations.

A summary of the results in each of the above areas is presented in this appendix.

II. DERIVATION OF BASIC EQUATIONS SUITABLE TO GIVEN GEOMETRY

A complete set of equilibrium equations in terms of displacements in spherical

and toroidal coordinates was derived and is given below. A singular point exists

at _ = O° The equations for this point are also presented.

Report No. 5654-02 FS Page GI

For orthogonal curvilinear coordinates (6l,__ 62, 63), the element of arc ds isdefined by

3 2ds2 = Z gii d6.

1i=l

(al)

where gii represents the metric coefficients

_/axis of sy_netry

R,4

III

xx I

axis of symmetry

2

\\

V

Fig. GI - Coordinate Axes

Spherical Coordinates Toroidal Coordinates

_i R r

62

63 e e

gll I i

R2 r2g22

R2 sin 2 _ (a + r sin _)2g33

g = _gllg22g33 R2 sin _ r(a + r sin _)

Note: Toroidal coordinates reduce to spherical coordinates in the limit

as a ---_ 0.

Report No. 5654-02 FS Page G2

Equations of equilibrium with zero body force take the form shownbelow.

= o (02)

where g _ _gllg22g33 and T.. and _.. are normal and shear components of stress, re-im mjspectively.

After substituting the respective components of 5.m and gii in Equation (G2)

and performing the indicated differentiations and summations_ there are obtained the

following equilibrium equations in terms of stresses for each coordinate system:

A. SPHERICAL COORDINATES

STRR i _ i STR@ 2TRR - + cot+ R + R sin _ _ + - T_ RTO@ TR_ - 0 (G3)

= + -+ R d9 + R sin _ 00 + (T_ T@@) cotR : 0 (G4)

STRe i _ i _@@ 3TR@ + 2T_@ cot-_ + R + R sin { -_ + R = 0 (G5)

B. TOROIDAL COORDINATES

rr

-_r +-

i _Tr8+

(a + r sin _)

+ (a + 2r sin _)Trr - (a + r sin _)_ - T@@ r sin _ + Tr_ r cos

r(a + r sin _) : 0 (06)

r + sin

+(20 + 3r sin _)Tr_ + (T_ - TS@) r cos

r(a + r sin _)=0 (07)


i

II

II

+-r +_a+ r sin_) -Y@-

(a + 3r sin _)Tr@ + 27_e r cos

+ r(a + r sin 9)-0 (G8)

C. HOOKE'S LAW INCLUDING TEMPERATURE TERMS

T

To

(G9)

where

• .. : 2_e.. (OlO)

0 = e_ + e_ + e_=

and I and g are the Lame constants defined in terms of Poisson's ratio _ and Young's

modulus E according to

_E=

(1 + _)(1 - 2u)

E } (all)

D. STRAIN - DISPLACEMENT RELATIONS

u 3 uki i Sgiie. = + _

@ii 2gii k=l _ _gkk(GI2)

e.. -

10(G13)


Let u, v, and w be componentsof displacement in the three principal

directions r or R, _ and @. Then substitution of these displacements in Equations

(GI2) and (GI3), with the metric coefficients of Page G2, yields the strain-dis-

placement relations for the coordinate systems shownin Sections II_E and II,F.

E. SPHERICALCOORDINATES

_U

eRR =

i _v u

:

i _w U v cot

e88 - R sin _ _+ _ + R

eR_ : _ _ - [+i

e_@ : _ _ R R sin

eRe : 2 R sin _ _ - [ +

(Ol4)

F. TOROIDAL COORDINATES

_uerr

_i _v u

e_ = --r _ + --r

i _w u sin

e88 = a + r sin _ _ + +a + r sinV COS

a + r sin

(o15)

if!er_ : _ _ + _r - r


e_@: _ la + r sin _ _ r _ a + r sin _I

zl l _u _w w sin_ 1

er9 = _la + r sin _ _ + _rr - a + r sin _I

G. EQUILIBRIUM EQUATIONS IN TERMS OF DISPLACEMENTS

Expressing the stresses in terms of displacement using Hooke's law(Equations G9) and (GIO) with the strain-displacement relations _quations (GI4)

and (GI5)], the equilibrium equations _quations (G3) through (GS)_ may be written

in terms of displacements in the form

-,2 -,2 -,2 -,2 -,2ou ouk _0_1 k _0_2 k _0_3 k OC_ldO_2 k dOC2dO_3

-u Ou Ou ou

_2 -,2 _2 -,2+ _- V -- 0 V -- V --- 0 V --- _2VU---Z+B U----z+c_ U---_+o y&-y.&.U+-_.

82+- v +- 8v - 8v +- 8v -

+ A ----_ + B ----_ + C ---'_ + D _----_--- + E

5

- -- _0_ ' - ' '_


(_15)

Page G6

(GI6)

TABLE GI

COEFFICIENTS OF EQUILIBRIUM EQUATIONS

SPHERICAL COORDINATES

(_io)

Ak

Bk

Ck

Dk

E k

Fk

Gk

Hk

Ik

Jk

%

%

%

%

%

%

%

£k

k=l

K+2_

_/_2

I_/(R 2 sin 2 ¢)

0

0

0

2(x + 2_)/R

I_ cot _/_-

0

-2( x + 2_)/R2

0

0

0

(x + _)IR

0

0

(;_ + k) cot ¢/R

-(X _- :_,,_/.,:,2

0

-(X + 5_) cot ¢/R 2

0

0

k=2

0

0

0

(x + _)IR

0

0

0

2(X + _ix_/_-

0

0

(X + 2_)IR 2

_/(R 2 sin 2 ¢)

0

0

0

21_/R

(),. + 2i_) cot ¢/R 2

0

-(X + 2_)I(R 2 si 2 ¢)

0

0

k=3

0

0

0

0

0

(X + _)/(R sin _)

0

0

2(X + 2_)/(R 2 sin ¢)

0

0

0

0

0

(;_ + _)I(R 2 sin ¢)

0

0

0

(;< + 5k) cot ¢I(R 2 sin ¢)

0

_/R 2


TABLE GI (cont.)



(_/0)

k=l

_k o

5k o

_k o

_k (X + _)/(R sin _)

_k 0

_k 0

Ik -(_ + 3")/(R2 sin _)

_k

1.... • _=2

: 0

0

(k + ,)/(R 2 sin ¢

0

0

0

-(k + }_) cot ¢/(R 2 sin ¢)

0 O

......... 7:-"ETI--_<-_-ii _:l. IU: i _ ----i_ .... _. "

TABLE G2


TOROIDAL COORDINATES

A k

Bk

Ck

D k

-k

Fk

Gk

Hk

k:l

_+2_

_/r 2

./(a+rsin¢ )2

0

0

0

(X+2_) (a+2rsin¢)/ Jr(a+rsin¢)_

_cos¢/Er( a+rsin¢ )]

k= 2

0

0

0

0

0

2( },+2_) rs in¢+ (k+5,)a

r2(a+rsin¢)

k=3

(4+ 2 2

0

0

0

2_/R

, cot ¢/R 2

0

-_/(R 2 sin _ ¢)

k:3

0

0

0

0

0

(k+_)/(a+rsin¢)

0

0


TABLE G2 (cont.)



Ik

Jk

\

%\

_k

Gk

Hk

\

k =i

-(_+2_) [i/r2+sin_/(a+rsin_)_

0

0

0

(X+_)/r

0

0

(k+_)cos¢/(a+rsin¢)

-(_+3_)/r 2

0

(X+3 )rsin cos + oos r(a+rsin¢ )2

0

0

0

0

0

(X+_)/(a+rsin¢)

0

0

-(k+}_)sin¢/(a+rsin¢) 2

0

_k

_k

_k

_k

_k

_k

_k

_k

3_

k = 2

0

k=3

r(a+rsin¢) 2

0

0

0

0

0

i (k+_)/_r( a+rsin¢ )]

)

' 0

(k+2_)acos_/ _( a+rsin_ )_

(X+2_)/r 2

_/( a+rsin_ )2

0

0

O

_(a+2rs in_ )/rr( a+rsin_ )]

(_+2_ )cos_/jr( a+rs in_ )]

0

0

0

(_+3_)cos_/( a+rsin_)2

(X+2_)r2+_a2+( X+3k)arsin{I

r 2(a+rs in¢) 2 ,

O

_/r 2

(k+2_ )/( a+rsin_ )2

0

( _+_)/[r(a+rsin_)]

0

0

0

0

0

_+3_)cos_/( a+rsin_ )2

0

0

(a+2rs in_ )/Jr( a+rsin_ )]

#cos¢/[r(_+rsin¢)]

i -_/(a+rsin¢) 2


TABLE 03

COEFFIC IENTS OF EQUILIBRIUM EQUATIONS

POLAR COORDINATES [FOR APPROXIMATE TWO-DIPIENSIONAL LOCAL

SOLUTION FOR ANY (8 = constant) CROSS-SECTIONAL PLANE]

Ak I _ + 2_ 0

Bk | _/R2 oDk

Gk

0

TT_-_]__

J_-

-_ i_ IR

R !u'i'! !o_/ ..... -"

\\

\

0

0

(x + _)/R

_ _2

(_ + _)/R

_ ;_ + X + 5_)/R2

Im___llbl , _., _m

(X + 2_)/R 2

0

-_ (_ + 2_)_

Note: Temperature-dependent material property derivativ_ t_rms are

also included. Only applicable coefficients are listed. By replac-

ing R with r, the above coefficients are applicable in the torus

cross-section region.

Report No. 5654-02 FS Page GIO

H° EQUATIONS FOR STRESSES IN TERMS OF DISPLACEMENTS

From Hooke's law_ Equations (G9) and (GIO)

[ f 1Tij : 2_eij + 8ij @ (3X + 2_) _(T)dT

9o

where 5.. is the Kronecker delta defined bym$

(o17)

and

6.. :i, i :jzj

:0, ilj

(_ -- + +ell e22 e33

Writin_ the strains in terms nf ai_1_r_m_+_ f_m =_+_=- _....+_- {_),_

or (G15) and shortening the nomenclature by defining the stresses

i

I

I

I

T1 : T orrr TRR

T5 : T_e

T6 = Tr8 or TRe

Equation (GI7) may be written in terms of displacements according to

T

_Z + h_ (3X + 2_) / _(T)dT = _u r + _u_ + y_u 0 + 8zu

To

+ _v r + _v_ + _v e + Y_v

+ _w r + _w_ + _w e + 6_w


(618)

Page GII

where

TABLE G4

COEFFICIENTS OF STRESS EQUATIONS


(_/O)

% 1

(%

B%

8

B%

7_

8%

%

_+2_

0

0

0

X/R

0

x oot ¢/R

0

0

X/(R sin ¢)

0

2

X

0

0

2(X + _)/_

0

(x + 2_)/R

0

X cot ¢/R

0

0

X/(R sin ¢)

0

3

0

0

0

X/R

0

(),. + 21_)cot ¢/R

0

0

(4 + 2b)/(R sin ¢1

0

4

0

dR

0

0

0

0

-_/R

0

O

0

O

5

0

0

0

0

0

0

p/(R sin ¢)

0

0

_/R

0

-_ cot ¢/R

6

0

0

_/(R sin ¢)

0

0

0

0

0

0

0

-_/R

...................................................... L.:LIZL 2:_;--- L 12 L'L:IJLI,yLLL'__'_ZZIL-I:Z..LL_.T.'-_L::[JIJ.',£,'ZL_.."

Report No. 5654-02 FS Page GI2

TABLE G5

COEFFICIENTS OF STRESS EQUATIONS


B

7_

8

F

0£

i

k+2_

0

0

X(a+2rsin¢)

r(a+rsin_)

0

X/r

0

_cos_

a+rsin_

0

o

_ kl(a+rsin_)

_ o

2

k

0

0

X+2_____+ ksin_

r a+rsin_

0

(k+2_)/r

0

a+rsin_

0

0

X

0

4

0

00

x_+ (X+2 )sin r a+rsin_

0

X/r

0

a+rsin_

0

0

k 1 X+2_

a+rsin_ I a+r_in_0

5

0

0

0

0

0

0 0

0 _/(a+rsin¢)

- t_/'r : 0

0 0

_/r

0 0

_cos_0 a+rsin_

6

0

O

_/(a+rsin_)

0

0

0

0

0

0

0

_sin¢- a+rsin_

TABLE G6


POLAR COORDINATES IFOR APPROXIMATE TWO-DIMENSIONAL LOCAL

SOLUTION FOR ANY (@ = constant) CROSS-SECTIONAL PLANE]

.......... i .....

I

8_

8_

I 2 4

k +2_

0

X/R

0

X/R

0

k

O

(k + 2_)/R

0

(X + 2_)/_

0

0

_/R

0

0

- [_/R

Note: Only applicable values of _ and coefficients are listed.

By replacing R with r_ the above coefficients are applicable in

the torus cross-section region.


I. EQUATIONSAT THEAXIS OFSYmmETRY

Certain of the coefficients in the displacement equilibrium and stress

equations becomesingular at the axis of symmetry (_ = O). For the nonaxially

symmetric case the axis of symmetry has no special physical significance and this

point can be avoided. For the axially symmetric case, however, the axis of symmetry

is generally quite important and the singular coefficients maybe evaluated by the

use of L'HSspital's rule. For example, the coefficient HI in the displacementequilibrium equations in spherical coordinates is _ cot _/R2 which becomesinfinite

as _ approaches zero. From Equation (GI6), this term multiplies the displacementcomponent_. The conditions for axial symmetryare

u_

_fw(R,¢,0) = _ = 0 (GI9)

where f is any function of R,_,@. From this it can be shown that

_,, 82- _ = 0 at _ = O.

v : _- _2(G20)

_U

Hence, since _approaches zero while HI approaches infinity, L'H_spitalXs rule is

applicable to the product

as _ ---_0. Taking the limit, there is obtained

cot _ _u __ lim

R2 "_ : R2 ¢ --_o

= _ . 82u

R2 _¢2

sin ¢

cos


Hence, for this case, the coefficient HI becomeszero and the coefficient BI which2

_u 2multiplies _2 is increased by _/R • Applying this limiting process to all thesingular te_s, the following sets of coefficients are obtained:

TABLE G7

COEFFICIENTS OF EQUILIBRIUM EQUATIONS ON AXIS

OF SYMMETRY (_ = O) FOR AXIALLY SYMMETRIC CASE

SPHERICAL COOHDINATES

k :, I i 2 ' 3 k ; i

Ak X+2_ i! 0 0 _k _ 0

Bk

Ck

Th

_k

Ek

Fk

Gk

Hk

Ik

Jk

2p./R 2

0

0

0

0

2(X+2 )/R

0

0

-2( X+2_)/R 2

I

1 0

0

0

0

0

0

0

0

0

itoI

I°0

0

0

0

0

0

0

i 2 .3 kt ................

\

u k

\

\

\

A

o o _k

0

0

;,-'t,_+lJ,) / ._

0

0

0

-2( X.+3 IJ.)/R 2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

_k

_k

_k

_k

_k

_k

_k

_k

Jk

....... ? .............

l i 2 3F ...........

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0


%

7%

F%

Y_

2%

TABLE G8

COEFFICIENTSOF STRESS_QUATIONSON AXIS OF

S_TR¥ (_ : 0) FOR AX_r,LY S:_XETEC CASE

SPH_aICAL C00RDINAT_,S

1 2

k+2_ _ k

0 . 0

j 0

2_/R 2(_ + _)/R

0 0

2X/R

0

0

0

0

0

0

2(x + _)/R

0

0

0

0

0

0

3

k

0

0

2(x + _)/R

0

2(_ + _)/R

0

0

0

0

0

0

4

0

_/R

0

0

k

0

0

-_/R

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

- -t ........

5 6

0

0

0

0

0

0

0

0

0

O

0

0

J. TEMPERATURE DEPENDENCE OF ELASTIC CONSTANTS

If, in addition to the coefficient of thermal expansion, the elastic con-

stants are strongly dependent on temperature, then additional terms must be included

in the displacement equilibrium equations to account for the special derivatives of

these constants. For example, in differentiating the stress component _.. withii


_Tii

1

_0 _X _e.

- X _-_ + @ _. + 2# _-- + 2 eii _.1 i 1 i

5T 5- (Sx -_2_) _(T) _- _7. -

1 1

(3X + 2#)

T

_('_) dTT O

T

= O _ + 2 e..mm _ - (3X + 2k) G_ i

To

50 5e..ll

+ X _. + 2_ _--. - (3k + 2_) _(T) 5T1 1 1

(G21)

where the first three terms to the right of the equal sign have not been accounted

primed quantities, Equation (GI6) becomes

(A k + AI_.) _+52u + Bk), _+52u ... = (3X + 2#) CZ(T) 5T

T

_ 5 5m f a(T) aT, k = l, 2, 3

0

(G22)

' are tabulated below for spherical and toroidal coordi-The coefficients A_, Bk, ...

nates, and for the special point in spherical coordinates on the axis of symmetry

for the case of axial symmetry.


TABLE G9

ADDITIONAL TERMS IN COEFFICIENTS OF EQUILIBRIUM EQUATIONS

FROM TEMPERATURE DEPENDENCE OF ELASTIC CONSTANTS


Ai

cl

El

-k

!

H k

I !

I k

!

Jk

k

--I

Ck

W,

E k

I F-;

k=l

0

0

0

0

0

0

(x+2_)

R

k=2

0

0

0

0

0

0

1 _ _T

R2s_n2 _ _ _-_ 0

0 0

0 0

0 0

0 , 0

0

0

0

0

8_ 8T

k=3

0

0

0

0

0

0

Rsin_ _Y

0

Rsing _

2 (_1 (_+_)_TR2_in¢

0

0

0

O

0

0

0


TABLE G9 (cont.)




I

Hk

Ik

I

k =2

Jk

m

z%

----I

Bk

Ck

-----I

Dk

Ek

----!

Fk

G k

----I

H k

0

0

0

0

0

0

0

Rsin_ _

0

Rsin_ _ _-R

=I

Ik

R2sin2¢ _-8

0

0

0

0

0

0

R2sin¢

_gR sine

k=3

R2sin_ _

R2sin¢

cot@ (_T)(%.+2_)8TR2sin_

0

0

0

0

0

0

(_T) (_+2_)3TR2sin2_

'_ 1 8_ _T _ _T

I -__- R


TABLEGIO




Ai

Cl

Di

El

O]K

-!

I k

Ji

B __!

A k

--I

Bk

I cD.| '

--!

D k

--!

Gk

k=l

0

0

0

0

0

0

OT" '_ -'-"_r

(a+rsin¢)2[_Y]_

0

0

0

0

0

0

k=2

0

0

0

0

0

0

0

sine [_k I_T

+ r(a+rsin_)_/_-'_

0

0

o

0

0

0

_ _T

ii ........

k=3

0

0

0

0

0

0

k_+rsin_ )IOT_ O_

0

(a+rsin_) _l_-rr

(a+rs in¢ ) 21 _"T-I ( X+2.1_

8k 8T

+ r(a+rsin#)

0

0

v

0

0

0

0


TABLE GlO (cont,.)

ADDITIONAL TEEMS IN COEFFICIENTS OF EQUILIBRIUM EQUATIONS



=I

A k

=I

Bk

=I

Ck

=!

Dk

=I

Ek

=!

i

Ik

I

!

0

0

0

0

0

0

0

0

0

0

0

0

0

r(a+rsin¢) t_l_

r(a+rsin¢) I_I_

/ _'_"_'_I _'_'_

(a+rsin¢)= _u" IU_

(a+rsin_) 2

0

8_ 8T

(_+r_n_)21_l<x _m

_k

_ sin_ Ic_p.l_T _" ___T( _d_ _T leo___

, + ,,91_l_-_ -'(a+rsin¢)_r "si--"' _ 8_| r IJ


Ko AXIS OFSYMMETRYWITHAXIAL SYMMETRY

The only non-zero terms in the coefficients of Table G9 on the axis of

symmetry in the axially symmetric case are the following:

' _T aTaI = (x + 2_)y_

The integral term in Equation (G22) is also non-zero for the equilibrium

equation corresponding to k = i.

L. EQUATIONS FOR THE SINGULAR POINT, _ = O, FOR THE NON-AXISYMMETRIC CASE

The components of displacement u, v, w in the R, _, e directions, re-

spectively, have the following properties at _ = O:

8u a3u 8u

_w _3w = - 2 aev

= v ae2a---_ y_

_v a3v= 2

_._.= w _e2.-.-..-_ oeop

T

= o _ = _(T) aT

T O .2

(G23)

These results are derived beginning on page G26 of this appendix.


M. EVALUATIONOF STRAINS

From Equation (G23) the singular terms in the strain-displacement equa-tions can be evaluated. Thus we find

eee = R sin _ _+ [+ R _ --_0)[ l_l + uR

and from Equation (G23)we note that the bracketed part is IO) and thus we have the

result

Similarly we find

+ (G24)

ece= Z w cot_ + _i-_-y

and again through Equation (G23) we obtain

e_e = 2-_ - _ +

i _2v

(_: o)

(¢ : o)

(G25)

Finally,

N.

ere = 2--R _ - w + R

ere = 2--R - w + R

EQUATIONS OF EQUILIBRIUM

The equations of equilibrium

(_= o)

TRR i _TR(_ i _ TR8

-Y_-+[ -_+R_in¢ --_-+2_RR - 'I'_ - '1"@@+ TR_ cot

R=0

(G26)

(G27)


+ R + R sin _e + _R_ ÷ (_¢ - _ee) eotR

-0 (G28)

-y_- + _ +3'rRe + 2,','_e cot

R=0 (G29)

have the indeterminate parts indicated below.

From Equation (G27), we have

i 8TR@ TR_ cot

Rsin_ -_+ R(a3o)

Putting in the values of TR@ and TR_ in terms of displacements, we get

L '1

(G31)

From Equation (G23) we find that the expression (G31) is zero at _ = O. Thus we may

write, for Equation (G30)

+

which is the evaluation of the indeterminate portion of Equation (G27).

tion (G28), the indeterminate part is

From Equa-

R sin _ + R cot (G32)

In terms of the displacements

sin_ + v


From Equation (G23), the terms in parenthesis are zero. Thus we have

which is zero by Equation (a23). Therefore we may write Equation (G32) as

which is the evaluation of the indeterminate part of Equation (G28).

From Equation (G29) , the indeterminate parts are

and

R sin + 2 cot (a33)

(034)

For Equation (33) we have (at _ --_0)

+R sin _ - w R sin

i 0It can be seen from Equation (G23) that the terms with _ have the form

and that the terms in u and _ are zero. Thus we can write

ST@@ i I $2V

-_-+ 2 %_ :s 12_Y_+ (x+ 2_) _3w

83___Z_w+ 2 _---._,-/8e28_

= 0 by Equation (G23)


Thus we maywrite Equation (G32) as

For Equation (G34) we expand the functions in powers of _ thus:

w =Wo +Wl_+W2 _2+ "'"

Then we may write (for _ -_ O)

_ [w w +wlg+w2 _2=_ i +_2_ o _ +

_vO

w - by2o= 2W_ + W_ "_

_V

And since wO = _ from Equation (G23), we have

e_-+ e_-_ +e_-¢ 2/_

_v2 1

:w2+ e_- (¢:0)

i{_ _3v 1

: 7 * : o)

Thus, the one term, __m , in the equilibrium equations is not evaluated by simple

differentiation with respect to _.*/-

0. RELATIONS BETWEEN DISPLACEMENTS AT _ : 0

To obtain the equations given in Equation (G23), we take the gradient

and the Laplacian of the displacement vector, i.e.,

See note on page G28.

(a35)


and

v2Y = v2 (_u+ _ + g)

Expanding in powers of _ viz,

u = Uo + Ul _ + u2 _2 + ...

and letting { = --> 0 after obtaining _ and V 2 _, we find the foil owing terms in-

I

volving _ as a factor:

_V

_ o_r dO 0

R _02 + uI

0 _chO2 + 2 c_@ ,/

Since _7__ and _72 -_must be finite wheu _ : O, the above expressions (having 1. _ _ _ as

a multiplier) must all vanish. Thus we have

bw0

: - V 0

_V

O

O_- : Wo


_uO

$2 Sv I

Wl - 2_e2

_2v I Sw I

The equations in (G25) are e_ivalent to these when _ = O. The equation

where

T

To

can be seen from the fact that at _ : O, there is only one point for all @_ so that

cannot vary with 8. (The same argument could have been applied to the quantity u.

It could not be applied to w or v since these are functions which have extension in 8.)

III. DERIVATION OF THE FINITE-DIFFERENCE MODEL OF THE EQUATIONS

The finite-difference model of the basic equations is presented below. The

difference analogs to the partial differential equations are constructed on a grid

network as shown in Figure G2, for which _I - _i_o_ lines ............... -_s_s _v_ th_ sub-

script i, _2 : constant lines by the subscript j, _ = constant lines by the subscript

k, and the intersection of grid lines (nodes) by the triple subscript i_ j; k.

Note: The further evaluation of Equation (G35) may reduce the expression to a simple

derivation process, but such a possibility will not be investigated since the

expression has already been made determinate.

Report No_ 5654-02 FS Page G28

Axis of Symmetry

Axis of Symmetry

j - i.

_ j+l

, • /

/ /_Q'_k 1

_s

k

-_k+l

k

Flg. G2 - Grid Notation for Finite-Difference Formulation

For the general case, the grid spacing will be irregular and the incre-

ments in the vicinity of a node will be designated by the following notations:

_l_: (_l)i+1" (%)i

h12--(_l)i+2- (_l)i

_13: (_l)i- (%)i-1

_x4: (_l)i- (_l)i-2

h21: (%)i+l- (_)i

I % g_' \

n22 = k_2Ji+2 - k,._2Ji

h23 = (%)i" (%)i-i

h31 = (_3)i+i - (a3) i

_132 _3Ji+2 _3'i

_33: (_3)i- (%)_-_

h34 = ((_3)i - (_3)i_2


Page G29

Let f (51, 52, 53) be any function of the coordinates such that it and its partialderivatives (up to any order required in the analysis) are continuous_ and expand

the function about the point i, j, k. Using a new coordinate system with origin at

i, j, k and with _i' _2' _3 directed along @i' 52' 53_ respectively, the function f(_i' _2' _3) is written

f (:i' :2 :3) = f + Bl:l + + + +' z, j ,k B2_2 B3_ 3 B4_I_2 B5_2_ 3

+ B6_3_ I + B7_I 2 + B8_22 + B9_32 + BI0_I_2_ 3

2+ BlZ_I:22 + B12:1_32 + B13:Z2:2 + Bl_:2:3 + ... (G36)

The first and second derivatives of f (@i' 52' 53 ) with respect to 51, 52, @3 are

obtained from Equation (G36) according to

i,j,k o,o,o)

o .9 _

= B4' _--_121 = 2 B7= BI' &_l &_21i_j,k i,j,k

i,j,k (o,o,o)

_2f I= % =2B 8

_f / = _f = B3, &a 3 6c_1 = B6, -- = 2 B9_7ti, j,k (o,o,o) i,j,k _c_32/i, j,k

By considering the values of f (_i' _2 _ _3 ) at the 12 nodes adjacent to i,j,k, the

constants B. are evaluated in terms of the function at these nodes and the gridz

spacings as shown in Figure G3.

(a3T)


2Node

q oI

Coordinate

12

(i+2,J,k)

(±-l,j,k)(i-2,j,k)(i,J+l,k)

(i,j÷2,k)

(i,J-l,k)

(_,j-z,k)(t,j,k+l)(i,J,k+2)

(i,j,k-1)

(i,j,k-2)

Fig. G3 - Coordinates of Irregular Mesh Intervals

Note that the grid spacing increments h.. do not, in general, have the dimensions of10

length but have the dimensions of Jl' _2 _ and _.

At points i and 3, Equation (G36) becomes

2

f(hll,o,o):f + +B_lli, j,k Blhll

f(-hl3,o,o): f. + B 1321,j_k - Blhl3

@38)

where terms of higher order are deleted. Solving for BI and B 7 from Equation (G38)

gives, for the first and second irregular central derivative with respect to _l'

= h132 fi+l,O, k (hll2 - h132) ;k

hllhl3 (hll + [_3i

" hll 2 fi-l;_;k

=2hl3fi+l,j,k

- (hll+ hl3)q,0,khllhl3 (hll + hl3)

i(a39)

li/


Substituting hll = h13 = hI into Equation (G39) gives for the first and second

angular central derivatives with respect to _i

Sf I fi+l,j,k - f"= z-l_j_k_2 i,j_k 2hlJ

$2f I fi+l,j_k - 2f. += z,j,k fi-l,$,k

$_12]i_j,k hl 2

(o4o)

By a similar procedure, the following first and second regular and irregular cen-

tral derivatives are obtained with respect to the coordinates _2 and 63:

First Regular Central Derivatives (h 2 = h21 = h23 , h3 = h_l = h3_ )

Sf I f" - f"z,j+l,k z_j-l,k (G41)_ = Phi

--2 !i.j,k

Sf 1 fi,j,k+l - f"= ,0,k-i (a 2)

_3 i,j_k 2h3J

First Irregular Central Derivatives

2 2 h212_f ' h2_-f" + ) fi, - f= z,j+l_k (h21 h23 j_k i_-l,k (G43)

i,j,k h21h23 (h21 + h23)

h_2 2 2 h312

_f_3) = fi,j,k+l + (h)l - h_3 ) fi, j,k - f',j,k-lz (G44)i,j,k h31h}3 (h}l + h33)

Second Regular Central Derivatives (h 2 : h2i : h23 _ h3 : h_l = h33 )

_2f ) fz,j+l,k - 2f. + f.

_ " z,j,k 1,j-l,k

_22 i,j,k h22


(c45)

Page G32

$2f I f. - 2f. + f.= m,j_k+l z,O,k z,j_k-i

_32Ii,j,k h32

(G46)

Second Irregular Central Derivatives

_2___I _ 2 Eh23fi,_+l,k- (h21 + h23 ) fi,j,k + h21fi,j_l,k_

_22Ii,j,k h21h23 (h21 + h23)

(G47)

_2-!f/ _ 2 [h3Si,j,_:+l+ (hi + b3) fi,j,_ + blfi,j,_-l] (a48)_ff32]i,j ,k h31h33 (h31. + h_3)

Forward and Backward Derivatives

By applying the same procedure as above with respect to two nodes located either

i._ _,__ • and second r_gl]lar and irregularforwer_ _r backward from the origim (i,j,_j, .... fmrst __ _

derivatives are obtained in terms of the function f(_1'__ _2' _3 ) evaluated at these

nodes. The results are summarized below for the three coordinate directions:

First Irregular Forward Derivatives

_f I - (h122 - hl12) fi,j,k + h122 fi+l,j,k - hll 2 fi+2,_,k

G_I i,j,k/ = hllhl2 (h12 - hll)(a_9)

2 _ h212 ) fi- (h22 ,j,k + h222 fi,j+l,k - h212 f':].,o+2,k (G50)

i,j,k h21h22 (h22 h21)

_f._.--I -(h322 - h312) fi,J, X + h322 f" 2= z,0,k+l - h31 fi,_,k+2 (G51)

%1 i,j,k h31h32 (h52 - hsl)

First Regular Forward Derivatives

For equal grid spacings in each of the three coordinate directions,

according to

defined


hll = h12/2 _ hI

h21 = h22/2 _ h2

_31 = _32/2 _ h3

(Q52)

Equations (G49) through (G51) reduce to

_f - 3fi,j, k + 4fi+l,$, k

i,j,k

- fi+2_j,k (G53)

- ]fi,j_k + 4fi,j+l,k - fi_$+2,k

,j,k

(_5_)

_ 5f±_O,k + 4_. _i,k+]-- #l, z,j ,k+2_

91].?g _L -#

(Q55)

Second Irregular Forward Derivatives

,j,k

= 2

_- hllhl2 _n12 - mllJ J

(Q56)

,j_k

=2 _ +(h -_ )f.-h22fi_j+l,k _ _22 _'21_ -m_, k

_---_121h22 (h22 - h21)

(a5T)

(G58)

Second Regular Forward Derivatives

With equal grid spacing_ according to Equation (G52), Equations (G56) through

(G58) reduce to

_2fl - 2fi+l,j_ + fi,j,____k+ fi+2,j,k

$_12_i, j ,k hl


(G59)

Page G34

_2__f.I - 2f. + f" + f" j+2,k= z,j+l,k m_j,k z,

_22/i_j,k h22

(G6o)

- 2f. + f" + f' $,k+2

_ m,j,k+l z,$,k z,

i,j,k h32

(G6I)

First Irregular Backward Derivatives

2h132 fi-2,j.,k + (h14 - h132) fi,j,k

h13h14 (h14 - h13)

h142 fi-l, j,k(G62)

,j,k

= h232 fi,o-2,k + (h242 - h232) fi,i,kh23h24 (h24 - h23

h242 f. z,j-l,k(063)

h33'2 2 h33'2 h 2fi,j.,k-2 + (h34 - ) fi,j_k 34

h93h34 (h34 - h93)

f°

z,j_k-i(G64)

First Regular Backward Derivatives (hl_ = h14/2 _ hi, etc.)

8%) = f±_2,j,k + 3#±,j,ki, j, k 2hl

- 4fi_l_, k(G65)

_ i_j-2,k i_j,k i,O-l,k

i,j ,k 2h2(G66)

8f____l f" +3f i -4f= z,_,k-2 _j_k i,j,k-i

i, j, k 2h5(G67)

Second Irregular Backward Derivatives

_12_2f ) = 2 I hl}fi-2'i,j,k

j,k + (h14 - h13) fi,0,k

h13h14 (h14 - h13)(G68)


=2h23fi,j-2,k+ (h24 - h23) fi,j,k_23h24 (h24 - h23)

(o69)

=2h33fi_0,k_ 2 + (h34 - h33) fi,O,k - h34 fi,0,k-i ]h33h34 (h34 - h33 ) J (aTo)

Second Regular Backward Derivatives

_2f I fi-2, j,k + f= i,j,k

_i 2ji,j,k h12

- 2fi_l,j, k(aT1)

_2f I = fl,j-2,k + fi,O,k - 2fi,0-l,k

_22ii,j,k h22

(G72)

_2f _ f. + f. - 2fl,j,k-2 z,j,k i,0,k-i

_321 = (G 73 )i,j,k h32

Mixed Derivatives

It can be shown from Equation (G36) that mixed derivatives require values of the

function at any six nodes in the vicinity of the point under consideration. Figure G4

shows various combinations of mixed derivatives with respect to the coordinate axes

_i and _2" It is noted that the mixed central derivatives involve the four corner

nodes as well as two adjacent nodes in either of the two coordinate directions. The

various combinations shown in Figure G4 are summarized below for the coordinate

directions Jl and 62:

Second Mixed Irregular Central Derivative with Respect to _i and _2

a) 1h21h23 (hll + hl3) (h21 + h23) 232 Ifi+l,j+l,k

-fi-l,j+l,k)- (h232- h2! 2) (fi+l,j,k- fi-l,j,k)


2- h21 Ifi+l,j-l,k - fi-l,j-l,k (G74)

Page G36

Central

Forward

Backwar_

Corner

_)

T

c)

e)

bh,,

P

O(Z

hi/

_03

f)

5

_ss

I.

(_Xi

A

_u

C_1

g)

.[_o< z

Fig. G4 - Irregular Mesh Intervals for Mixed Central, Forward

Backward, and Corner Derivatives


b) a2t" /hZZhl_ _ll + hl3) (h2:L + h23) 1,32 (f±+]_,j+l,k

2 hl12)

2

(fi-l,j+l_k - fi-i -l,k)]- hll _J

)m,j-l,k

(oT_)

Second Mixed Irregular Forward Derivative with Respect to _i and _2

c)

d)

i

h21h2) (hll - h12) (h21 + h2_) Ih232 Ifi+l,j+l_k

- - - h 2 (fi+l,j_l, kfi+l,j,_ fi+2,j+l,k + fi+2,j,k I 21

- fi+l_j,k - fi+2,j-l_k + fi+2,j,k

hllhl3 (h21 h22) (hll + h13) 132 (fi÷l,j+l_k

- f.m,j+l,k

- f.m,j+l_k

- fi+l_j+2,k

- fi-l,j+2_k

fi,j+2_k I - hll 2 (fi-l,j+l,k

+ fi,j+2,k) 1

Second Mixed Irregular Backward Derivative with Respect to 51 and G2

e)h21h23 {h13- h14) (h21 + h23) 232 Ifi_l,j+l,k

fi-2, j+l,k + fi-2,j _k

fi-l_j,k - fi-2,j.-l_k

h 2- fi-l_j_k) - 21 Ifi_l,j_l,k

I]l-2,j,k

(o_6)

(aTr)


_)-i lh

- *f" -f" 1fi+l,j-2,k ' z,j-2_k z,j-l_kh 2ii (fi-l, j-l,k

N- f. - + f II

z,j-l,k fi-l,j-2_k i_j-2,k U

Second Mixed Irregular Corner Derivative with Respect to 51 and (_-2

(G79)

_)_2f i lh= ll l.2hah22 (%2 - h21) 12h222{%+l.,j+l,k - fi+l,j,k

- f.z, j+l ,k

- f.z,j+2,k

+ fi,j,k} - hllh212 (fi+2,j+2_k

+ fi_j,k_

fi+2,j,k

@8o)

Second Mixed Regular Derivatives

All of the above results can be reduced to regular derivatives with respect to

either 51_ 52 or both coordinates by making the substitutions

h12 h14

hl/ : hl] - 2 - 2 = hi (a8l)

h22 h24 _

h21 : h25 = 2 - 2 : h2 (G82)

hI : h2 : h (a83)

The various derivatives are s14_m_marized below for the case in which all grid spacings

are equal (i.e., hI - h2)°

Report No. 565&-02 FS Page G39

Second Mixed Regular Central Derivative with Respect to _i and _2

$2f ) _ ia)_ b) _i $_2 i,j,k 4 h2 Ifi+l'j+l_k - fi+l,j-l,k - fi-l,j+l,k + f'l-l,j-l,k I

(G84)

Second Mixed Regular Forward Derivative with Respect to _I and _2

_2f ) -i Ifi+l'j+l'kc) 8_i $_2 i_j,k - 2 h2 - fi+2,j+l_k - fi+l,j-l_k + fi+2,j-l,k} (G85)

_2f. I

i,j_k

- -i2 h2 {fi+l,j+l,k '" fi+l_j+2,k - fi-l,j+l,k + fi_l,j+2,kl (G86)

Second Mixed Regular Backward Derivative with Respect to _i and _2

$2f 1 _ i (fi-l, j+l_ke) __Ji,j,_ 2h2 - fi-2,j+l,k - fi-l,j-l,k l-2,j-l,k

f) $2f 1 _ i (fi+l, j-l,k_i _2}i,j,k 2 h 2

fi+l_j-2,k fi-l,j-l,k +f. }l-l,j-2,k

Second Mixed Re6ular Corner Derivative with Respect to _i and _2

g) li, ,k 2 I - fi+l,j,k- f.

l,j+l,k

Ifi+2,j+2,k - fi+2,j,k -' fi,j+2_k

(G87)

(G88)

(G89)


IV. DEVELOPMENT OF THIN SHELL MODEL FOR THIN LAYER AND

ITS FINITE-DIFFERENCE EQUIVALENT

A. THE THIN SHELL IN SPHERICAL AND TOROIDAL COORDINATES

The thin-shell analysis used here has been derived from a rigorous_

direct expansion in appropriate powers of z (the distance from shell center

surface). The equations are derived in the toroidal system since the spherical

case can be obtained from these by putting a = 0.

(a95).

Strain-displacement relations are shown in Equations (G90) through

i (_v v l_u)er_ = _ _rr - r + -r_ (G91)

i !m w sin _ + 1 _u ! (092)_r@ = _ l_Tr a + r sin _ a + r sin _ _-@

i _v u (G93)_ : r _ + r

¢@@ = a + r sin _ + u sin _ + v cos _ i (G94)

1 {1 _w 1 6v w cos _ ) (G95)_@:7 _r_+_ +r sin} _ a+rsin¢

Report No. !_654-02 FS Page G41

Thus_

The stress-strain relations are as follows:

_ rr

_rr _ _ o_¢+ %0 +off

= s -_ _rr +%0 +os

!

¢00- E E r o-_

The first-order expansion is

r =p + z

i i z

r p 2P

z

ll = l_ +-- U 1o O

•._' f

P ,

! "• " // /

etc., including _, E_ G_ _ T since these are variables.

(G96)

(097)

(G98)

(G99)

(GIO0)

(GIOI)

\

\

Median surface

(not necessarily

center)

From Equations (G90), (G91), and (G92), we have

uI

rr po

(GI02)

Wherever terms are to be differentiated with respect to r and then combinedz

with first-order terms_ they must be carried to second-order in _, since thedifferentiation reduces the order by one.


Put

V_ V _Uo 1 o

_r_ o P P P

wI w sin _ _uo i o

_r@ p s sO

z

e¢¢: _¢¢o+-0 _¢¢ietc.

(s : a + sin _)

(alo3)

(aio4)

From Equations (G93), (G94), and (G95

_v ui o o

_¢¢o= _ _- * --P

i _Vl i _v uI uO o

o

_ O

=71 _Y_-+ _osin _ + v cos

o

_i : s - _- + _isin¢ + _icos¢

p sin _ i _Wo2 I- _-_- + u sin _ + v

S _ 0 0

cos¢)

aw w cos _ _vi o o i o

p a_ s s @_-_-

p + ]4 cos-- _ 0

s

(GI05)

Gi06)

OiO?)

GI08)

Gi09)

GIIO)

get_O-

i rr go i 0-_o +% i+_Ul = Pl E E I o o

_ o 0 0 O_

_tl (7o r_o

vI :v - + -o _ P G0

(Gill)

(GII2)


_UoI %%wI =P Wo sin _ + @_@-I + P G--_-

(all3)

For boundary stresses; second-order terms are needed because of

later differentiations.

2

z -IO- =(3- +-- O-- + I Z

rr rr° p rr I _p °rr2

{h)At upper and lower surfaces + _ , we have the pressures

- P2 : O-rro + _p _rr I + _r 2 (for upper surface)

h- P7 :o- G

rro 2P rrlo-

rr 2(for lower surface)

Thus,

= D

°-rrl (Pl P2 ) _h (GII4)

(pl + p2' (_12 (alzs)rr 0 2 ) 20;, Crrr2

For _r_' Gr@' in terms of the upper and lower shear stresses, it is seen that

: / - P-- (all6)

_r@ I : (T@I T@ 2_- (OZlT)h

(D1 + _'¢j /h /2%¢0 - * !_) %¢2 (Gll8)

T@ I + T@_I i__ 12

To put o-_ O + %@0 in Equation (Gill) in terms of displacements, use Equations

(GI00) and (GIOI); thus

Re)ort No. 5654-02 FS Page G44

i- _ 2J

°i )÷O O O O

Using Equations (GI05) and (GI07), we have then

O

EO

i _v u %wo o i o

+%e 7_ -+-to o 0 s

o-rr

o

u sin _ + v cos _ 2_O O O

+ +=--- o- -25 Ts E rr o o

O O

+ 2_ TO O

(GI20 )

Then by putting Equation (GI20) into Equation (Gill), we get

uI -o +p_

i -_o _- + u° s

bw0

- @_--+U 0

(i- 2_)(io+ %)+_..

E (i - _)) o-rr +PO O O

(i+_} ]0

i _ %T- 00

(GI21)

Equations of equilibrium are as follows:

rr i i _C_r@

-_r +- - =0r a +r sin_ -_

+-- - : 0r a +r sin_

-57-r +- 7J- - =0r a +rsin_ -_

These result in six equations in zero-order and first-order terms; thus

_%_o p o

_rr I +--_ s _-_ : 0

%¢1 + -_-m -o

_¢5_° _%_O o

%_i +-_ s _- :0

(G122)

(Gi23)

(G124)

Repoz't No. 5654-02 FS Page G45

2 o-

rr 2÷-_r--s _ _ -S -+ 2 _-_-_=o

S

2 %¢2

2_¢¢_ _ _o_% _¢¢o _ sin+_ - s-%_---%-g-+ 2

S

(a125)

O

-_ - 0 (G126)

(a127)

Equations (G99), (GI00), and (GI01) give o-_, _@, _8@ in terms of

the strains; thus

0-_ 1 - 9 2 ¢_ + _ ¢e@ i -9 + i_ Crrr (G128)

E

These relationships give six equations for the quantities cr_¢O 0"_¢1, etc. Thezero-order expressions are

o-_ ° : Equation (G128) with all quantities given zero subscript

(e.g., Eo, _o' 0_°' etc.) (GI_I)

O

= Equation (G129) with all quantities given zero subscript

, •(e.g., Eo, e¢¢o _o' ete ) (GI]2)

O

= Equation (GI]0) with all quantities given zero subscript

, •

(e.g., Eo, ¢_o C_°' etc ) (G133)

The first-order expressions are much more complicated. Thus,


E__) E 9 [ E10 o 0

2 ¢@@I + e +___2 ¢¢1 1-9 210 0 0

2 E0

(1-

+I EI_ ° E _)i (i +k) 2)]2 + o o

(1 - k) 2)2 _oi - _o o

E cz TI E oal T E1 _ T E c_ T k)IO O + O O O O O O oi- _ + i- _ +

1 - _)O O O (i - 9 )2O

]+ "i -'9 °-rrl + )2 o-(i Q rr- 0

E E k)O O O

l v 2 _dd1 +- i - 9 2 e@@ IO o

E1 2 E° o 91

+ 1-9 2+ 2)2(j-90 0 _

'¢¢o

+

X} 2 + 2)2 I eee(i - V _1 00 0

_, _ TI E %T El%_Oo o + o o+

i - 9 0 i - _)o 1 - 9o

+Eo % To 9z|

l

:Vo)2J

o ° ] 0

i

+ I o- + 90)2 o-]i _ _oj rrl (i rr- 0

( G155 )

E i E E _)IO O O

1 - Qo)2 ,0_@ 1 1 + 90 e¢@l + + 9o (1 + _ 0

(G136)

Equations (G!12), (Gl13), and (G121) are substituted into Equations

(G!06), (GI08), and (GII0); then Equations (Gl05) through (GIIO) are substituted

into Equations (GI31) through (G136). The resulting equations give _@@o' _@i'

_¢o' _i' G_@b' _@i in terms of Uo, uI, vO, Vl, Wo, Wl, _ro' _rl' _9o' _o"


Thus,

_¢o is obtained from Equations (GI05), (GIO7), and (GI31)(G137)

_i is obtained from Equations (GIO5), (GIO6), (GIO7), (GIOS),(GII2), (GII3), (GI21), and (G135). (Q138)

_@@ is obtained from Equations (GIO5), (GIO7)_ and (G132)O

(G139)

%@1 is obtained from Equations(G134),(GIO5), (GIO6),(GIO7),(G108), (azz2), (azz3), and (aZ2l) (al_)

_e ° is obtained from Equations (GI09) and (GI)])(G141)

is obtained from Equations (GIOg), (GllO), (GII2), (GII3),_@i and (GI}6)

When Equations (G137) through (G142) are substituted into Equations (G122) through

(G127), the result is six differential equations in the six quantities u ,O Vo_ Wo_

E_ ' _r ' etc. are already given in terms of surface stresses,°-rr2' _r_2' _r@2 rl o

Equations (GlZ4)through (Gll9)J. The corresponding equations for the spherical systemi

are obtained by letting a = O.

The first-order expansions of the tangential stresses are

(a143)

IZl _ku-_-P )

where _o' °-_l' _@ ' c_@@l" _e ' and o_@ 1 have been derived in Equations (G117)O O

through (G142). Substituting these equations into Equations (GI4}), (G144), and (G145),

the stress equations in toroidal coordinates are obtained in terms of u° _ Vo_ Wo, and

o- I

rr 2

Report Noo 5654-02 FS Page G48

v0

11o

_ o o..__o

o

o + -

+ c2 _ - cl. o _ %

2 _2 1 _eo]sin _ u uo 1 o

o s " s 602 o

t# ol+ c:_ o -S- + _o%_

+ 21_l_c_I_suo

qs- °Ivz +. o,,..£_

+c 1 p ]]:L L

+ tl0

+ V0

+ (1-2%)(>%)!£_ o-E° 1- _)o rro

o

i 1+ p:_":P2- _ i-%)2J

Zero-

order

terms

First-

order

terms

_J

(G146)


Page G49

o

Fs.i_!_ pD__o] EaT

o 0 0

+ C1 + uL s o i-_)0

2_rr2

21 0 " 0 1A + 0+c 3 T¢-- cl. 2 o _¢2p C2 z o- e_-Go

_ C1 u +P- ___E+£-o s _e2 GO

P/ o pl---:v - _'--__l o T¢- ' o reO O

ps j o

0 Is (z-2%)(1* %1_--_o _d"rr

o

s_ ( sin+ L10 + II0

I .q rJL " '_J I../J

C'% "]

L'J L "oiL ._.1

z

P

Zero-

order

terms

First-

order

terms

/

(GI47)

Report No o 5654-02 FS Page G50

Where

_V

oob_-+ c5W

O

c_v

oo;_-V-, c5 z

+C 6 !_] o

+GO

+ C5 s Wo

. 0 iTj--+- s

W o

_2U.

Wo

+l$u ol

_ o i

oJ

Zero.

Order

terms

First_

Order

terms

S = a + r Sin

E

CI = o___2___

0

C2E 2E OV_

=_+ 0 0 i

i- %2 (I- %%2

c 3

E U

0

%%h %%%c 4 = _ +o i - x)

o

E'(_T E_c_ T V+_+_

o (i. _ )20

@i5o)

(GiSl)

Report No. 5654.02 FS

Page GSl

E

c5 = 1+5° (a154)o

E 1 E _;o i

C6 = i + _o (i + 4o)2 (G155)

B. THIN SHELL SOLUTION: FINITE DIFFERENCE FORMULATION

The generalized finite-difference analog formulation is directly applicable

to the thin-shell solution with the following changes: coordinate _lWill be eliminated

since it corresponds to the radial direction,* and subscript i will be dropped to con-

form with the above statement.

The finite-difference solution of the equilibrium equations will first be

obtained for the general case where the grid spacing is assumed to be irregular. Then

a solution will also be obtained for regular grid spacing. In both cases the first,

second, and mixed derivatives are needed for the central, forward, and backward grid

combinations. A typical general grid spacing is shown in Figure G5.

This coordinate

leliminated

hs_ _ h33

t I

Fig.G5 - Coordinates of Irregular Mesh Intervals

The partial derivatives in the equilibrium equations given in Equation (G36) are

taken with respect to _2 and _3 (¢ and e, respectively).


The increments in the vicinity of a node will be designated by the following nota-

tions, in accordance with Figure G5.

h21: (_2)i+l- (_2)i

h22: (_2)i+2- (_2)i

h23= (_2)i- (_2)i_I

h24= (_2)i- (_2)i_2

h3_: (%)i+l- (%)i

h32= (%)i+2- (%)i

%3 : (%)i - (%)i-1

h34: (%)i - (%)_-2

(G156)

From Equation (G36) it can be seen that f({l_ {2' {])' the function of the coordinates

with the origin at i, j, k, is

f({l' {2' {3) = f" + + + + + B5{2{} +z,j_k Bl{l B2{2 B3{] B4{I{2 B6{3{I

+ B7{I 2 + B8{22 + B9{32 + BIO{I{2{ 3 + BII_{22

+ BI2{I{ 2 + + BI 23 B13_12_-o _2_3 + ...

The first and second derivatives of f(C_l, 52, _3) with respect to 52 and (_3 are ob-

tained from Equation (G36).

Sf = _f :

i_j_k io,o_o i_j,k o,o,o

$2f _2f i = B5 _ $2f

o _22

_ _2f = 2B8 ,

_2f

O_2fI

- _-T-21 = 2B 9 (G157)i,j,k ()_3 o,o,o

The constants B. are evaluated in terms of the function at these nodes and the gridi

spacing as shown in Figure G5 by considering the values of f(gl' g2' g3 ) at the eight


nodes adjacent to j_k, and proceeding in a mannersimilar to that outlined on pages

G30 and GS!.

I. General Case - Irregular Grid Spacin_

a. Central Derivatives

I 2f + (h212

_f _ h2> j+l,k

_Y_2 j_k - h2ih25

P

h23-) f______k- h2!2fj-l,k

(h21 + h2})

(a158)

I 2 _ h332) fi

_f h3_2fj_k+l + (h31 . _ ,k

= t h3j,_ h_h_._ (h) 1 :5"

- h312f_,k_l(G159)

9i h_ - (fj+l,k+l - fj_l,k+Z)

(h552 2 _ f ) 2 )]- - h.51 )(fj+l.,k j-l,k - hs1. (fj,-_,k-1 fj-l_k-1 ._(Gz6o)

_2f i l [h__!j_k - h21h25 (h21 + h25)(hs1 + hss) _232 (fj+l,k+l - fj+l,k-1 )

- [h2_ - h2l "(_j,k+.i j_k-1 - h21 fj-l,k+Z - fj_l,k_l )]

2 ]h25fj+l,k_ (hgl + h2}) fj.,k

h21h2} (hpl _ h2))

÷h f l_]211j-z,

I

8c_521J

- b :)_).I,k*l [hsl PP" o,k

h)lhs) (h31 + h>_3)

-1

* h)lrj ,k-zJ

(GI6i)

(G162)

(oi 6_)

b. Forward Derivatives

h222fj+l,k - (h222 - hol 2) fj.,k

h21h22 (h22 - h2l)

9

- h21-fj+2, k(G164)


_ _,k+l

_n_3j, k -

2 9

h sih32 hR_] 2 - h.31)

- h312fj,k+ 0

(G165)

_2f I i

_lj,k = h31h3} (h2_h31 + h337 _332 (_'j<z,k+i " fj+l,k

j+S,k+.l + fj+2,k ) - h3L- (fj+l,k-1 fj_l,k -fj+2,k-1+ fj+2, k)_

(G166)

82f t _

- fj,k+l

- fj+l,k+2 + fj,k+2 ) - h21- (fj-l,ktl " fj,k+l - :fj-l,k+2+ fj,k+2) 1

(C167)

'52f J 2[--h2f + + ] f ,k+2]__ :_,k z (h32 - h3z" j k + hN/j8o_ 2i

.3 Ij,k h31h}2 (h}p - h31 ) (G169)

c. Backward Derivatives

r_-" l -h-P_fi_2"k + ['h2h " hps2) fj,k 2q- j-l,_, = - _ . - - h_l- f

.... (_i,7o)

_/ : h332fj'k-2

!j,k

+ (h3L2 [ _3 2) fj,a - h342fi,k...1

(Gi7i)

Report No, 5654-t,2 FS

Page C55

+ fj-2,k - fj-l,k ) - h_l (fj-l,k-i - fj-l,k - fj-2,k-i + fj-2, (G172)

_2f I = __ i lhj,k h21h25 (hg} - h54--_hgl + h23 7 252 (fj+l,k-1 - fj+l,k-2

- " h 2 - - + _I+ fj_k-2 fj,k-i ) - 21 (fj-l,k-i fj,k-i fj-l,k-2 fj,k-2 ,j

(G173)

j, k

!--

2 mh2_rj_2,k . (h24 _ h23 ) rj, k

(h2L - h2 )(G174)

2 [h_fj,k_ 2 + (h34 - h33 ) f¢,k

h}_h34 _h34 - h33)

- h34fj,k_l](Gm?5)

, General Case - Regular Grid Spacing

When the grid spacing is regular_ then

i i

h2 = h2l = h23 = Zh22 =[_2_

I i

hi : h]l : hi] = _ hi2 : _ h]L_

(G176)

Substitutim_ the eooaitions of EQuation (G176) into Equations (G158)

through (Gi75) the first, second, and mixed derivatives for the central, forward, and

backward regular grid spacing combinations are obtained.

a. Central Derivatives

_f I fj+l,k - fj-l,k

I_2j,k 2h2(GITT)

Report Noo 5654-02 FS Page G56

(G178)

k = h2_3 j+l_k+l - fj+l,k-i - fj-l,k+l(GZTg)

j,k = h_ j+l,k - 2fj,k

(ozSo)

_cz32 I J ,k h32 j, k+l j., k , k-

(G18Z)

b. Forward Derivatives

_f I = i _fj+l,k- 3fj_- fj+2_k]_2 j,k 2h-_ k(G182)

j,k_ l 14fj k j,k+212h3 ,k+l - 3fj, - f

(az83)

= 2h2h _ j+l,k+l - fj+2,k+l - fj+l,k-i + fj+2,k-(oz84)

-i _ fj _ fj + fj_l,k+212h2h 3 j+l,k+l +l,k+2 -!,k+l(Gz85)

= h_ 2fj+l'k + f"j,k + fj+2, kl

(G186)


k = i____h32E 2fj'k+l + fj,k + fj,k÷21(G187)

C. Backward Derivatives

E J_f = 1 - 4fj,k 2h--_ j-2,k + 3fj,k j-l,k

(al88)

_f' _l_ -4f J_31j, k - 2h--_ j,k-2 + 3fj,k j,k-(G189)

-i Ifii,,.

2h2h3 LJ-l'k+l fj-2,k+l - fj-l, k-i + fj_2,k_ll(GI90)

_2f i _ -1 if 2] (GI91)

_2f i 1 _ fjj,k-2 ,k

_32 j,k h32- 2fj,k_l] (G193)

Vo OVERALL THREE-DIMENSIONAL BOUNDARY CONDITIONS

Given the notation and equations previously developed, and with Figures G6a,

G6b, and G6c illustrating the geometry involved, the following described boundary and

..... ÷_k_7_÷ ..... a_+{m_ _ hhn_e which will _overn the solution to the full three-

dimensional heat shield problem for the two significant cases ("fixed" and "free-free"

conditions, respectively, at the structural juncture surface).

A. On inside and outside boundary surfaces OB and IB for both cases:

Trr = Tr_ = Tr@ = 0


B. On interface surfaces Ii and 12 for both cases (index denotes which

material mediumis indicated):

\

i = 1,2

C. In the r - _ plane for which @ = 0° for both cases:

W --

8e 2 - _--@-= _ = o

D. In the r - _ plane for which @ = 90° for both cases:

u(r,_,90 °) = u(r,-_,90 °)

v(r,_,90°) = v(r,-_,90 °)

w(r,_,90 °) = .-w(r,-_,90 °)


E. On the structural juncture plane (SJ) for the "fixed" case:

U = V =W = 0

F. On the structural juncture plane (SJ) for the "free-free" case:

'1"_ = Tr_ = ff'@_ = 0

G. Along the circles at the intersections between the structural juncture

plane and each interface surface for the "free-free" case (IPI, IP2), the conditions

of Condition F are replaced by (index denotes which material medium is indicated):

V]i+l = _rVli

+

li -I+ 3+ ad - 3+ O_T =O

T TL_ 0 I.._ 0-_ ±_± L_ -- _i

i = 1,2

H. Along the circles at the imtersections between the structural juncture

plane and each of the boundary surfaces OB and IB for the "free-free" case (PI,PO),

the normal stress condition (T_ = O) of Condition F is replaced by

_u _v

u=r_rr- _


AB is edge view of planeof temperature and thickness

distributional symmetry.

\\ o

\

\oo \" \

e=9°° !

B

Fig. G6a - Front View of Shield

OB

OB

Fig. G6b - Cross section of Shield

PO

Fig. G6c - Portion of Structural Juncture Surface SJ

Definitions:

r - Radial direction (both sphere and torus)IB - Inside boundary surface

OB - Outside boundary surface

SJ - Structural Juncture plane

IP1, IP2 - Intersections between the structural Juncture plane and eachsurface interface

PI, PO - Intersections between the structural juncture plane and each

of the boundary surfaces OB and IB


VI. PROGRAMING OF THE FINITE-DIFFERENCE MODEL(S) AND ATTEMPTS TO SOLVE THEEQUATIONS BY THE OVER-RELAXATION APPROACH AND BY DIRECT MATRIX INVERSION

A. OVER-RELAXATION APPROACH

The first approach in an attempt to solve the displacement and stress

equations was point relaxation. The reason for this choice was the apparent success

of this method in the previous work by Morgan and Christensen. Repeated attempts

with different values of the over-relaxation experiments were not successful. Evi-

dently the reason for the difficulty in point relaxation was the incapability of this

method to bring in the effects of boundary conditions.

An alternative form (i.e., line relaxation) was then attempted. The

first trial with line relaxation was made utilizing a radial line to the boundary.

Repeated attempts were made with this method with various boundary conditions. It

was demonstrated that this method is also incapable of meeting the remaining

boumd_ly cmliuLb±uLio. A "_ __'- "

that all boundary conditions could not be met simultaneously; the solution diverged

as it proceeded radially outward.

The basic difficulty in these methods seems to lie in the formulation of

an acceptable and consistent system of boundary conditions. Similar difficulties

have been reported in the literature for very much simpler cases.

B. DIRECT MATRIX IICV-ERSION

A second approach (i.e., direct matrix inversion) was then attempted. Due

to the relatively small size of the computer memory, the mesh size was too large to

achieve a successful solution. However, direct matrix inversion cannot be ruled out

as a method for the solution of this problem. It should be noted that the very

short time allotted to the attempts at the solution using relaxation and direct in-

version methods did not allow a complete exploration of these procedures.

VII. ALTERNATIVE METHODS AND RECOMMENDATIONS

Additional methods for the so±utlon of the complete non_xi_y_H_1_L_lc c_ w_

investigated. It was determined that the possibility exists for a direct matrix in-

version solution of the whole problem provided machine language is used throughout for

the programing. An additional memory capability utilizing magnetic tape or memory

disks would be employed.


A second approach was also investigated which would utilize equivalent analog

circuits to transform the equations into a set_ the behavior of which is well known.

This approach seemsto offer another possibility of solving the complete three-

dimensional nonaxisymmetric problem.


Documents

Q G VONKARMANCENTER - NASA · @ q g_ vonkarmancenter thermal strain analysis of advanced manned spacecraft heat shields final report to national aeronautics and space administration