AD-753 040

OPTIMAL DESIGN OF LOCALLY ORTHOTROPICELASTIC FLAT BODIES WITH WEAK BINDING

IA

G. I. Bryzgaliný

Foreign Technol'ogy Division'Wright-,Patterson Air Forc.e Base,, Ohio

17 November 1972

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FTD-HT-23-1245-72

0FOREIGN TECHNOLOGY DIVISION

-J OPTIMAL DESIGN OF LOCALLY ORTHOTROPIC ELASTICFLAT BODIES WITH WEAK BINDING

by

G. I. Bryzgalin

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I-ORIGINATING ACTI VITY (V ePereeeUMS1) RaU. t REPORTLSSFIATO

Foreign Technology Division mFAir Force Systems Command .GRU

OPTMALDESIGN OF LOCALLY ORT1I0T!OPIC ELASTIC FLAT BODIES WiTHWEAK BINDINGI.. 9CRPTV NOT66Ms , otisp al el eaGnd incluseive dotes)

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't I I SUPPLEMINITARY NOTES 12. SPONSORING L4ILITARY ACTiVITY

Foreign Technolosvy DivisionI.rir-ht-Patterson APB3, Ohio

IS- ABSTRACT

optimal design of elastic bodies made of fibergylassreinforced D1astics or metals reinforced b', fibers with ahigh elasticity modulus. It is assumed that the comrositecomponents are in a nlane strain or a plane stress state andthat the reinforcement is directed alonv two mutuallyorthogonal families of' plane curves. The equations of these

curves and the reinforcement level at each point of thebulk body are determined for a riven boundary profile andIspecified conditions existinp on this boundary.

PIC."1k D ,Nov s51473, / U'!ILA.3.31 FIED

Security Classification

N 17- T- "I" - - - 1ý-W7 -I- -, .T

UNCLASSIFIED

rrh W OI 6 ooita

Fiberglass 3R•einforced Plastic

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MetalElastic ModulusStress AnalsisI

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9.cunty Cla1sification -i

UCLASS~IFIED,

FTD-HT- 23-1245-72

EDITED TRANSLATIONFTD-HT-23-1245-72

OPTIMAL DESIGN OF LOCALLY ORTHOTROPIC ELASTICFLAT BODIES WITH WEAK BINDING

By: G. I. Bryzgalin

English pages: 12

Source: All SSSR. Izvtstiya. MekhanikaTverdogo Tela, No. 3, 1971, pp. 169-175

Requester: ASD

Translated by: TSgt V. Mesenzeff

Approved for public release;distribution unlimited.

THIS TRANSLA riON IS A RENDITION OF THE ORIGI.NAL FOREIGN TEXT WITHOUT ANY ANALYTICAL OREDITORIAL COMMENT. STATEMENTS OR THEORIES PREPARED BY:ADVOCATED OR IMPLIED ARE THOSE OF THE SOURCEAND DO NOT NECESSARILY REFLECT THE POSITION TRANSLATIOh DIVISIONOR OPINION OF THE FOREIGN TECHNOLOGY DI. FOREIGN TECHNOLOGY DIVISIONVISION. WP-AFB, OHIO.

M TH .23-1245-72 Date 17 Nov 1972

All figures, graphs, tables, equations, etc.

merged into this translation were extracted

from the best quality copy available.

S~FTD-HT-23-1245-72

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I5

U. S. BOARD ON GEOGRAPHIC NAMES TRANSLITERATIONK SSEM jBlock Italic Transliteration Block Italic Transliteration

B 6 S 6 B, b C c C C ,S, sB a B 8 V, v T Tr m T, tr r 1 8 G, g Y Y Y y U, u

Aa D, d 0 0 0 9b F, fE E Ee Ye, ye; E, e* X x X x Kh, kh

*K x IV Zh, zh 4 U Ll 4 Ts, ts3 s Z'z 4i 4 Y Ch, ch

H H U I, i U11 w L Sh, sh,k" ,, y III a U Shch, shch

SK i'V K, k b: -0 1S iI 1 3A L, 1 bi YA & y, yM M M M, m b &0 0 00 O,o 0 0 10 )0 Yu, yu

Ti n /7 n P, p Y,

* ye initially, after vowels, and after ', b; e eLsewh.-r'e.When written as L in Russian, transliterate 1s yd or. i.The use of diacritical murks is preferred, but -.ach marx;•may be omitted when expediency dictates.

I 3

•< ~FTD-.HT-?3-12'45-72

j

FOLLOWING ARE THE CORRESPONDING RUSSIAN AND ENGLISH

DESIGNATIONS OF THE TRIGONOMETRIC FUNCTIONS

Russian English

sin sinCos Costg tanctg cotsec seccosec coe

sh siAbch coshth tanheth cothsch sechcach cach

arc sin sin"1areC COB C08"1

arc tg tan-1arc ctg cot-1

arc sec sec"1arc cooec coc"1

arc sh sinh"1

arc ch cosh-1arc th tanh"1

arc oth coth-1arc sch sech" 1arc each cso.h-1

rot curlig log

FTD-HT-23-1245-72 ii

TO I 771"7- ,-IR

OPTIMAL DESIGN OF LOCALLY ORTHOTROPICELASTIC FLAT BODIES WITH WEAK BINDING

G. I. Bryzgalin

(Volgograd)

Solved are problems of optimal design for elastic bodies made

from a material such as fiberglass reinforced plastics or metal

reinforced by fibers with a high elasticity modulus [1, 2]. It is

assumed that the composite components are in a plane strain or a

plane stress state and the reinforcement is directed along two

mutually orthogonal families of plane curves. The equations for

these curves, the reinforcement strength at each point, and the

body thickness are determined during the designing for a given

boundary profile and conditions on it.

Let us present a rather small body element cut by planes

normal to the directions of reinforcement. On the two opposite

edges orthogonal to the first direction of reinforcemenb we will

apply expanding or compressing normal stresses a1. We will assumethat the material is deformed as a monolith, i.e., deformations e

along the stress, on the average, are equal at all points. Let

al be the stress in the reinforcement fibers of the first direction;

sI - cross sectional area of these fibers per unit area of material;

" and E' - elasticity modulus of fibers.

FTD-HT-23-1245-72

If the edge of the examined element has a unit area, then the

"remaining part of material, composed of binder layers and fibers

oriented in a transverse direction, will occupy on it area 1 - s1.

The average elasticity modulus of this remaining part is designated

in terms of E", and the average stress - all. Using the well-known

formula of mechanical mixing we obtain

-, ¶g'Sg ,"•- ,-,'. n,' £',,. i.,"• ". (1 )

Considering that sI does not equal zero and E" is considerably

less than E', we can write the approximate equality

SE(2)

Calculations based on more specific formulas [3] indicate that

if the modulus of elasticity of binder Ec is ten times smaller than

the reinforcement modulus, then the replacement of equality (1) with

equality (2), at worst, yields an error (contributing to the safety

factor) on the order of 30% of the maximally possible stress in

the material.

The following experimental factors also speak in favor of this

substitution:

a) the elasticity modulus of the binder of fiberglass rein-

forced plastics is 10-20 times lower than the elasticity modulus

of reinforcement;

b) usually, binders are subject to considerable creep, and

coefficient E" in the preceding formulas, in essence, is not a

constant value but an operator, such that with eI = const lim E"s =

= 0 when t - - (t - time). So that for parts working under condi-

- tions of prolonged action of slightly changing stresses, relation-

ship (2) can prove to be more acceptable than (1);

c) the range of the elastic and strength curves of composite

materials is tens of percent from sample tc sample.

FTD-HT-23-1245-72 2

-J~~* "- ~ "~-.-

.7I-,:1.

The elasticity modulus of such an element can be considered

as identical during expansion and compression [4], and Poisson

coefficients during deformation along the fibers as small (so small

that they can be disregarded); therefore, we extend the relation-

ship of type (2) to the general case of plane stressed state.

We will limit ourselves to tLhe designs in which the main axes

of stress and strain tensors coincide with the fiber directions of

"reinforcement at every composite point (rational designs [2]).

Then in the curvilinear orthogonal system of coordinates a$ along

whose coordinate lines the reinforcing fibers are placed, the

following law governing the connection between strain and stresses

can be written as:a. - Esot,, a- - (3)

Here a1, o 2 - principle stresses, C •2 - principle strains,

E - constant selected on the basis of experiments on the compositematerial.

The problem concerning the strength criterion of such hypo-

thetical material, in other words, concerning the limits of appli-

cability of relationships (3) is rather complex and it is doubtful

whether it can have a unique answer suitable for description of

all actual materials of a similar type.

Taking advantage of the fact that tensile strengths of com-

posite materials during extension and compression are values of

the same order, we will assume that the permissible state of an

element is characterized by deformation values not exceeding a

certain maximum value ofI',I ,. I:1• ,(4)

For reinforcement which remains elastic right up to destruction

(of = E'i, i = 1, 2), this means that the maximum value of internal

stress in fibers determines the material strength.

FTD-HT-23-1245-72 3

_emN''C!- -n

iOf course, during compression or during a complex stressed

state, destruction occurs not necessarily due to a break in fibers;

however, criterion (4) can be used safely if the materials tested

have different S1 , s2, if we construct families of strength curves

[ at diV'ferent stressed states in plane E l•, and select the squar, Iof the permissible states such that none of the

strength curves should intersect it (Fig. 1). ISince there are no fibers in a direction normal

__ , to the coordinate lines of system aO, then within

i the accepted permissible limits we can assume that

Fig. 1. the third principal stress is equal to zero 3 = 0.

The stress components with mixed indexes equal zero due to the fact

that this is precisely how the rational designs are constructed;

so that directions a = const and a = const were the principal

tensions of stress and szrain tensors a1 2 = 013 = a23 = 0.

In view of this fact there will not be any fundamental differ-

ence between the plane stress and plane strain states. We will

only note that with a plane strain state the design is for a com-

posite layer with a unitary thickness, while with a plane stress

st:..te - the thickness can be arbitrary and even vary from point to

point; thus, only its minimally possible value is selected during

designing. The problems will become wholly identical, if in the

latter case in place of stresses and levels of reinforcement we

introduce values

7. '' (5)(h - composite thickness at a given point)

If we consider that the above examined volume element has

ight h and unitary transverse dimensions, then T1 , T2 - forces

acting on this element along directions a and ý; S1, S 2 - fiber

volumes of corresponding direction in it. (For simplicity, T1 ,

T will also be called stresses.)2.

The equations of equilibrium of such volume element ([5],

Chapter 2) can be obtained from (8.2), if we assume that, there,

12 0 0 -- ,•:r,)---r: -o0. (••- T ,- (6)

(HI, H2 - Lamd parameters of system a$)

Formula (8.2) in work [66 (Chapter 2), when e)2 0, leads tothe following conditions of deformation ccnsistency:

d*o

+ ( ' )+ " I'd - -I,'( I+ 2 : }H ,', , + 2I,)

Equations (6), (7) should include boundary conditions on body

profile, conditions (4), and positive conditions of reinforcement

levels S1 •. 0, S2 Ž 0.

Let the problem be solved with regard to the theory of elas-

ticity for an isotropic body whose middle plane occupies area ~

bound by contour L on plane cz8. Then for constructing a reinforced

body design with the same boundary profile and same conditions on

it, we can assume that the reinforcement fibers are directed along

the lines of principal stresses of the isotropic body (theorem on

the existence of rational design [2]). For the selected system of

coordinates ci8 (whose coordinate lines coincide with the isostatic

tgrid of the isotropic body) we have nine equations (3), (5)-(7)

and ten functions

The initial form of these functions and the body thickness are

determined according to the i odicated theorem, and beyond this it

is possible to state a variation problem, i.e., require that

functions (8) yield an extremum to a certain functional. In this

work, the design withstanding given loads and having the smallest

5

571; ..7P.-

volume of reinforcement among all rational designs we will consider

as optimum. In other words, for the optimum design, functions (8)

satisfy all the conditions written earlier and yield a minimum for

the following functional:

In a specific case, when both principal stresses in an iso-

tropic body have the same sign, the design is constructed on thebasis of the corresponding theorem of existence, in this case

(10)

Another specific case corresponding to the following conditions:

T -S , - ( 11 )

is indicated in work [7]. The possibility of constructing a design

which would satisfy equalities (11) is restricted by the condition

of deformation consistency. Actually, assuming that eI = const,

2 = const in (6) and using the condition of equality to zero of thetensor of curvature of space a8 in the initial state, we find

S,,:11, ,'11, ,11It'll, ,'11, el1.-• . .,. , . _ --- u.0 (12)

In this case the second equality proves to be a symmetrical

substitution of indexes and coordinates; furthermore, from it we

can easily obtain

, ,' -" /is" (113 )where f(a) - arbitrary function. Using the known expressions for

the curvatures of families of plane curves

8,1,4 11 ,11, ,.z

we rewrite (13) thus-N,ff, - I(°) (i14 )

6

?i

Condition (14) was shown in [7]; geometrically it mpans thatthe increase in the inclination'angle bf a tangent to line a per

single increase of coordinate a does not depend on 0. ,

Generally, we have to solve the variation problem; moreover,

t we should assume that functions T2 , S2 can have a break along line,

a = const, and functions TI, S1 -along line 8 = const. After wedetermine functions •i' S2, thickness h can be'found in the

following manner. Let

At point (a0, 80), where this maximum is achieved', we'wi',l1

assume that s + s2 = s. (s* -greatest posdible. reinforcement

density, s. <1 ). Then

45)4

There are .other possibilities for selecting the thickness and

level of reinforcement! since addition or removal of ,xcess binder

does not (within the framework of, this system) affect the carrying

ability of the design.

Further, we are presenting examples illustrat~ing various possi-

bilities of spatial optimum designs.

A. Half-plane loaded on the edger section by a normal uniform'

force of intensity P. First of all let us investigate a stressed ]state in the corresponding is:otropic body. From:formulas (4, 4'1,

5) of work [8], (page 352) we can obtain the following expression

for the tensor components of stresses in an elastic,isotropic

half-plane loaded in the same way (the're is an error in the corre-

sponding equalities [8))

T, ,- n.,; :i . 'l'f -'I,- 1:) 4-2a,..'-(B, + O•) /r, rer. • 24V. - e - 0, + e..,/.,r] (16)

T. ,,e - I -2.¶PSY Sin (el + it.) fit?

(notations are in Fig. 2). in this case it turns out that the

7I I:

81 Figure 2. -

principal direction a forms angle y - 1/2(01 +- 02) with axis x, and

the principal stresses are

7,- .•'PI-0+•a~ .... 4 7,- -'PI-e-•v/r,,,4 (17)

Here

0-6 ,-Oe. 0, ,-aWtI[-v/(i-f( .M) 02 S1-,- (Z+&)1 (18)

Expression (17) can be presented in the form

re - ,-r 1-o -,*in61. r. - a-P-,t +.el (19)

It is obvious from (19) that both principal stresses are

always negative. Consequently, it is possible to construct a

design with a reinforcement which is uniformly stressed and with

identical principal strains

, ,. -. .. :(20)

The trajectories of principal stresses, along which the rein-

- forcement fibers are oriented, in this case, are families of co-

focal ellipses and hyperbolae with foci at points

S' here, the hyperbolae determine direction a.

. .On the x-axis segment [-a, a] the principal stresses are

maximum, under condition (20) this indicates the maximality of

value S1 + ,- then, according to (15), we find that

B10'~~epo a ... C091 -- ( 1Dei

It is interesting to note that if the load on segment [-a. a]

is not constant and the constant displacement v = const (smooth

rigid stamp) with the same total force 2Pa, then the optimum design

coincides with the constructed. Actually, the found deformation

field permits us to satisfy condition

It is evident from (19), (21) that when rl, r.2 s.• Sls0, consequently, at the sufficient distance from the loaded

segment the reinforcement can be terminated (when the contribution

of depleted reinforcement to the total material strength proves to

be small in comparison with the actual strength of the binder).

B. Circular plate loaded with a uniform tangential force along

the internal and external contour. We construct a design which

satisfies conditions (11). As a matter of course we introduce

polar coordinates rO and accept for the reinforcement directions

condition

V 0

(plus for a and minus for 8)

specifying that direction a forms angle 1/ 4n with the radius-vector,

and angle y with axis x. In this case the equation of line a in

system xy has the form

A solution of the latter, written in polar coordinates e = n cr

(logarithmic spiral), is c = const. Equation of lines 8: e =

= -ln cr.

Families of such curves s fy-em•rittin (14). The tensor

components of stressesY. FB A T. To

Here F - force per unit length of the internal contour, a -

internal plate radius, the external radius can be arbitrary.

The principal stresses, thickness, and reinforcement levels

are expressed by formulas

r?,--T:•.- .a,a. A FiAts.. i, . ,=..,

From conditions (11) we can also determine the displacements

b along the radius and in the circumferential direction

""-O. u' -- 2trIn 0,/I)

C . Circular plate under internal pressture. The radial and

circumferential directions w ke to be the reinforcement direc-

tions, then the equations of equilibrium and compatibility willi"a s su m e t h e fo rm r N .( 2Sr- (,r,)'. ,,. - ) ( 22 )

Here suoscript 1 pertains to the radial conditions and sub-

script 2 - to the circumferential; the prime indicates differentia-

tion with respect to r.

It is shown in [l] that in the absence of external pressure

the construction of a design with £1 = E2 const is impossible

according to (10). It is readily seen that construction of a

design which satisfies conditions (11) is also impossible due to

the fact that the second condition (22) is not satisfied.

We will assume that the reinforcement fibers are oriented,

this leads to the following condition concerning the reinforcement

level in radial direction [l]:

S. a/r (,-cons, (23)

Boundary conditions

7,, -P when ,r a. r -, 4,tcn.- I

Expressing all variables Ift'tmerms o7' t21 we find specifically

that• • ,,-=" ",.,•,:(24)

FTD-HT-23-1245-7 2 10

77

We select S2 such that volume (9) of reinforcement is the

smallest, i.e.,

S ++. 2r:' it (25)

The Euler variation equation has the form

+ 2e2#,3 (:')' -- 0

Its general solution is e2 = Cl exp (c 2 /r); where T1 = 0, S1

• 0 when r = b; consequently (see (3), (5)), we have el(b) = 0.

From this c 2 = b. With aid of (22) we obtain

r: -- teXp (b/P) t, - C,(0- bI /) 0 p CX / I r) (26)

Let b < 2a (for b > 2a the investigation is similar but the

expression for constant c 1 is different).

Functions (26) have the greatest values when r = a, here

We will assume that this maximum is equal to the maximum

deformation value c and thus determine the constant

r ezp (-bl/)

Now, from (24) we determine the level of reinforcement S2 (r),

which yields the extremum to the examined functional

The remaining boundary condition we use to find constant

finally,

S o Pal-!Et(b-e)rr So-Pa-b:/Et(b.-.a),J

At =- P(fl -I+ b+)/Ees.(b- a)s

The design is constructed. We can show that the second

variation of functional (25) is an extremely positive functional

1SFTD-HtT-23-1245-72 11

at "point" e£2 a Cl 1 xp (b/r); consequently, the found extremum Isa minimum.

For reinforoement volume (9) we obtain the rollowing formula:

W-2~q(+b)

In conclusion, we will examine the behavior of representativepoint (ci, £2) describing the stressed state of the body elementsalong a certain plate radius. For this we will express el in termsof C2(b a 2a) with the aid of (26)

elm -91e(1+10(sl)

A corresponding curve is shown in Fig. 2. When r changes froma to 2a the representing point travels the curve from B to Q.

As we can see, there is no analogy here with the optimumdesign of the rigidly elastic bodies, where it is most convenientand always possible to use the points on a restricting surface.In the examined case, the points at which the limiting curve inter-sects the axes of coordinates (for example) are less suitable thanmany internal points, since one of the reinforcement directionsfor them is not loaded.

The author expresses his appreciation to Yu. N.t Rabotnova

for his valuable suggestions.

Received 15 December 1969

BIBLIOGRAPHY

#,of. ofle NE fil J9. I , 1. "lot' I #A, Col.'..* Ii,

f. Al '9. . l f5 II ,I ,, I Ve Ci 110' 0 i f 1 101111 4I lti IidN e '11s. it'll ,,. " is , ,, 9,pi'' 1,99.''g 1;, le'.1 1 j til so, D I %to 4 , .9.9''le iij .

It 'lolo, M. 411lIts. 10u 6,el . " is e il

FPTD-IIT-23-12145-72 12