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STRESSES IN AN ORTHOTROPIC ELASTIC CYLINDER DUETO A PLANE TEMPERATURE DISTRIBUTION T(r, θ)M. A. Kalam a & T. R. Tauchert aa Department of Engineering Mechanics, University of Kentucky, Lexington, KY, 40506Published online: 05 Apr 2007.

To cite this article: M. A. Kalam & T. R. Tauchert (1978) STRESSES IN AN ORTHOTROPIC ELASTIC CYLINDER DUE TO A PLANETEMPERATURE DISTRIBUTION T(r, θ), Journal of Thermal Stresses, 1:1, 13-24, DOI: 10.1080/01495737808926927

To link to this article: http://dx.doi.org/10.1080/01495737808926927

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STRESSES IN AN OR THOTROPIC ELASTIC CYLINDERDUE TO A PLANE TEMPERATURE DISTRIBUTION tv. 0)

M. A. Kalam and T. R. TauchertDepartment of Engineering Mechanics

University of KentuckyLexington, KY 40506

Stresses in a hollow orthotropic elastic cylinder due to a steady-state planetemperature distribution T(r, (J) are analyzed for tractionfree boundary condi­tions. An exact solution is developed using the Airy stress function in aFourier-series form. Numerical results are given for a fiber-reinforced com­posite material.

INTRODUCTION

Exact solutions are available for the thermal stresses produced by a radial temperaturefield 1'(r) in a hollow circular cylinder of either isotropic (1) or orthotropic [2, 3)material. In the case of a plane temperature variation 1'(r, (J), Boley and Weiner [l ]have evaluated the stress distributions assuming isotropic behavior. More recentlyParida and Das [4) have considered a plane thermal-stress problem involving a solidcylindrically orthotropic disc. The present research treats a hollow orthotropic circularcylinder. Here the cylindrical surfaces are taken to be tractionfree and subject tononaxisymmetric thermal boundary conditions. The results are applicable in instancessuch as the exposure of circumferentially reinforced pipes and pressure vessels tothermal environments.

Following the approach used in preceding investigations [1, 4), the presentsolution is obtained using the Airy stress function in the form of a Fourier series. Thenecessary compatibility relations for the doubly connected region are developed in theappendix.

FORMULATION

According to the linear, uncoupled thermoelastic theory, the steady-state temperaturedistribution 1'(r, (J) in an orthotropic cylinder is assumed to satisfy the heat-conductionequation

(I)

Journal of Thermal Stresses, I: 13-24, 1978Copyright © 1978 by Hemisphere Publishing Corporation

0149-5739/78/0101-0013$2.25

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14 M. A. KALAM AND T. R. TAUCHERT

where K; and K9 are the coefficients of thermal conductivity in the r and /J directions,respectively.

. The stresses induced by this temperature will satisfy the two-dimensional equationsof equilibrium if they are related to the Airy stress function ct>(r, 0) as follows:

a 9 = _ ~ (.!.. act»r ar r ao (2)

For an orthotropic elastic material, the associated: strain components in the case ofplane stress are

e99 = Q12 ar r + Q22 a99 + a2 T

_ Q66 a-eer9 ---

2

(3)

where the ali are the elastic compliances, and the al are the coefficients of thermalexpansion. For the problem of plane strain, alj and al should be replaced, respectively,by

t, j =·1, 2 (4)

in which the subscript 3 denotes the z direction.Since the hollow cylinder is a doubly connected body, the strain components must

satisfy the compatibility condition

plus the Michell conditions at r = Q (see the appendix)

(2IT ~e + 2 aer 9 _ r ae9 9) r sin 0 d/J = 0In rr a/J aro

12 " ~ + 2 aer9 - r ae9 9) r cos /J d/J = 0rr ao ar

o

12 IT

~ + aer 9 ae99) d/J - 0. err - e99 ae - r -a;- -o

(6)

(7)

(8)

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STRESSES IN AN ORTHOTROPIC ELASTIC CYLINDER 15

f211 (, 3T)=-E 0 \HT- r 3r r sin I) ae (10)

{211 (-r 3

3<1> + A + C 3<1> + A + B 32<1>

_ C + D 33<1> ~rcos I) so

3r3 r 3r r2 31)2 r 3r 31)2o

= -E1211

(HT - r~~)r cos I) dl) (II)

{211 ( 3T)= -E 0 HT-T-ra; dl) (12)

where

(13)

For traction free surfaces at the outer radius r = b and the inner radius r = a, weget the following conditions in terms of <I>(r, I)) (see [11, p. 114):

At r = b:

At r = a:

3<1><1>=-=0

3r

<I> = aIr cos I) + a2r sin I) + bo

~~ = a I cos I) + a2 sin I)

(14)

(IS)

(16)

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16 M. A. KALAMAND T. R. TAUCHERT

where aI, az, and bo are arbitrary constants that are introduced to satisfy the Michellconditions, Eqs. (10)-(12).

SOLUTION

Considering a solution for TCr, 0) in the Fourier-series form

cc cc

T(r, 0) = L Fn(r) cos nO + L Gn(r) sin nOn==O n=1

we obtain from Eq, (I) the following equations for the coefficients Fn and Gn :

(17)

P ' zkzF,,+---.!! __n_ F =0

n r ,2 n

G' zkzG,,+-!!.-_n_ G =0

n r r2 n

n = 0, 1,2, ...

n=I,2, ...

(18)

(19)

where the primes denote differentiation with respect to r, and k Z = Ke/K,. Thesolutions to Eqs. (18) and (19) are

Fo = PI (0) In r + pz(O)

Fn = PI (n)r nk + Pz(n)r-nk n ;;. I(20)

(21)

in which the constants of integration p/O) , p/n), and Q/n) (i = 1,2;1 n = 1,2, ... ,00)are determined from the thermal boundary conditions. Substituting Eqs. (20) and (21)into (17) gives the desired temperature distribution:

ce

T(r, 0) = PI (0) In r + Pz (0) + Ln=l

ee

(PI (n)rnk +Pz(n)r-nk) cos nO + Ln=1

+ Qz (n)r -nk) sin nO (22)

Following the approach of Boley and Weiner [1], we now introduce the stressfunction

ce cc

<I>(r, 0) = L fn(r) cos ne + L gn(r) sin nOn=O n=1

(23)

Substituting Eq. (23) into Eq. (9) leads to the following equations for the coefficientsfn and s-:

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STRESSES IN AN ORTHOTROPIC ELASTIC CYLINDER 17

n=0,1,2, ... (24)

2 ~ ,) 2iv + 2 '" _ A + n B "_ gn + n (n2A - 2A - B)s; rs« ,2 gn r ,. s«

= -E (G" + 2- H G' _n2

H G)\ n r n r2 n

The boundary conditions (14)-(16) imply that

n = I, 2, . .. (25)

fo(a) = bo fl(a)=ala fn(a) = 0 n;;;'2

f~(a) = 0 f;(a) = a, f~(a) = 0 n;;;'2

gl (a) = as« gn(a) = 0 n;;;'2

g; (a) = a2 g~(a) = 0 n;;;'2

fn(b) = f~(b) = 0 n=0,1,2, ...

n=I,2, ...

(26)

(27)

(28)

(29)

(30)

(31)

whereas the Michell conditions (10)-(12) require that, at , = a,

,ro" + f~' - :if~ = E(HFo - Fo - ,F~),,t." - A : B~; - ~) = E(HF1 - ,F;)

,gt- A :B~; _~) =E(HG1 -,G;)

(32)

(33)

By solving Eqs. (24) and (25), subject to the boundary conditions (26)-(31) andthe Michell conditions (32) and (33), it is found that

cP(', 0) = j~l C/O),mi(o) + c, (0),2 In, + n~{~ C/n),mj(n) + Cp , (n),nk+2

+ C (n),2-nk) cos nO + ~ ({- D.(n),mi(n) + D (n),nk+2P, ~ ~ , PI

n= 1 i= 1 + Dp

, (n),2-n) sin nO (34)

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18 M. A. KALAM AND T. R. TAUCHERT

where

C (0) = E(H-I) P (0)

p 2(1 - A) •

C (n) = -Enk [nk + I - H(n/k + I)] P (n)P, nk(nk+2) [(nk+ I)Z -L] +M •

C (n) = -Enk [nk - I - H(n/k - I)] Pz(n)p, nk(nk - 2) [(nk - IF - L] + M

D (n) = -Enk [nk + I - H(n/k + I)] Q (n)P, nk(nk + 2) [(nk + IF -L] +M I

D (n) = -Enk [nk - I - H(n/k - I)] Qz (n)p, nk(nk - 2) [(nk - 1)2 - L] + M

and the m;(n) are the roots of the equation

m4- 4m 3 + (5 - L)mz + 2(L - I)m + M = 0

in which

(35)

(36)

(37)

The constants of integration C/O), C/n ) , and D/n ) are obtained from Eqs. (26)-(33).For n =0 there are five unknown constants, C/O) (i = I, 2, 3, 4) and ho, to bedetermined from the five equations involving 10' For n = I there are 10 constants,C/·), DP), Q., and Qz, to be determined from 10 equations involving 11 and g i - Andfor each n;;' 2 there exists a set of eight constants C/n ) and D/n ) to be obtainedfrom the eight boundary conditions involving In and Sn- These constants can beevaluated easily on a digital computer once the coefficients Cp (0), Cp , (n), . • . ,Dp , (n)

have been determined from Eqs. (35) for a particular choice of thermal boundaryconditions.

The components of stress, obtained by substituting Eqs. (34) into Eq. (2), are

Orr = c,(0) (I + 2 In ,) + it. mi(O)C/0),mi(0)-2 + n~! {it! [mien) - nZ]

• C/"),mi(n)-2 + (2 + nk - nZ) Cp , (n),nk + (2 - nk - nZ)Cp

, (n), _nk} cos n()

+ n~! {it! [mien) - nZ] D/n),mi(n)-2 + (2 + nk - nZ)Dp • (n),nk

+ (~ - nk -r- nZ)Dp , (n),-nk} sin n() (38)

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(38)(Cant.)

STRESSES IN AN ORTHOTROPIC ELASTIC CYLINDER 19

4 ~

000 = c, (0) (3 + 2 In r) + l m;(O) [m;(O) - I] C/O)rm;(0)-2 + li= 1 n= 1

.ttl m;(n) [m;(n) - I] C/n)r m ;(n )- 2 + (nk +2)(nk + I)Cp , (n)rnk

+ (2 - nk)(1 - nk)Cp , (n)r-nk} cos nO + n~1 ttl m;(n) [m;(n) - 1]

• D/n )r m ;(n )- 2 + (nk + 2)(nk + I)Dp , (n)r nk + (2 - nk)

. (I - nk)Dp , (n)r-nk} sin nO

orO = n~1 n ttl [m;(n) - I] C/n

)rm

;(n )- 2 + (I + nk)Cp , (n)rnk + (I - nk)

• Cp , (n)r-n k} sin nO - n~1 n ttl [m;(n) - I] D/n)r

m; (n)- 2

+ (I + nk)Dp, (n)r nk + (I - nk)Dp , (n)r- nk} cos nO

The preceding results can be reduced to the solution given in [I] for an isotropiccylinder. For isotropy, k =A =H= I and B = 2, in which case Eqs. (35) give Cp(O) =Cp , (n) =Cp , (n) =Dp , (n) = Dp , (n) =O. Furthermore, for n ~ 2, the differential equa­tions (24) and (25) are homogeneous with homogeneous boundary conditions (26)-(31).Hence In =gn = 0 for n ~ 2, and the resulting expression for ct>(r, 0) becomes

4

ct>(r, 0) = LC/O)rm;(O) + C/l)rm ; ( I ) cos 0 + D/!)rm;(I) sin 0 (39)

i= 1

where the m;(n) (n =0, 1) are the roots of Eq. (36) with

L = 1 + 2n 2 (40)

These results agree with those derived in [1].

NUMERICAL EXAMPLE

To illustrate the applicability of the solution given above, we consider the plane-straincase of a hollow cylinder with insulated outer surface and a nonaxisymmetrictemperature distribution along the inner surface; in particular, we let

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20 M. A. KALAM AND T. R. TAUCHERT

-Kr

aT(b, IJ) = 0ar

{

ToT(a, IJ) =

To - To sin IJ

(41)

(42)

PI (0) = 0

where To is a constant. For these thermal boundary conditions the constants ofintegration in Eq. (22) become

P2 (0 ) = ~ -~)To

P (n) _ fOI - 2To

1I(n2 - I lank [I + (b/a)2nk]

P2 (n) = ~ 0 2b 2nkTo

(1I(n2 - l)ank[I + (b/aFnk]

and

Q (I) _ -ToI - 2ak (1 + (b/a)2k]

Q, (I) = Q2(n) = 0

n = 1,3, ...

n = 2,4, ...

n = 1,3, ...

n = 2, 4, ...

n;;;'2

(43)

(44)

We consider now the particular case of a cylinder having a ratio of outer to innerradius b/a = 1.5, and a ratio of thermal conductivities k 2 =Ko/K, = 10. The tempera­ture distribution computed from Eq. (22), (43), and (44) are plotted as isothermallines in Fig. 1. Because of symmetry, only the region -rr/2';;; IJ .;;; 11/2 has' beenconsidered. The results shown are based upon the first 20 terms (n = 1, ... ,20) of theseries in Eq. (22); however, it was verified that inclusion of additional terms had noappreciable effect on the results. All the required computations were performed on anIBM System/370 Model 165 11.

For comparison, the isothermal lines for a cylinder with isotropic thermalproperties, k = 1, are also shown (dotted lines) in Fig. 1.

The thermoelastic constants selected for this example are

s, = 2.068 X 1010 Pa (3 X 106 psi)

Eo = 20.68 X 1010 Pa (30 X 106 psi)

E, = 2.068 X 1010 Pa (3 X 106 psi)

G,o = 1.034 X 1010 Pa (1.5 X 106 psi)

VOr == v,." = Vo, = 0.25

ex, = 51 X 1O-6 CC-I )

CXo = 8.4 X 1O-6('C- I)

ex, = 51 X 1O-6 ('c- ' )

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STRESSES IN AN ORTHOTROPIC ELASTIC CYLINDER 21

8= .!!...I 2

t

I8= - ~

Orthotropic

Isotropic

8=71"I 2

I+

8= - ;

__ Orthotropic____ Isotropic

!T8e 022 _'T - 0.0

02 0

8=0

Figure I Isothermal lines T/T o in a cylinderof orthotropic (Ke/Kr = 10) or.isotropic(Ko/Kr = I) material.

Figure 2 Lines of equal circumferential stress088Q~1/C1.; To in a cylinder of orthotropic orisotropic material.

Based upon these values, which are representative of a boron-epoxy fiber-reinforcedcomposite material, stress distributions were computed using Eqs. (35)-(38). Again itwas found that inclusion of only the first 20 terms in each series provided adequateconvergence to the solution. Figure 2 shows lines of constant circumferential stress(oooa;2/0.;To). Also shown are the corresponding stress levels for a cylinder ofisotropic material, for which

k=1

B; = Eo = s, = 2.068 X 1010 Pa (3 X 106 psi)

G = 0.827 X 1010 Pa (1.2 X 106 psi)

v = 0.25

0. = 36 X 1O-6(C- 1 )

It is seen that throughout most of the cylinder the stresses are considerably larger fororthotropy than for isotropy. The maximum stress occurs at the point (r =a,e= -rr12), where oooa;2/0.;To=-1.30 for the orthotropic material and -{).I33 forthe isotropic material.

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22 M. A. KALAM AND T. R. TAUCHERT

APPENDIX

To the authors' knowledge, the complete compatibility equations for multiply con­nected regions, written explicitly in terms of cylindrical coordinates r, IJ, and z, havenot appeared in the literature. However, for general curvilinear coordinates, thenecessary and sufficient conditions for a single-valued and continuous displacementfield are given by (5)

Elj,rs + Erstii - €js,ir - Elr,is = 9

f i( . -. Ism >k ) d>r - 0g elr "tIki "t ~ ers.m ~-

Cp

(45)

(46)

(47)

where ers = covariant strain tensorgi, gl = contravariant, covariant base vectors

"t 'sm. "tjkl = corresponding permutation tensorsand a comma denotes covariant differentiation. Also, the Cp (f3 = I, ... , N) are simpleclosed curves that surround single cavities, and ~k is the position vector to a point on~c~~. .

Conditions (45) alone are necessary and sufficient to ensure smoothness of thedisplaceinent field in a simply connected body; these equations are expressed incylindrical coordinates in (5). For multiply connected bodies, Eq. (47) ensures thatthe infinitesimal rotation vector is continuous at every point along the curve Cp, andEq. (46) then ensures that the displacement vector is single-valued. Equations (46) and(47) are referred to as the Michell conditions.

Following the covariant differentiation indicated in Eqs. (46) and (47), and theintroduction of physical strain components err, eOO, ezz, erO, erz, and eoz, one obtains

J{~~ - aerr + aerz) d + ~ _ aerO + aeo~ dlJ + ~ _ aerzerr z-- z -- r erO z - z -- r e z z -Cp ar ar az ar r az

+ aezz) d J. + ~(aerr aerO aerO + z aerz z \ dz a;:- z er ~ ao - erO - r ---a;:- - z az -; ao - ,eOZ) r

+ (e + aerO - r aeoo - z aeoo + !... +!.. aeoz\ dlJrr alJ ar az r erz r alJ j r

+ (aerz _ aeoz _ aeoz + z aezz) dJao z az r a;:- , alJ J eo (48)

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STRESSES IN AN OR THO TROPIC ELASTIC CYLINDER 23

+ E - r aErz) dr + (r aEre + E - r aEe~ rdOrz ar az Bz ar /

+ (r aErz+ Ezz - r aEzz\ dZJ ez} = 0 (48)\: az ar} (Cont.)

f {[(a;;e -~ a;;z +~ Eez) dr + ~ a;;e - a;;z - Erz) dO + (~:ez -~ a;~z) d~ e,c(J

+ [laE,z _ aEr~ dr + (r aEez _ r aEre) ae + (aEZZ_ aE,Z) dz1ee~ ar az} ar az ar az J

+ I{ I aE" _ 2 _ aEre) d + l _ + aEre _ OEeo) dOL\;:- ao ;:- e-e --a;:- r \" eee 00 r ar

+(1 aErz_ I _aEez)dJ } - 0 (49)rao r Eez ar z ez -

In the case of plane strain, Eqs. (48) and (49) reduce to

I ~(I aE" - 2 - aEre) d +-- -Ee - rr ao r r ar

C(J

(50)

(51)

For the doubly connected hollow cylinder a';; r';; b considered in this paper, the curveC. can be taken as a circle of radius a. Integrating Eq. (50) by parts and rewriting theequations in terms of fixed base vectors i, j and k yield the following conditions alongr = a:

f_ (E" + 2 oere - r oee~r sin 0 dO i + (E" + 2 aE,e - r aEee\ r cos 0 ae jl = 0L ao ar) ao ar ) ~

(52)

12 " ~ - + OEre - oeeeJ ae k - 0Err teO - r - -an aro

(53)

Equations (52) and (53) are the appropriate Michell conditions for the problem treatedhere; they are equivalent to the conditions (6)-(8) given previously.

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24 M. A. KALAM AND T. R. TAUCHERT

REFERENCES

I. B. A. Boley and J. H. Weiner, Theory of Thermal Stresses, Wiley, New York, 1960.2. J. Nowinski and W. Olszak, Thermal Stresses in Thick-Walled Anisotropic Cylinders, Arch. Mech.

Stosow., vol. 5, pp, 221-236, 1953 (in Polish).3. T. R. Tauchert, Thermal Stresses in an Orthotropic Cylinder Subject to a Radial, Steady-State

Temperature Field, Developments in Theoretical and Applied Mechanics, vol. 7, pp. I-II,Catholic University of America Press, Washington, D.C., 1974.

4. J. Parida and A. K. Das, Thermal Stresses in a Thin Circular Disc of Orthotropic Material Due toan Instantaneous Point Heat Source, Acta Mechanica, vol. 13, pp. 205-214, 1972.

5. N. N. Hsu and T. R. Tauchert, Conditions of Zero Thermal Stresses in Anisotropic MultiplyConnected Thermoelastic Bodies, Acta Mech., vol. 24, pp. 179-190, 1976.

Received September 8. 1977

Request reprints from M. A. Kalarn,

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