Compostte Structures 15 (1990) 259-273

Effect of Aerodynamic Heating on Deformation of Composite Cylindrical Panels in a Gas Flow*

Victor Birman

Engineering Educatton Center. Umverslty of Mtssouri-Rolla, St Lores, Mtssoun 63121- 4499, USA

Charles W. Bert

School of Aerospace and Mechamcal Engmeermg, University of Oklahoma, Norman, Oklahoma 73019, USA

&

Isaac Elishakoff

Department of Mechamcal Engmeenng, Florida Atlanttc Umvers~ty, Boca Raton, Florida 33431-099 I, USA

ABSTRACT

In this paper, motions of composite cylindrical panels in a gas flow are constdered. It is shown that the mam factor contributing to large static deformations is a nonuniform aerodynamtc heating, while aerodynamic pressure is of secondary tmportance, at high Mach number.

It was found that the main factor resulting in the increase of deforma- tions is the nonuniform distribution of temperature along the curved edges. Deformations decrease rapidly in shallower panels. Nonuniformly heated panels become unstable at the values of axial compressive load, which are much smaller than the static buckling value calculated in the absence of heating. The condition of panel flutter of nonuniformly heated composite panels in a gas flow is also formulated.

1 INTRODUCTION

Problems of panel flutter of composite cylindrical shells have been studied since the 1970s. t-3 At high flight velocities and with modem

*This paper was presented at the 16th Congress of the International Council of the Aeronautical Sciences, Jerusalem, Israel, 28 August-2 September, 1988.

259 Composite Structures 0263-8223/90/S03.50 O 1990 Elsevier Science Pubhshers Ltd, England. Printed in Great Britain

260 V. Btrman, C W Bert, L Ehshakoff

materials, aerodynamic heating has to be included m the analysis. The effect of temperature on supersonic panel flutter of aerospace structures has also been studied. ~-9

In this paper, the effect of aerodynamic heating on the behavior of a stmply supported composite cylindrical panel in a supersonic gas flow is considered. Both aerodynamic and material damping effects are neglected. The aerodynamic pressure is described by the Ackeret theory. The temperature distribution is assumed to be a general quadratac function of both the x- and the y-position coordinates:

T=(ao +atx +aEx2)(bo + b l y + b 2 y 2) T o (1)

This assumption permits reduction to uniform, linear and parabolic distributions of the temperature in particular cases.

It is shown that cylindrical panels experience static deformations in a nonuniform thermal field. These deformations can exist even ff the gas flow is absent. The presence of a gas flow can result in panel flutter, superimposed on static deformations. The effects of different geo- metries, temperature distributions and axial compression on static deformations of nonuniforrnly heated panels are considered in numerical examples.

2 ANALYSIS

Consider a cylindrical panel in a supersonic gas flow (Fig. 1 ). The axial length of the panel is a, the arc length of the curved edges is b, and the radius of the middle surface is R. The panel is formed of layers symmetrically oriented with respect to the middle surface. The number of layers is large so that the analysis can utilize the assumption that the panel is orthotropic. The analysis is based on Donnell's shell theory, which was shown to be sufficiently accurate for thin, shallow composite shells that are not too long) °

The equations of motion of the panel can then be written in terms of displacements and thermal and external forces as follows:

L u u + Ll2v + Li3 w -- N~, x - N r = 0 6,y

L,2u + L22v + L 2 3 w - N ~ x -N~yf f iO (2)

LI3u +L23v +L33 W-N 1 w,~ - N 2 w yy • q -/ohiO

where u, v and w denote the axial, circumferential and radial displace- ments, respectively; iO is the second derivative of w with respect to time; N r, N r, N6 r are thermal stress resultants; N~ and N 2 are the stress

Deformanon o f composite cylindrical panels in a gas f l o w

,,L(#/," ////:VB © ' "

b/, Fig. 1.

261

Cylindrical panel in a gas flow.

resultants of external loads acting in the x and y directions, respectively; p is the mean density of the material; h is the total panel thickness; and q is the intensity of aerodynamic loading. The differential operators L,j corresponding to the generalization of Donnelrs theory to orthotropic shells are ~°

-- 2 2. Li 2 _-(Ai2 +A66 ) dxdy Lll - A ll dx + A66dy, -- 2 2. Lls =(At2/R)dx; L22 -A66dx+A22dy, L23 =(A22]R)dy (3)

Laa-_Dnd4+2(DI2+2D66) dxdy2 2 + OEEdy4+A22]R2

In eqn (3), A, I and D,/are extensional and bending stiffnesses, respect- ively; dx and dy denote differentiations with respect to the corresponding argument.

Following the well-known Ackeret supersonic theory, II one can represent the aerodynamic pressure as

/gacEM 2 q= __W,x (4)

1

where p. is the gas density, c is the sound velocity in the gas, and M is the Mach number. The vector of thermal stress resultants is given by

N N r-- Z Q(k)a(k)hkT (5)

k=l

where {o} N r= N , a (k)= a 2

tN:J ,+°,+

262 v Btrman, C W Bert, 1 Eltshakoff

and

,[Q" Q': Q' l Q'k=IQ,: Q,.,. Q,.61 (6)

[Q,6 Q_-6 Q66J/,

Here, Q~kt is the matrix of the transformed plane-stress reduced stiff- nesses of the k-th layer, a ~k~ is the vector of the transformed thermal expansion coefficients of the same layer and T is the temperature, which is assumed to be independent of time. The total number of layers is N, the thickness of the k-th layer is hk.

The relationships between the elements of the vector a l~ and at. and a r, the coefficients of thermal expansion m the principal orthotropic directions of the layer, are

O~ I = O~ L COS20-'[ - a T s i n 2 0

a2 = at. sin20 + ar cosZO (7)

a~ =(at . - a t ) sin 0 cos 0

where 0 is the lamination angle. The temperature distribution is assumed to be a general quadratic

function of the middle-surface coordinates given by eqn (1). This assumption permits consideration of uniform, linear and parabolic distributions, in particular cases. Under these circumstances

N r = f l T (8)

where

;/ f l= [fl~/ (9)

~s the vector of reduced thermal stiffness coefficients

N

r = Z tnIk)aIklh (i,j = 1, 2, 6) (10) ~ q / k k~ l

The boundary conditions at the edges x = 0 , x =a, y--0 and y = b correspond to $2 conditions in the terminology of Almroth 12 or SS3 conditions in the terminology of Hoff: ~3

x=O,a N 1 = v = w = M l = O (11)

y=O,b N 2 = u = w = M 2 = O

Deformaaon of composite cylindncal panels m a gas flow 263

These conditions are satisfied if the displacements field is represented by the following series:

~x 2ztxl n~y u= ~ U'" c°s - - + U2" c°s ---a--) sin

azx 2a~X ) nzty V =~n VIn Sin ~a + V2. sin cos --if- (1 2)

The substitution of eqn (12) into the first two parts of eqn (2) and apply- hag the Galerkin procedure yields the following two independent sets of algebraic equations:

kl Or. + k2 I;'ln = k3 ~VIn + Tl (13)

k20I. + k, 171. = k s ~-V I. + T2

and

where

k~ 02. + k~ ;',. = k~ ~v,_. + r3

kTfJ2n +kgV2n--klo~V2n + T 4 (14)

(U,,,, 12l, ,, U2., 172,,) =(U,., V,., U2., V2.)/h (15)

are the nondimensional amplitudes of in-surface displacements. The nondimensional coefficients k, in eqns (1 3) and (1 4) are

k I =(n/~)2An +(nat;t/~)ZA66; k 2 = n2(az/~)2(.,~ 12 +A66); k3 = atdl~2Ai2

k 4 =(~/~)2A66 +(n~Ah)2A22; ks =nzt2~hZAz2; k6 =(2nh)2A. +(n~2/~)2A66 (16)

k7 =2k2; ks =2k3; k9 =(2~rh)2A66 +(n:rt,;th)2A22; kl0 - k 5

where

d=a]R, h = h ]a, ,~ =a/b (17)

and ft,/ ---A~/Erh (18)

264 V Btrman, C W Bert, L Ehshakoff

E r being the transverse-direction modulus of elasticity of a layer. The nondimensional thermal terms in eqns (13) and (14) are given by the following relations:

Tt = 8/~{2(a2 a2) [bol l (n)+ b,bfz(n) + b2bZf3(n)] t ,

+ 2(a I a + a2 a2) [b t bfl(n ) + 2b2b2f2(n)]/3a}/z~ 2

T, = - 8h{ Xf1212a o + a I a + a2a2( 1 - 4/rt2)] [ b2b2 f~( n)] (19)

+/~3( - a, a + a2a'-) [ b , bfa( n) + b2b2 f5( n)]}/zr

T 3 = - 2(2/~ 2) h(a_, a-')[b I bf~(n) + 2b2b2f2(n)] t3

T~ =4h{2f12( a I a + a2a 2) [ b2b2 fa( n)] +/~3(a2 a2) [b t bf4( n ) + b2b2 fs( n )]}/ ~

where

(/3,, /32, /33)-- (fl,, f12, f l , ) ( T o / E r h ) (20)

are nondimensional parameters, and

f ,( n) =(1 - c o s nrt )/n:r

f2(n) = - ( cos nzr)/n (21)

f~(n) = -2/(nz~) 3 - [ 1 / ( n ~ ) - 2/(nzl) 3] cos n:r

f~(n) =(cos n:r - 1)/( n~)2; fs( n) - 2 cos nzc ]( nzl ) ~-

Note that the products a~a, a2 a2, b t b and b2 b2 a r e nondimensional parameters.

From eqns (13) and (14) the nondimensional in-surface amplitudes can be expressed in terms of 1~ t. and 1~2,,:

gr t. = S~ I~,, + S.

I7"I" =$3 l~'l" +St (22)

O~.=ss fv2. + s~

f/2. =sT Iv,_. + s8

where

S~ =(k3k4 - k2ks ) ] ( k~ k4 - k~)

S: --(k~ r , - k~ 7",_)/(k, k, - k~)

S 3 --(k~ k 5 - k 2 k 3 ) ] ( k I k4 - k~)

s~ =(k , r2 - k,_ T , ) / ( k , k~ - k~)

Deformanon of composue cyhndrtcal panels m a gas flow 265

S 5 =(k8k 9 - k7klo) / (k 6 k 9 - k7) (23)

& ~(k 9T 3 - k 7T4)/(k 6k9-k72)

$7 =(k~k.., - kTks)/( k6k9 - k ?~)

$8 =(k6 T, - k7/'3)/( k6k9 - kT)

The substitution of eqns (4) and (12) into the third part of eqn (2) and applying again the Galerkin procedure yields the set of differential equations:

( # h 2 / E r ) ~/In '}- kll Oln + k12 ~Trln + k13 Win = k14 ~/2n/~ (24)

(ph2/ET) ~/2n + k15 02n + k16 l'~2n + k17 I4/2n =k18 Vl/ln/~

In eqn (24)

/~t = M2/M, FMT~-- 1 (25)

The nondimensional coefficients are given by

k . = - ztdh2 A t2 = - k3

k12 = - n~,~d]~2/~22 = _ k 5

k13 =(~h)4[/)ll + 2(/)12 +2/)66)(n).) 2 +/)2z(n;t) ~]

+(dh)2A22 +/9, +(n2)2/92 (26)

k,4 =(8/3) h ( p a c 2 / E r )

kt5 = 2 k . ; k16 =k12 kit =(~rh )4[16/)u + 8(/)12 + 2/)66 ) (nil.)2 + O22(n; t )4]

+ ( d h ) z d z 2 +4/9, +(n,~.)2/92

kl8 = _ kl4

where

and

/)~j = D , f l E r h 3 (27)

(a) 2 29,= N , h / E r (i= 1,2) (28)

is the nondimensional axial load.

266 V Btrman, C. W Bert, I Ehshakoff

Equation (24) can be written in terms of radial deflections only:

( P h " / E r ) W1. + g l Wl,, +g., = kt4/~14/,.n

( P h Z / E r ) i f2 . + R3 W2. + R , = kls/QW,.

w h e r e

(29)

Rl = k , S t +kl2S 3 +kl3

R2 = kitS2 + kl2S4 (30)

R 3 = k t s S 5 + kl6S 7 + kit

R 4--kl5S 6 +kl6S 8

The solution of eqn (29) consists of static and dynamic parts:

17I/I . = 171~1" + ITV~. sin w t (31)

I~_.. = I~2 . + W~. sin tot

where to is the frequency of harmonic vibrations of the fluttering panel. The substitution of eqn (31) into eqn (29) yields two independent sets

of algebraic equations. The solution of the set including the static deflections is

1712~. = (Rz + k,~R4/R3)

( k~4kts/R3) ffl'- - R t (32)

l~2. = ( k,8.1Vl /R3) I;V~. - R 4 / R a

The set including the dynamic deformations can be written in the form:

( R ~ - 0 ) 2 ) - , - - , W l,, - kt4 M W 2 , = O (33)

_ klsif41;V~" +(R3 _0)2) 1~.-- 0

where

0)2 = ph to2 / E r (34)

The Mach number parameter M corresponding to the flutter oscillations of the panel is obtained from eqn (32) as

.M= ,j(/~] _ o5 2)(R3 - 0) 2)/(kl 4 k, s) (3 5)

Note that (kl4kl8) < 0. Therefore, the frequency of motion of the flutter- ing panel is in the interval

min(R t , R3)< 0) 2 < max(Rt, R3) (36)

Deformanon of compostte cylmdncal panels m a gas flow 267

The flutter velocity parameter corresponding to the minimum value of flight velocity at which flutter becomes unavoidable is given by

N/.= JR, -R31 (37) 2k14

It can be concluded that the panel may have two types of deformations. Static deformations given by eqn (32) exist if the distribution of tempera- ture is nonuniform. If the temperature is uniformly distributed over the platform of the panel, these deformations vanish within the Galerkin- type analysis. Panel flutter can be superimposed on static deformations at the Mach number given by eqn (35). In the numerical example, attention is concentrated on static deflecuons of the panel at the center (see eqn (32)):

17V= E l;V% sin nrt/2 (38) /7

which is the largest deflection.

3 NUMERICAL EXAMPLES

The cylindrical panel considered in the numerical examples is assumed to be made from boron fiber/aluminum 7178-T6 matrix composite, ~ with room temperature properties: EL=214 GPa, Er--131 GPa, Gt`r-- 44 GPa, vt.r= 0.255, at. = 9"2 pm/m K and a r = 18"2/xm/m K.

The calculations were carded out by assuming that To--371"C and the effect of high temperature on mechanical properties is uniform throughout the plate. This assumption imposes limits on the magnitude of ala, a2a, b~b, bE bE, which cannot be too large. Degradation of mechanical properties due to high temperature was calculated, based on data on the mechanical properties of boron/aluminum at T O --0* and 371"C in Ref. 14: EL = 154 GPa, Er--35"9 GPa and Gt`r = 11"7 GPa. The Poisson ratios and the coefficients of thermal expansion were assumed to be unaffected by temperature. Circumferential load N2 was zero in all examples. The panel was formed of 30 layers symmetrically oriented with the lamination scheme 0"+ 45* (two layers adjacent to the middle surface have the lamination angle 0").

The effect of the Mach number on static deflections of the panel appeared to be negligible. This can be explained by the analysis of the coefficient pac2/Er in k14 and k~8 which has an order of 10 -6. There-

268 V Btrman, C W Bert, I Ehshakoff

fore, staUc deformations are due to aerodynamic heating and only to a negligible degree to aerodynamic pressure. Note that in geometrically nonlinear problems this conclusion may be invalid. In the following examples, the Mach number was always taken as M-- 4.

As can be seen from the solution, temperature change causes trans- verse deflections only if it is nonuniformly distributed over the platform. In the examples, it was supposed that a 0--b0 =1, b~b =b2b 2= 1, i.e. temperature was nonuniform in the circumferential direction. The values of a I a and a2a were taken equal to either zero or 0.1.

The effect of the panel central (included) angle on static deflections of cylindrical panels is shown in Figs 2 and 3. Deflections are always larger in narrower panels; they decrease at larger panel angles (except for the case of very thick panels as illustrated by curve 3 in Fig. 2) and converge to a constant value. This indicates that wide panels behave as closed circular shells of the same geometry. Similar conclusions were obtained by Sobel and coworkers in problems of static buckling of isotropic panels.i 5,16 Physically, the decrease of the deflection in wider panels carl be explained by the fact that the temperature is 'more uniformly' dis- tributed in such panels if b~b - - b 2 b-" = a constant. Apparently, this affects the magnitude of deflections more than the tendency of wider plates to have large deformations due to the lesser effect of the boundary conditions along the straight edges.

The influence of the length-to-radius ratio is illustrated in Figs 4 and 5. Deflections of the panels tend to approach zero if their radius increases (small d). The explanation for this fact is that if the radius increases, the panel approaches a flat plate, which has a very small deflection due to nonuniform temperature distribution.

The effect of axial compression is shown in Fig. 6. The deflection curve in all cases considered consists of two branches. At certain values N~/Ncr < 1, the deflection approaches infinity, i.e. the panel becomes unstable. The right branch can be reached only if the temperature is applied after the application of compressive load, provided that in- stability in the transient regime is avoided.

The estimation of the influence of temperature on deflections is shown in Figs 7 and 8. The calculations for these figures were carried out by assuming that the mechanical properties of the material remain constant within the range of temperatures considered. The effect of T O on deflections of relatively thick panels is not very large. However, deflections of thin panels increase at a much higher rate with increase in temperature (curve I in Figs 7 and 8).

Finally, the influence of spatial distribution of temperature is shown in Fig. 9. The nonuniform distribution of temperature along the straight

Deformanon of composue cyhndncal panels ,n a gas flow 269

I0:

lOW

o s

3 f

o J o ° do ° 9 b ° ,~o ° Wo ° A NGLE

Fig. 2. Effect of panel included angle on stauc deflection (d=0"5, ala--a2a 2= 1; N t =0). Curves 1, 2 and 3 correspond to dtmensionless thmknesses/~ of 0 01, 0-02 and

0.05, respecuvely

tO

tOW

05

£o ° go* go* o , ~ o ° , g o °

ANGLE

Fig. 3. Effect of panel included angle on static deflection (d=0.5; ala=a2a2=O; N~ =0). Curves 1 and 2 correspond to dimensionless thmknesses h of 0.01 and 0 02,

respectively.

o o~ ,b ,# 5

Fig. 4. Effect of length-to-radius ratio on static deflection (/~=0-01; 2=2 ; Ntffi0). Curves 1 and 2 correspond to values of

at a = a2 a 2 of 0 and 0.1, respectively.

270 V Btrman, C W Bert. 1 Ehshakoff

Fig. 5. Effect of length-to-radms ratio on stanc deflect!on ( /~=002, 2 = 2 , N t = 0 ) Curves 1 and 2 correspond to values of

at a =a,a'- of 0 and 0-1, respectively

0 75

~0 ~/

0 50

0 2 5

I

0 05 I 0 15 0

Fig. 6. Effect of dimensionless axial com- pression on static deflecnon (2 = 2). Curves 1 and 2 correspond to a,a =a2a 2 values of 0 and 0-I, with/~ = 0 01 for both; curves 3 and 4 correspond to a[a =a2 a2 values of 0 and

0.1. with h = 0-02 for both.

IO

!

0 5 4

0 5 ! 0 ,,.,N'/"cr

- 0 5

Fig. 7. Effect of temperature on static deflection (d = 0.5, 2 =2 ; ala =aza2=O; N l =0). Curves 1, 2 and 3 correspond to dimensionless thicknesses of 0-010,

0.015 and 0.020, respectively

m

w

O.4

0 2 2

o , ~ 0 soo 660 zbo 860 T*

Deformation of compostte cyhndrical panels m a gas f l ow 271

03 W

02

J 3

0 500 600 700 800 T*

Fig. 8. Effect of temperature on stattc deflectton (d=0.05, 2--2; a t a = a , a 2 = O . 1 ; N I =0). Curves 1, 2 and 3 correspond to dtmensionless thicknesses of

0.010, 0.015 and 0.020, respectwely.

0 3 w

02

o!

o too[, I 2 3 4 5

Fig. 9. Effect of spatial distribuuon of tem- perature on static deflection (d=0-5; 2--2; N t --0). Curves 1 and 2 correspond to values

of a ia = a 2az of 0 and 0.1, respectively.

edges does not change the character of the deflection versus relative thickness /~ relationship. However, if bl b -- b 2b2 _---0 while ala --a2a2--0"l, the magnitude of 1~ is of the order of 10 -3 only (this curve is not shown in Fig. 9). This means that significant static deflec- tions can exist only if the temperature is nonuniformly distributed along the curved edges.

4 CONCLUSIONS

Deformations of nonuniformly heated composite cylindrical panels in a supersonic gas flow have been considered. It was found that static deflec- tions can be considerable if temperature is nonuniformly distributed over the planform. The effect of aerodynamic pressure on static deforma- tions is negligible compared with the effect of nonuniform aerodynamic heating at high velocities of flight. Significant static deformations can be reached only if temperature is nonuniformly distributed along the curved edges.

272 v. Btrman. C. W Bert, 1 Eltshakoff

As the panel angle ts increased, the static deformations approach a constant value, which is equal to deformation of a circular cylindrical shell of the same geometry. The deformations are much larger in curved panels than in the shallower panels. There exists, on heating, a critical value of the axial compressive load, which ~s usually much smaller than the classical buckling load calculated in the absence of heating. The deflection of the panel increases as the temperature is increased. This increase is more pronounced in thinner panels.

REFERENCES

1. Ambartsumaan, S. A., General Theory of Antsotroptc Shells. Nauka, Moscow, 1974, pp. 430-40. (In Russian.)

2. Llbrescu, L., Elastostatics and Kmencs of Anisotroptc and Heterogeneous Shell-Type Structures. Noordhoff, Leyden, The Netherlands, 1975.

3. Kosichenko, A. A., Pochtman, Yu. M. & Rikards, R. B., Design of composite cylindrical shells of minimum weight in supersomc gas flow. Soviet Aeronautics, 23 (1980) 49-52.

4. Dixon, S. C., Experimental investigation at Mach number 3"0 of effects of thermal stresses and buckling on flutter characteristics of fiat single-bay panels of length-width ratio 0"96 NASA TN D-1485, 1962.

5. Garrick, I. E., A survey of aerothermoelastlcity. Aerospace Engng, 22(1) (1963) 140-7.

6. Novichkov, Yu. N., Nonstationary flutter of cylindncal panels. Proc. 4th All- Union Conf. Theory of Shells and Plates, Erevan, 1964. (In Russian.)

7. Schaeffer, H. G. & Walter, L. H. Jr, Flutter of a flat panel subjected to a nonlinear temperature distribution. A/AA J., 3 (1965) 1918-23.

8. Bolotin, V. V. & Novichkov, Yu. N., Buckling and steady flutter of heat- compressed panels in a supersonic flow, NASA TT F-10106, 1966.

9. Soong, T.-C., Thermoelastic effect on delta-wing flutter with arbitrary stiffness orientation. J. Aircraft, 9 (1972) 835-43.

10. Bert, C. W. & Reddy, V. S., Cylindrical shells of bimodulus composite material. J. EngngMech. Div., ASCE, 108 (1982) 675-88.

11. Ackeret, J., Luftkr~ifte an Fliageln, die mat gr6sserer als Schallgesch- windigkeit bewegt werden. Zeitschrfitfiir Flugtech. Motorluftsch., 16 (1925) 72-4; Ackeret, J., Air forces on airfoils moving faster than sound. NACA TM 317, 1925.

12. Almroth, B. O., Influence of edge conditions on the stability of axially- compressed cylindrical shells. A/AA J., 4 (1966) 134-40.

13. Hoff, N. J., Buckling of axially compressed circular cylindrical shells at stresses smaller than the classical value. ASME J. Appl. Mech., 32 (1965) 542-6.

14. Advanced Composite Design Guide, Vol. IV, 3rd edn. Structures Division, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, OH, 1973.

Deformanon of composzte cyhndncal panels zn a gas flow 273

15. Sobel, L. H., Weller, T. & Agarwal, B. L., Buckling of cyhndrical panels under axial compression. Computers and Structures, 6 (1976) 29-35.

16. Weller, T., Combined effects of in-plane boundary conditions and stiffening on buckling of eccentrically stringer-stiffened cylindrical panels. Computers and Structures, 14 (1981) 427-42.