Thermal buckling of shallow shells

hr. .I. Solids Strucrures, 1966, Vol. 2, pp. 167 to 180. Pergamon Press Ltd. Printed in Great Britain

THERMAL BUCKLING

OF SHALLOW SHELLS

M. A. MAHAYNI Department of Mechanics and Aerospace Engineering, The University of Kansas, Lawrence, Kansas

Abstract-The large deflection equations for a thin shallow shell are modified to include thermal effects. The temperature is considered to be an arbitrary function of the space coordinates. The resulting equations are used to study thermal buckling and post-buckling behavior of a simply supported cylindrical shell panel, subjected to a parabolic temperature distribution along its axial direction. It is assumed that the temperature is constant across the thickness of the shell and along its circumferential direction. Buckling charts for various curvatures, KY, and aspect ratios, I, are presented for two cases of edge conditions. In one case the edges of the panel are free to displace, and in the second case the edges are restrained. The critical temperature is found to fall outside the practical temperature range of the material for KY 2 200 in the free-edge condition and for K, 2 50 in the restrained-edge condition.

NOTATION A”

a, b

at,h c D

undermined coefficients in equation (9) panel dimensions functions of the parameters 1, K,, p, K, P, and P, defined in the Appendix coefficient of thermal expansion Eh” 12(1-/la)

E,P

elastic constants of the shell

e

e,, _ ep _

T/x=0 average displacements in the x and y directions

iif*

TI,=, functions of the parameters ,I, K,, p, K defined in the Appendix

e,, ey

f

cfa’ hZ

h

thickness of the shell

k

AT 2=7

k k,

curvatures in x and y direction

K,

k$= -=h

K

a2b2 -pk

MT

h/2 Tz dz CE s-b/2

NT

cE

e-f

b’ Rh

b/Z Tdz

I-L,Z

167

168

M. A.

PX

ab2 Eh*

PY

-Da2 Eh’

Q

Ii T

4 0, w

Z”

MAHAYNI

error function intensity of lateral load radius of curvature temperature function displacement components in the x, y, z directions respectively A” h

Airy stress function average in-plane forces at the edges of the panel

INTRODUCTION BECAUSE thermal stresses are assuming major importance in aerospace structures, their determination has attracted widespread interest. Numerous references on thermal stability of shells are reviewed by Anderson [l]. Most of the existing work is concerned with the effect of thermal stresses on the stability of cylinders and cones [2-61. The present investigation considers the stability of a simply supported, shallow, cylindrical shell panel subjected to non-uniform temperature distribution. The temperature varies along the axial direction of the shell only. Since the post-buckling behavior of the shell is of primary interest in this investigation, the governing differential equations are based on the non-linear theory. An exact solution of the resulting non-linear problem is not possible. However, an approximate solution is obtained by applying the Galerkin’s method Numerical results for maximum temperature differences and displacements are determined for two cases of edge conditions. The influence of the curvature parameter and the aspect ratio on the critical temperature is investigated.

DIFFERENTIAL

EQUATIONS

Let x and y be measured along the axial and the circumferential direction in the middle surface of the undeformed shallow shell, Fig 1. With u, u and w as the displacement components of a point in the middle surface, the internal forces can be represented in the following forms, including displacement terms up to second order,

s h/2

N,

=

a,dz = $+[;-k,w+;@

-h/2

+,{$-k,,w+;@$l]-

5

h/2 N, =

a,dz = $[;-k,,w+;($)’ s

-h/2

(14

169

Thermal buckling of shallow shells

Nx dy

FIG. 1. Geometry of the shell.

+p{&~xW+;(gy}] h/2 Nxy = s -h/2

-5

1

Eh a~ au aw aw Txydz = 2(1 +p) [ &+ax+Z’ay

and M,=

-D($+p$)-

f$

My = -D($+p$)

Mxy = U-PP-

(lb)

-3

azw axay

where D= 2

Eh3

12(1-p2)’

W

N, = cE

h/2

Tdz; s -h/2

Tz dz.

M,=cE s -h/2

170

M.

A.

MAHAYNI

The equilibrium equations of an element of the shell are

aNx aNxY _

()

XC+ ay aN “Y+aN,=O ay

ax

aM >+

aM _.-_?-

aM

aM

ax

ay

ax

ay

--??+~__Q,

Q,

=

0

for the change of curvatures

2 xy= dw

azw

XX=p

(2)

= 0

~+~+N,(kx+$,)+N,~,+$)+2N,~+q

where the following simplified expressions twist of the middle surface are used

= 0

and

aY2

and the unit

a%

xXy =axay*

(3)

The first two equilibrium equations are satisfied identically if N x

NC!.?

=!f!!.

ay2 9

y

ax2

and

N

XY

=--!f!!k

(4)

axay

where 4 is the Airy stress function. Eliminating the variables u and u in equations (la) and (4), the compatibility equation of the shell is obtained

a+ __a% ._4-x!?_,y$_&V2,,. ax2 ay2 ay2

(5)

Expressing the shear forces Q, and Q, by the bending moments and taking into account the last three relations (lb) and expressions (4), we obtain from the last equation of group (2) the differential equation I)v?w

a2w a24 a% a%p ax2 ay2 ay2 ax2

&“MT.

= _._+-.--2

(6)

For the cylindrical shell panel considered, k, = 0 and k, = l/R, the governing non-linear equations reduce to

ah a2W i .__&‘“‘N, a2w a2

ay2

R

ax2

Eh

1

= ()

1

-- (1_P)V2M,

(74

1

= 0. (7b)

171

Thermal buckling of shallow shells

SOLUTION

OF THE PROBLEM

Consider a thin cylindrical shell panel, occupying the space b -py<-

-a
h

h

b

and -z’z’z 2 22’ The shell is free of transverse load and is subjected to a temperature distribution governed by the following equation T = [f+k(x-a)21

(8)

where

k=~fdy AT = Temperature

difference

f = T/x=, e = Tlx=,, The edges of the shell are restrained in the transverse direction by simple rigid supports. A solution of equations (7) which satisfies the boundary conditions is presented in the form of the trignometric series n

w=

=

co

c

A,$,

(9)

n=1,2.3...

where A, are undetermined coefficients. For a first-order solution only the first term of equation (9) is used The resulting differential equation for t$ becomes

2

+f

0 ;

A,co+o+ck.

The solution of eov: :ion (10) is

(11) cry2 /3x2

Ehck

where a and /I are the in-plane forces N, and N, at the edges of the panel

4

172

M. A.

The Galerkin method In general, the left hand resulting function on the undetermined coefficient

MAHAYNI

prescribes substituting equations (9) and (11) into equation (7b). side of equation (7b) will not be zero after this operation The left hand side is the error function Q. To minimize the error the of the deflection w must satisfy the integral QY, dx dy = 0.

After carrying out the integration following dimensionless quantities,

of equation (12) simplifying, and introducing

K&’

the

2, =+r

/i =;,

a2b2 KC-.-..h2 cky

(12)

(13) P, = $

P, = 5

and

a cubic equation in 2, is obtained

&($+“2)z:pY(i+;,l)“+&~j z:+[gG(n+:,i)2 2

7c6

() ‘i-1

+ 192(1 -,U2) /1

-$(P,+P&

(

G-t

(14)

nZR z, )I

= 0.

To check the convergence of the series equation (9), the second-order solution is obtained and compared to the first-order solution, Fig. 2. In this case the first two terms of equation (9) are considered. The resulting equation for 4 becomes

-cos~cos~~cos%os-a

+ Ai

a

b

27rx 2XY cos~cos-cos%os b

a

271Y

a

b

271x . 22ZY sm b - 4 cos a cos2F cos b271Ycos’x co~~cos~+2A~cos~cos~~ a

The solution of equation (15) is

a

1

-ck.

-

b

(15)

173

Thermal buckling of shallow shells

3

cos y

b

cos

ZE a

1

cos3xycos3Kx b

a

nx

7ry

1 CoS;CoSb

16000

12000

0000 i?=+l 4000

0

-4000

-0000 -40

FIG. 2. Load-deflection

-20

0

20

I

I

I

I

/ I

I I

i!Yl!k 40

60

curve for p = 0.3, I = 1 and P, = Py= 0.

M. A. MAHAYNI

114

The integration of equation (7b), using the first two terms in equation (9) for w and equation (16) for 4, leads with simplification to the following simultaneous cubic equations in non-dimensional form a,z:+a,z~z,+a,z,z, b,z: + b,z:z,

+ b,z,z,

+a,z,z:+a~z:+aszf+a,z~+a,z, + b,z,z:

+a,~,

= f,(AT)

+ b,z: -t b,z; + b,z: i- b,z, + b,z,

in which the coefficients Ui, bi, fi and ft are functions of the parameters P, and P,,, defined in the Appendix. 2, = A,/h and Z, = AZ/h. NUMERICAL

(17)

= fi(A\)

,J, K,, ,u, R,

RESULTS

(a) The edges of the panel are free to displace in its plane

For this case the end forces P, and P, are equal to zero. Hence equation (14) reduces to

The behavior of the shell is best described by plotting graphs of deflection in terms of the maximum temperature difference for various values of the curvature K, and aspect ratio 1. The results are shown in Figs. 2 and 3. The convergence of the series is checked by solving equations (17) for the case where R.= 1 and K, = 100. The results are plotted in Fig 2 for comparison. (b) The edges of the panel are restrained from displacement in its plane

The average displacement of the middle surface in the x-direction is e,=

--

1 +a/22 dx u s -012 (3x *

(19)

This average varies in the y-direction. Its average over the y-coordinate

is

b/2

1 fjX= e, dy. b s -b/2

However

(20)

The substitution of the values of the displacement w and stress function 4 in equation (21) and integrating in accordance with equations (19) and (20) yield in non-dimensional form

0= U2

2X ; where

P,i2-@Py+$z:

+$C~z,(12-_p).

1 (l/U+@

-5R24

[

f+i$

1

(22)

Thermal buckling of shallow shells

175

20000 ,-

16000 /

12000 , -

0000

R:*ck

\

4000

0

-4000

-40

FIG. 3. Load-deflection

-20

0

20

40

60

curve for p = 0.3, K, = 100 and P, = P, = 0.

Similarly, the average displacement in the y-direction is _ b= eY

f+Kf [

i; 0

For the panel with restrained y-coordinates must vanish.

edges, the average displacement

1

.(23)

along the x and

e, = SY= 0. The solution of equations (22) and (23) for this condition yields

(24)

176

M. A.

MAHAYNI

The governing equation for the panel with restrained edges is finally obtained by substituting P, and P,, in equation (14). The resulting equation is

+KK,,A’

5-1nJ (24

+K

Y&2&+4

(3+9

=”

14000

12000

10000

0000 K-

I

6000

II

2000

0

-2000

-8

-4

FIG. 4. Load-deflection

0

4 z,= -Wmox h

6

12

16

curve for ; = O-3, L = 1, f = 0 and 0, = I, = 0.

177

Thermal buckling of shallow shells 10000

I I

I I I

0000

I I I

I

6000

FIG 5.Load-deflection

X=I

I I I

curve for p = 0.3, K, = 30, f = 0 and 8, = 8, = 0.

The behavior of the shell with restrained edges is studied by plotting the above equation for various values of the curvature K, and aspect ratio il. The resulting graphs are shown in Figs. 4 and 5.

CONCLUSIONS The buckling temperature of a thin shallow cylindrical shell panel is found to be a function of the curvature parameter, the dimensions of the shell, and the degree of restraint of the edges. The graphs obtained show that if the shell is subjected to a temperature difference, which is larger than the critical value, snapbuckling may occur once sufficient perturbations are present to overcome the energy barrier separating the stable equilibrium states. From the results of the analysis, the following conclusions can be drawn: (a) For the shell, with the edges free to displace in the middle surface, thermal buckling may occur in the practical temperature range even for moderately high curvature parameter; whereas for the shell with restrained edges buckling may occur in the practical temperature range only for small curvature parameters. (b) The value of the maximum deflection W,,, which corresponds to the critical temperature increases with increased curvature parameter.

178

M. A. MAHAYNI

(c) For the free-edge case the critical temperature difference value increases slightly for higher aspect ratios; whereas for the shell with restrained edges the critical value decreases considerably when the value of the aspect ratio is doubled. This decrease is due to the larger influence of the in-plane edge forces. (d) The use of the first-order solution is quite sufficient for engineering purposes. Acknbwledgment-This work was supported in part by the National Aeronautics and Space Administration under Research Grant Ns G-298-62. The author expresses his gratitude to Dr. H. Zorski of The Institute for Fundamental Engineering Research, Polish Academy of Sciences, for suggesting the problem.

REFERENCES [l] M. S. ANDERSON,NASA tech. Note, D-1510, pp. 255-266, Dec. (1962). [2] N. J. HOFF, J. appl. Mech. 24, 405 (1956). [3] N. J. HOFF, J. R. Aeronaut. Sot. 61, 756 (1957). [4] D. AMR and S. V. NARW, J. Aeronaut. Sci. 26, 803 (1959). [5] D. J. JOHNS,D. S. HOUGHTONand J. P. H. W~BBER,COIL Aero. Cranfield, Report (61 J. SINGER and N. J. HOFF, AFOSR, TR 59-203, p. 82, December (1959).

147, p. 29, May (1961).

APPENDIX The values of the coeficients ai, bi, fi and fi

The following coefficients are used in equations (17), for the case where P, = PY= 0.

Thermal buckling of shallow shells

179

(Received 4 March 1965)

R&JO&--Les equations de flexion large dune enveloppe mince peu profonde y sent modifiees pour inclure les effets thermiques. La temperature est consider&e en tant que fonction arbitraire des coordonn& dans l’espace. Les equations qui en resuhent sont employees pour ttudier le complement du gondolement thermique et l’etat de l’apr~s-stability d’un panneau d’enveloppe ~ylind~que a support simple et soumis a une distribution parabolique de la temperature le long de sa direction axiale. L’on a assume que la temperature est constante A travers I’tpaisseur de i’enveloppe et le long de sa direction circonferencielle. Des diagrammes de stab&e pour diverses courbures, 4 ainsi que des rapports d’aspect 1, sont prisentCs darts deux cas de conditions de rebord. Dans l’un des cas, les rebords du panneau sont libres de se dtplacer, et dans k second les rebords sont retenus. L’on a trouve que la temperature critique baisse en dehors de la gamme pratique de temperature de la mat&e pour K,, 1 200 dans les conditions de rebords libres, et pour Ky Z 50 dans lea conditions de rebords retenus.

180

M. A.

MAHAYNI

Zusammenfasaung-Die grossen Biegungsgleichungen fur eine dtinne flache Schale sind abgelndert urn Wlrmeeinwirkungen einzuschliessen. Die Temperatur ist als eine willkiirliche Funktion der Raumkoordinaten erwogen. Die resultierenden Gleichungen werden ftir die Untersuchung von Warmeknickfestigkeit und NachKnickfestigkeitsverhalten eines einfach untersttitzten zylindrischen Schalen sttiches, welches zu einer parabolischen Temperaturverteilung entlang seiner Achsrichtung unterworfen ist, verwendet. Es wird angenommen, dass die Temperatur, quer durch die Dicke der Schale und entlang ihrer Umkreisrichtung konstant ist. Knickfestigkeitstabellen fur verschiedene Krtimmungen KY und Streckungsverhlltnissen, 1’ fur zwei Falle von Randbedingungen sind prlsentiert. In einem Fall, die Rlnder der Schale sind frei fur Verschiebung und im zweiten Fall werden die Rander zurtickgehalten. Es wurde gefunden, dass die kritische Temperatur ausserhalb des praktischen Temperaturbereiches des Materiales fallt fur KY t 200 im Falle der freien-Randbedingung und fur KY > 50 im Falle der zuriickgehaltenen Randbedingung.

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