Article Review Thermal Buckling Plates Shells

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Thermal buck l ing of plates and shel lsEarl A ThorntonDept ofMecli, Aerospace and Nuclear E ngineeringUniversity of Virginia, Charlottesville VA 22903-2442This review discusses research on thermal buckling of plates and shells since the first work inthe 1950s. Elastic thermal buckling of metallic as well as composite plates and shells isdescribed. The role of material thermal properties on thickness and spatial temperaturegradients is demonstrated first. Then thermal buckling and postbuckling research for plates,shallow shells and curved panels, cylindrical and conical shells is presented. Analytical,computational and experimental studies are described. Governing equations and formulas forcritical buckling temperatures are presented for several practical applications. An assessmentof past research is made, and future research needs are highlighted.

1 . INTR ODUC TIONIn the fourth AIAA von Karman lecture, Nicholas J Hoff,Hoff (1967), describes the evolution of thin-walledaircraft structures from the early days of aviation to thehalcyon days of the Apollo program. A recurrent themethroughout his description of aerospace structures historyis the engineer grappling with buckling problems of thin-walled structures. An early paper of Professor Hoff, Hoff(1946), traces aerospace buckling problems back to thewood-steel-fabric biplane era where the Euler bucklingformula and beam-column theory were used to designstruts of biplane wings and wing spars subjected tocompressive loads. In the late 1920s solutions tobuckling problems played an important role in theadvancement of all-metal semi-monocoque construction.For several decades thereafter, engineers struggled withbuckling problems of metallic thin-walled plates andshells under mechanical loads. The results of their workled to safe and reliable aircraft that made commercialaviation a worldwide economic success. Indeed, thebasic construction ideas of riveted skin-stringer metallicstructures made possible by the buckling analyses of the1920s-1930s prevail today in modern commercialtransports.

The importance of thermal effects on aircraft beganafter World War II with supersonic flight. Aerodynamicheating from supersonic flight induced elevatedtemperatures that affected material selection andsignificantly altered structural design practices. Thermalgradients and restrained thermal expansion introducedthermal buckling for the first time. A recent paper,(Thornton, 1992), describes development of thermalstructures from the early days of supersonic flight to themore recent challenges presented by hypersonic flight.The development of the National Aerospace Plane

(NASP) presents engineers with their greatest thermalstructural challenges yet. New advanced metal matrixcomposites were developed for severe elevated temperature applications. Complex convectively-cooledthin-walled structures subjected to intense local heatingprovided unprecedented materials and structuralmodeling challenges. One example of such an extremeinstance of a thin-walled structure subjected to intenselocal heating is the problem of shock-shock interactions.As hypersonic vehicles accelerate at high speeds inthe atmosphere, shocks sweep across the vehicleinteracting with local shocks and boundary layers. Theseinteractions expose structural surfaces to severe localpressures and heat fluxes. The first known instance ofshock-shock interactions in high speed flight occurred onthe X-15 in 1967. In that flight, the intense local heatinginduced severe local damage on a supporting pylon for adummy ramjet engine, Thornton (1992). On the NASPthese interactions occur on leading edges of integratedengine structures which experience intense, highlylocalized aerothermal loads. Dechaumphai et al (1989)studied issues relevant to the thermal-structure responseof hydrogen cooled, super thermal-conducting leadingedges subject to intense aerodynamic heating. A thermo-viscoplastic analysis of hypersonic engine structures,Thornton et al (1990), showed similar trends with thin-walled convectively cooled structures subjected to highinternal pressures and intense local heating. Poleski et al(1992) describe a three-dimensional thermal structuralanalysis of a swept cowl leading edge subjected toskewed shock-shock interference heating. The analysisshows that due to the intense localized heating, the thinelastic leading edge experiences very large localcompressive stresses. The high level of thesecompressive stresses suggests the possibility of localizedinelastic behavior and/or local buckling. Clearly these

Transmitted by Associate Editor RBHetnarski.AS ME Reprint NoAMR133 $20Appl Mech Rev vol 46 , no 10, October 1993 485 1993American Society of Mechanical Engineers

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486 Appl Mech Rev vol 46, no 10, October 1993 Thornton: Thermal buckling of plates and shellsnew, complex problems present structural analysts withnew, unprecedented thermal-buckling challenges.This article describes research on thermal buckling ofplates and shells. Elastic thermal buckling is emphasizedalthough recent experience on NASP problems hasshown the importance of inelastic buckling. However,the literature available on elastic buckling is much moreextensive than for inelastic buckling, and an assessmentof these studies was given priority for this review.Thermal buckling of metallic as well as composite platesand shells is considered. Thermal buckling studies ofcomposites are much more recent than for metallics, andrecent progress with composites is described.

Temperature levels and their distributions have amajor role in thermal buckling. The article begins with abrief description of temperature distributions in thin-walled structures. Then past literature on thermalbuckling of plates and shells is described. In eachinstance, past research is assessed, and research needsare discussed. The article concludes with briefsummarizing remarks.

2. T E M P E R A T U R E S I N T H I N - W A L L E DS TR UC TUR ESThe determination of temperatures in a thin-walledstructure begins with conservation of energy. In the mostgeneral form with thermomechanical coupling, Allen(1991), the conservation of energy equation includesterms that represent the conversion of mechanical tothermal energy. In practical aerospace structural heattransfer, thermomechanical coupling is usuallyneglected. Then the conservation of energy equationinvolves temperature as the only dependent variable. Itmay be solved for the temperature distributionindependently from the structural problem.

i2.1 Structural Heatl TransferHeat transfer in thin-walled structures is conductiondominated, and the classical heat conduction equation isused:

where T(xp x2, xy t) is temperature, k. . are the components of the material's thermal conductivity tensor, pis the density, c is the specific heat, and Q is the internalheat generation rate per unit volume. The components ofthe thermal conductivity tensor and the specific heat aretemperature dependent. The heat conduction equation issolved subject to an initial condition and boundaryconditions on the structure's surface. The initialcondition specifies the temperature distribution at timezero. The boundary conditions may consist of specifiedsurface temperature, specified heat flow, convective heatexchange, and radiation heat exchange. These may bewritten as

TB =T, (x 1 ,x 2 ,x a , t ) on.Slq i n i = -


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Appl Mech Rev vo l 46, no 10, Oc tober 1993 Thorn t on : The rmal buck l ing of p la tes and shel ls 487

Table 1Thickness Temperature Gradients for Various Materials0.1 BTU/in2-s (16.4 W/cm2) h= 0.125" (0.318 cm)

123456789

10

MaterialGraph i te -EpoxyTi tan ium (MIL-T-9047)Carbon-CarbonStainless (17-8PH)Has te l loyXCopper (G3-Heat Treat )Graph i te -MagnesiumMagnesium (AZ80A-F)Graph i te -A luminumAluminum (6061-T6)

Thermal conductivi tyk(BTU/in-s-'F)'2 . 2 x 1 0 -5

1.1 x lO- 42 . 0 x 1 0 -42 . 2 x 1 0 -42 . 6 x 1 0 -45 . 6 x 1 0 -45 . 9 x 1 0 -41.0x10-31.1 x10- 32 . 2 x 1 0 - 3

(W/cm-'C)1.6x10-28 . 3 x 1 0 -21.5 x10- 11.7x10-11.9 x10- 14 . 2 x 1 0 - 14 . 4 x 1 0 - 17 . 6 x 1 0 -18 . 3 x 1 0 -11.7

Thermal DiffusivityK

(in2/s)1.5x10-45 . 3 x 1 0 -31.1 x10-27 . 4 x 1 0 - 36 . 7 x 1 0 -32 .1 x10 - 22 . 8 x 1 0 -26 . 3 x 1 0 -24 . 9 x 1 0 -29 . 9 x 1 0 -2

(cm2/s)9 . 9 x 1 0 -43 . 4 x 1 0 -27 . 2 x 1 0 -24 . 8 x 1 0 - 24 . 3 x 1 0 - 21.3x10-11.8x10-14.1 X10-13.1 X10-16 . 4 x 1 0 - 1

TemperatureGradient, ATm280

5631282411116.1

5.62.8

CC)16 030171513663.43.11.6

orbital position, Thornton and Paul (1985). Structuraltemperatures may experience extremes from +250F to -250F for an orbit where the spacecraft passes throughthe Earth's shadow. Temperatures vary spatially inorbiting structures because of nonuniform incident heatfluxes, structural conduction and radiation heat lossesfrom surfaces. In the vacuum of space, radiation heattransfer is particularly significant. For example, thincylindrical shells may have major circumferential temperature variations since typically one-half of thecylindrical surface experiences heating from the solarflux while the remaining portion of the surface is shaded.All surfaces of the cylinder, heated and unheated,experience thermal energy losses to space by radiation.

Finally, thermal properties of materials have a majoreffect on temperature distributions. Metallic andcomposite materials may have significant differences inthermal properties that yield substantially differenttemperature distributions.To gain insight into how the properties of severalcomm on aerospac e m aterials affect the temperature of athin-walled structure, it is helpful to examine a simpleheat conduction problem. Consider a thin plate ofthickness /; with the x,y,z coordinate system located atthe middle surface. The initial temperature of the plate iszero. The surface z = -h/2 is perfectly insulated, and fortime t > 0 the surface z = h/2 is uniformly heated by aconstant heat flux qs. To understand how materialproperties affect the temperature gradient through theplate thickness, consider the situation where there is notemperature variation with x and y, ie T(z,t), Thermalproperties are assumed constant. The solution to the initial/boundary value problem is given by Boley andWeiner (1960) as

\ 2 iT(z,t) = ^ h 2 2 l h 2-^( - l ) n ex p ( -n 2 7 r 2 r t / h )co s r i 7 r i +

where the thermal diffusivity K = K/pc. This solutionpredicts that after an initial transient, temperatures of thetop and bottom surfaces of the plate increase linearlywith time according to the first term in the squarebrackets. Thus, the infinite series contributes to thetemperature distribution only during the initial transient,and after early time the series contribution becomesnegligible. Neglecting the series permits the thicknesstemperature difference or gradient AT = T(h/2,t)-T(-h/2,t) to be expressed by the simple equation,

AT 2k (4 )Equation 4 shows that the thickness temperaturegradient varies linearly with the applied heat flux andplate thickness but inversely with the thermalconductivity. Table I gives thermal properties andthickness temperature gradients predicted for tenmetallic and composite plates. Figures 1-2 show

Graphite-Epoxy

2 3 4 5 6 7.Time (seconds)

(3 )FIG 1 Thermal response of suddenly heated, thin graphite-epoxy plate, qs = 0.1 BTU/in2"s (16 W/cm2) and h = 0.125 in(0.318 cm).



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488 App l M ec h Rev vol 4 6 , no 10, Oc tober 1993 Thorn t on : The rm a l buc k li ng of plates and shel lsA16061-T6

4. 0 3.5-3 .0 f

0.0-

Heated SurfaceGradientInsulated Surface

0.000 0.025 0.050Time (seconds)

2.0

1.5 a&.1.0 J

0.5

~ t i % 0FIG 2 Thermal response of suddenly heated, thin aluminumalloy plate, qs = 0.1 BTU/in2"s (16 W/cm2) and h = 0.125 in(0.318 cm).transient temperatures for graphite-epoxy and aluminumplates. Graphite-epoxy and aluminum have the smallestand largest values of thermal conductivity, respectively,and produce extreme values for the temperature gradient.Graphite-epoxy has a very large thickness temperaturegradient, and aluminum has a very small temperaturegradient. Table I shows, in fact, that the graphite-epoxycomposite is exceptional; its thermal conductivity is solow that an extraordinarily large thickness temperaturegradient is predicted. Since the practical temperaturerange of this material is limited to less than 250F(120C), it is doubtful that such a large AT could everoccur in practice. The other metallics and metallic-basedcomposites show relatively small thickness temperaturegradients with titanium showing the largest andaluminum the smallest gradients, respectively. Thethickness temperature gradients shown may be regarded

Graphite-Epoxy450

t> = 5.0in (13 cm). , BTU (, , W"\

in - s ^ cm J

250

200

150 g32

100 E

- 50

0.04 0.06 0.08

CoolantCoolant

FIG 3 Centrally-heated, edge-cooledHeldenfels and Roberts (1952). plate studied by

as upper bounds since often thermal conductivityincreases with increasing temperature, and the insulatedboundary condition on the bottom surface will not berealized as energy losses will inevitably occur due tosurface convection or radiation heat transfer. In addition,any heat conduction in the x-y plane due to platesupports or non-uniform heating will lessen the thicknesstemperature gradient. For thin-walled metallic structures,customary practice has been to neglect thicknesstemperature gradients.Insight into the effect of thermal properties on thespatial variation of the thin plate temperature, T(x,y), isprovided by solution of the linear, unsteady initial/boundary value problem of a plate locally heated,F ig 3. The edges of the plate at y = b are maintained atconstant temperature of zero by specified coolant flows.The bottom and top surfaces of the plate are perfectly

insulated except along the centerline of the upper platesurface. For time t > 0, the plate experiences uniformheating qs on a narrow strip along the x axis.Hastelloy-X

0.10

500

400

300

200

-100

0.6

I1.0

y/bFIG 4 Temperature distribution for locally heated graphite-epoxy plate, qs = 0.1 BTU/in2-s (16. W/cm2), b=5.0 in (13 cm).

y/bFIG 5 Temperature distribution for locally heated HastelloyX plate, qs = 0.1 BTU/in2-s (16. W/cm 2), b = 5.0 in (13 cm).



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Appl Mech Rev vol 46, no 10, October 1993 Thornton: Thermal buckling of plates and shells 489Temperature is assumed uniform through the plate thickness. The solution to the symmetric, one-dimensionalconduction problem is given by Carslaw and Jaeger(1980) as,

Tfv t ) - q ( b ~ y ) 8 q b v H ) nU' ' 2 k * * k ( 2 n + l )1 J 2b (5)

where qo = qswl2h with w as the width of the heatedstrip, and as before, K is the therm al diffusivity. For largetime, plate temperatures approach steady-state with thepeak temperature at the centerline and a linear decreaseto the specified zero values on the plate edges. As withthe previous solution, Eq 5 predicts plate transienttemperature distributions that depend inversely on thematerial's thermal conductivity.Using the thermal properties shown in Table I,transient thermal responses were computed for graphite-

epoxy, Hastelloy X, and aluminum, (Figs 4-6). Theseresults indicate the responses of low, medium and highthermal conductivity materia] plates to local heating.The results show for low thermal conductivity graphite-epoxy (Fig 4) that temperatures rise rapidly with verysteep temperature gradients and a highly-localizedheated region. On the other hand, for aluminum withhigh thermal conductivity (Fig 6) temperatures rise moreslowly with lower gradients as thermal energy isconducted into the cooler boundaries. The response ofthe Hastelloy X, a high temperature nickel-based alloy,falls somewhere in between these two extremes, but thematerial demonstrates high local temperatures, steepgradients and a relatively small heated region. Thinwalled structures of this alloy subjected to localizedheating are likely to experience inelastic bucklingbehavior due to the high local temperatures.

These examples demonstrate two types of thermalgradient responses characteristic of thin-walled structures. For low thermal conductivity materials such asgraphite-epoxy, high thickness temperature gradientswill occur. However, for metals and metal matrixcomposites, thickness temperature gradients are negligible, and spatial temperature gradients are most likely tooccur in applications.

3. T H E R M A L BUCKLING O F P L A T E SThis section reviews past research on thermal bucklingof rectangular plates. The review is divided into threeparts: (1) the plane stress problem, (2) the bucklingproblem, and (3) the postbuckling problem. Within eachpart papers are described in chronological order. Asurvey of thermally induced flexure, buckling andvibration of plates, Tauchert (1991), provided some ofthe references in this review. A recent review article byNoorand Burton (1992) describes computational models

for high temperature multilayered composite plates andshells and lists 448 references.3.1 The Plane Stress ProblemThe classical plane stress problem consists of a thin,perfectly flat, isotropic rectangular plate heated so thatthe temperature is a function only of the spatialcoordinates and time, ie T(x, y, t). Since the temperaturedoes not vary through the plate thickness the thermalmoment is zero, and the plate remains flat as thetemperature changes. A state of plane stress is assumed,and the non-zero stresses Ov, 0V, xv are functions only ofx y xy Jthe spatial coordinates and time. Assuming elastic quasi-static behavior with constant properties, the elasticityproblem may be formulated in terms of the two non-zeroequilibrium equations, the strain-displacement relationsand Hooke's law.Some of the earliest thermal-stress analyses relevantto thermal buckling were concerned with solving theplane stress problem for a steady temperaturedistribution T(x, y). A popular approach was to introducean Airy stress function F(x, y) such that


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490 Appl Mech Rev vol 46, no 10, October 1993 Thornton: Thermal buckling of plates and shellsw h e r e E i s the modulus of elas t ici ty , and a is the coeff icien t o f thermal expansion . The s t ress funct ion F, ofcourse , mus t sat i sfy appro pr iate boun dary condi t ions onthe p late edges.

Heldenfels and Rober ts (1952) invest igated the p lanes t res s p ro b l em th eo re t i ca l l y an d ex p er im en ta l l y . Th eth eo re t i ca l s t u d i es p ro d u ced an ap p ro x im ateco m p lem en ta ry en erg y so lu t i o n fo r t h e Ai ry s t r es sfunct ion . In the exper imental s tud ies , s imple " ten t l ike"s t ead y - s t a t e t em p era tu re d i s t r i b u t i o n s were i n t ro d u ced b yh ea t in g a r ec t an g u la r p l a t e a lo n g a cen t e r l i n e wi th ah ea t i n g wi re an d m ain t a in in g co n s t an t t em p era tu re a lo n gparal lel edges by water f low through coolan t tubes (Fig3 ) . Top and bot tom surfaces of the p late were insu latedto p ro d u ce u n i fo rm , o n e-d im e n s io n a l , l i n ea r t em p era tu revar iat ions between the heated cen ter l ine and cooledparal lel edge s. In the exper ime ntal s tudy , in -p lane p lated i sp l acem en t s were p erm i t t ed t o o ccu r f r ee ly , b u t out-of-p lan e d i sp l acem en t s were p rev en t ed b y r es t r a in t s t h a tforced the p late to remain f lat . Thermocouples and s t raing ag es were u sed t o m easu re t em p era tu res an d s t r a in s ,respect ively . Exper imental resu l ts were found to be insa t i s f ac to ry ag reem en t wi th t h e ap p ro x im ate t h eo re t i ca lresu l ts . The resu l ts showed im porta n t character is t ics ofthe s t ress d is t r ibu t ion . The ten t l ike temperatured is t r ibu t ion (Fig 7) causes the cen t ral por t ion of the p lateto b e i n co m p ress io n . Fo r ex am p le , t h e s t r es s ojo,y) isco m p ress iv e fo r -b/2 < y < b/2 wh ere t h e t r an sv er sewidth of the p late i s 2b . For an unrest rained p late , thesecompressive s t resses must be equi l ib rated by tensi les t resses O x along the ou ter reg ions of the p late . The mostimportan t po in t , however , i s that the unrest rained p latem ay ex p er i en ce t h e rm al b u ck l in g d u e t o co m p ress iv est resses induced by the spat ial temperature grad ien ts .

A f ew y ear s l a t e r P rzem ien i eck i (1 9 5 9 ) u sedch arac t e r i s t i c b eam v ib ra t i o n m o d es t o d e r iv e an ap proximate so lu t ion for the p lane s t ress problem fora rb i t r a ry t em p era tu re d i s t r i b u t i o n s . Two y ear s l a t e r , Raoand Johns (1961) compared var ious so lu t ions for the

p l an e s t r es s p ro b l em assu m in g a sy m m et r i ca lt em p era tu re v a r i a t i o n ac ro ss t h e wid th o f t h e p l a t e .Th ese i n v es t i g a t i o n s p ro v id ed i n s ig h t i n to t h e m em b ran ethermal s t ress d is t r ibu t ions that es tab l ish the possib i l i tyo f t h e rm al b u ck l in g d u e t o t h e l o ca l co m p ress iv es t r es ses . Th e p l an e- s t r es s p ro b l e m rem ain s im p o r t an ttoday . In a s tudy of the v iscoplast ic response of p latesd u e t o u n s t ead y h ea t i n g , Th o rn to n an d Ko len sk i (1 9 9 1 )invest igated the p lane s t ress problem using a f in i tee l em en t ap p ro ach . P l a t e e l as t i c an d i n e l as t i c m em b ran est resses were invest igated for prescr ibed t ransien tt em p era tu re d i s t r i b u t io n s . Lo ca l y i e ld in g s ig n i f ican t lyal ters membrane s t ress d is t r ibu t ions, and for rap id r i seso f t em p era tu re , t h e n i ck e l -b ased a l l o y m ate r i a l s d i sp l ayin i t ial ly h igher y ield s t resses due to s t rain rate effects . Ast em p era tu res ap p ro ach e l ev a t ed v a lu es , y i e ld s t r es s an dst i f fness degrade rap id ly and pronounced p last icd efo rm at io n o ccu r s .3 . 2 B u c k l i n gTo d e t e rm in e t h e c r i t i ca l b u ck l in g t em p era tu re , sm al lt r an sv er se b en d in g d i sp l acem en t s o f t h e p l a t e a r eassumed. The s t resses ob tained f rom the so lu t ion of theuncoupled p lane s t ress problem is used to def ine theinplane s t ress resu l tan ts JVy N. N wh ich a re o b t a in edby mul t ip ly ing the s t resses by the p late th ickness h . Thed i s p l a c e m e n t w(x,y) of the buckled p late i s governed bythe l inear d i f feren t ial equa t ion

xdx 2 *ydxdy y ay 2 (8 )w h e r e D i s the f lexural r ig id i ty of the p late , D =Eh 3/12(l-x?), and v i s Poisson 's rat io . One of the basicthermal buckl ing so lu t ions to Eq 8 i s for a fu l ly-rest rained p late wi th a un i form temperature r i se . For ar ec t an g u la r p l a t e wi th d im en s io n s (2a x 2b) subjected tou n i fo rm co m p ress iv e m em b ran e fo rces N x and N,Tim o sh en k o an d Gere (1 9 6 1 ) g iv e t h e c r i t i ca l co n d i t i o nas

N v ^ V + N ^(2a) W = D2 2 2 2m % n %

w h e r e m, n denote the number of hal f -waves in the x andy d i rec t i o n s , r esp ec t i v e ly . Du e t o a u n i fo rm t em p era tu rerise AT, Nx=Ny = hEccAT/(l-v) for the fully restrainedp la t e . Th u s t h e c r i t i ca l b u ck l in g t em p era tu re i sdetermined by subst i tu t ing for N x and N and so lv ing forAT

AT. 7 r2 D ( l - u ) f m 2 , rr

. ~ Z n ' -


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Appl M ech Rev vol 46, no 10, October 1993 Thornton: Thermal buckling of plates and shells 491where the minimum value occurs for mSubstituting for 2? yields,

n nAT = 4 8 ( l + u ) a V a1 1

= n = 1.

(9)Equation 9 shows that the critical buckling temperature is independent of the material's modulus ofelasticity. For thin plates fully-restrained against in-plane

displacements, Eq 9 predicts quite small critical bucklingtemperatures. For example, an aluminum plate with a =18 in, b = 12 in and h = 0.25 in buckles at a criticaltemperature ATcr= 8.4F.Shortly after the Heldenfels and Roberts paper on theplane stress problem was published at NASA Langley, aclosely related paper by Gossard, Seide and Roberts(1952) described the buckling and post bucklingbehavior of the same plate. The paper made twoimportant contributions. First, the critical bucklingtemperature for the unrestrained plate with the tentliketemperature distribution was determined analytically.Secondly, the post-buckling nonlinear, out-of-planebending displacement w(x,y) was studied analyticallyand experimentally.The critical buckling temperature was obtained forsimple supports by the principle of minimum potentialenergy using a series of trigonometric functions for thetransverse displacem ent. The in-plane displacements areunrestrained. For a panel with aspect ratio oicilb = 1.57,the critical buckling temperature was determined to be

ling temperatures were computed in these two papers fora plate of dimensions (2a x 2b) and the parabolictemperature distribution,T = T0 + T, 1-1 i-l 1-1 i_z (11)

The results of these two papers showed that thecritical buckling temperature is given by Eq 10 in thegenera] form

l h 2T _-!_w 1 - v2 a b 2 (12)

where T is the critical value of the temperature rise.The thermal buckling coefficient k T depends on the plateaspect ratio and in-plane boundary restraints. Equation12 assumes fully restrained simple supports, and that theplate buckles in a single half-wave in each direction. Thevariation of the critical buckling temperature with theaspect ratio alb for the parabolic temperature distributionis shown in Fig 8.

Thermal buckling of a simply-supported rectangularpanel unrestrained in the plane is considered by Boleyand Weiner (1960). The edges x = 0,2a and y = 0,2b areassumed to remain straight. A temperature distribution TT0-T 1 cos ny/b is assum ed. An analytical solution is

Tw = 5 . 3 9 ^ l h 212 1 - t / a b 2 (10)In Eq 10, Tlcr is the critical value of the temperature

differential, the difference T1 between the center andedge temperatures in a tentlike temperature distribution,Fig 7. The paper also studied the plate's postbucklingbehavior which will be described in the next section.After the Gossard, Seide and Roberts paper, otherauthors investigated bifurcation thermal buckling ofrectangular plates. Hoff (1956) investigated thermalbuckling of supersonic wing panels. Temperature andthermal stress distributions due to aerodynamic heatingwere analyzed for supersonic wing structures. Ananalytical approach for determining critical thermalbuckling stresses for wing cover plates was developed,van der Neut (1957) investigated approximate analysis

methods for determining critical buckling temperaturefor panels with assumed distributions of thermal stress.Klosner and Forray (1958) studied buckling of simplysupported plates under an arbitrary symmetricaltemperature distribution. In-plane displacements of theplate are assumed to be elastically restrained bysupporting edge members. The critical bucklingtemperature was determined by a Rayleigh-Ritzprocedure. Miura (1961) used a similar procedure toobtain the critical buckling temperature when in-planedisplacements are completely restrained. Critical buck-

u.o-

0 . 5-

0.4-

0.3-

0.2-

0 . 1 -

no-

T,

CR k T 1 h2

1 - v 2 a b 2

T0/T1 = 0

0. 512

_ 35

1 1 1a/b

FIG 8 Buckling coefficient for simply-su pported , fully-restrained plate with parabolic temperature distribution,Klosner and Forray (195 2).



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492 Appl Mech Rev vol 46, no 10, October 1993 Thornton: Thermal buckling of plates and shellsdeveloped by first determining the stress function F thatsatisfies Eq 7. Then the transverse displacement w(x,y) isexpanded in an infinite series where a typical term hasthe formA mn sin (m7ix/2a) sln(nnyl2b). Satisfaction of Eq8 leads to an infinite determinant for computing thecritical values of the buckling temperature T} (Note, aswith the unrestrained plate of Gossard and Seide, thatbuckling does not depend on 7\). The criticaltemperature has the form

AT _ ky i i h2l" 24 1 - D 2 a b ! / 1 3 )where kj is a function of in and the aspect ratio a/b; in isthe number of half-waves in the x direction. A curve isgiven showing the variation of kj with aspect ratio. Theminimum value of k} is 3.848. For an aspect ratio a/b


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Appl Mech Rev vol 46, no 10, October 1993 Thornton: Thermal buckling of plates and shells 493

hV 4F = Eh


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494 Appl Mech Rev vo l 4 6 , no 10, Oc tober 1993 Thorn t on : The rm a l buc k li ng of plates and shel lsAn analytical approach using Galcrkin's method isemployed. Numerical results are presented for thepostbuckling response of composite panels with anassumed sinusoidal temperature distribution. Meyers andHyer (1991) analytically study the thermal buckling offlat, symmetrically laminated composite plates for auniform temperature rise. For the symmetric layup, theplates demonstrate bifurcation buckling behavior similarto an isotropic plate. Effects of support conditions andmaterial axis orientation are studied para metrically. Noorand Peters (1992a) study the postbuckling response ofcomposite plates subjected to combined axial load and auniform temperature distribution. The analysis is basedon a first-order shear deformation, von Karman nonlinearplate theory and used a mixed finite elementformulation. Sensitivity derivatives are used to study thesensitivity of postbuckling to composite plateparameters. Noor and Peters (1992b) also study thethermal postbuckling of thin-walled composite stiffeners.Using the same approach employed in the previouspaper, numerical studies are presented for anisotropicstiffeners with Zee and channel sections.

In one of the few recent experimental programs, apaper by Teare and Fields (1992) describes bucklinganalysis and test correlations of high temperaturestructural panels. A titanium, stiffened panel wassimultaneously heated and subjected to axial compressive forces in a testing machine. Good correlationbetween test and analysis provided both a validation ofthe computational methods and verification of thestructural concept. A postbuckling test indicated that thepanel was capable of withstanding more than 200% ofthe initial buckling load without permanent deformation.The paper concludes that vehicle weight can be reducedby taking advantage of the postbuckled strength asexhibited in the test.3.4 AssessmentSince the original investigations of Heldenfels andRoberts (1952) and Gossard, Seide, and Roberts (1952),numerous investigators have studied thermal bucklingand thermal postbuckling of plates. Until the 1980s, most

THERMAL BUCKLING OF PLATES

TotalNumber 10of Papers

FIG 9 Relative number of_ papers on thermal buckling ofmetallic and composite plates.

studies were performed for isotropic plates, but in the1970s research was initiated for laminated compositeplates. Figure 9 shows the relative number of papers onthermal buckling of metallic and composite plates overthe last four decades. Almost all of the work hasassumed: (1) perfectly flat initial configurations and (2)elastic behavior. There is a need for further study ofbuckling of plates with initial imperfections and inelasticbehavior. Aerospace designers also need rapid postbuckling analysis methods for both mechanical andthermal loads. With the exception of the original (1952)papers and the recent paper by Teare and Fields, theinvestigations have been analytical or computational;there have been no further experimental studies. There isa need for further experimental studies of thermallyinduced buckling for both isotropic materials andlaminated composites. Applications to aircraft structuresstrongly suggest that experimental programs beconducted for plates with spatial temperature variations.

4. THERMAL BUCKLING OF SHELLSBuckling of shells has been the subject of extensiveresearch in the aerospace industry because of thewidespread use of shell structures in flight vehicles.Analytical, computational and experimental studies haveinvestigated the behavior of shell structures subjected tomechanical loads. Much less research has occurred forbuckling and postbuckling of shells in a thermalenvironment. And yet future designs of airframes forhigh speed flight and spacecraft structures will have toconsider carefully the effect of the thermal environmenton structural and material behavior.

There are several excellent textbooks on the theoryand buckling of shells, but most classic books do notinclude temperature in the formulation of the equationsnor address thermal buckling. Thus, with a fewexceptions, solutions of shell thermal buckling problemsappear in papers and reports rather than texts.Furthermore, in contrast to the study of shell bucklingunder mechanical loads, studies of thermal buckling arerelatively more recent. For example, solutions for criticalbuckling loads of cylinders under axial loads werepresented in the early 1900s, but studies of criticalbuckling temperatures for cylindrical shells were notundertaken until the 1950s. At that time, the design oflightweight structures for high speed flight of aircraftand missiles provided strong motivation for the firststudies of thermal buckling. During the early days ofsupersonic flight, researchers learned that aerodynamicheating raised surface temperatures significantly andinduced strong temperature gradients. Consideration ofthermal effects on material and structural performancestimulated the first thermal buckling research for shellsin the late 1950s and early 1960s. A paper on buckling athigh temperatures by NJ Hoff (1957a) describes initialwork on thermal buckling of thin circular cylindricalshells. Over thirty years later a survey article by Ziegler



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Appl Meoh Rev vo l 46, no 10, Oc tober 1993 Thorn t on : Therm al buck l ing of p la tes and shel ls 495and Rammerstorfer (1989) summarizes thermoelast icstabili ty research for elastic and viscoelastic structuresincluding shal low and cylindrical shells. In the samevolume, Lukasiewicz (1989) presents a comprehensivereview of thermal stresses in shells and discusses thermalbuckl ing. A book b y Bushnell (1989) describes acomputer ized buckl ing analysis approach of shells fo rmechanical loads and presents thermal buckl ingexamples . A comprehensive bibl iography b y Keene andHetnarski (1990) l ists 7 7 8 references to publications onthermal stresses in shel ls .

This section focuses o n thermal buckl ing of shellswi th part icular emphasis o n aerospace appl ica t ions. T h ereview considers: (1 ) shallow shells, (2) cylindricalshells and (3) conical shells. Within each section papersare described in chronological order .4 .1 Shal low Shel ls a n d C u r v e d P a n e l sThe shallow shell idealization is used to consider the effects of curvature on the structural response of shellsegments and pane l s . T h e idealization permits use of asimplified set of equat ions simi lar to the von Karmanequations fo r flat plates rather than th e full shell equations. T h e classical shallow shell problem consists of athin, doubly curved surface with curvatures &v and kv inth e x and y direct ions, respect ively. T h e temperature Tmay vary wi th th e spatial coordinates x and y, throughthe thickness h of the shell , and with time. Typicallyquasistatic behavior with constant thermal properties isassumed , and published analyses have assumed steadytemperature dist r ibut ions. T h e non-zero stresses a , a ,T a re expressed in te rms of an Airy stress functionF(x,y) by Eq 6 such that the in-plane equilibrium equations a re satisfied. Satisfaction of membrane st ra in compatibili ty and the remaining t ransverse equi l ibr ium equation produces for an isotropic material th e nonlineargoverning equat ionsh V 4 F

= E h

D V 4w


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496 Appl Meoh Rev vo l 46, no 10, Oc tober 1993 Thornton: Thermal buckling of plates and she llsprob lem is so lved to determ ine cr i t ical buckl in gt em p era tu res . Th e e f f ec t s o f b en d in g -ex t en s io n co u p l in g ,laminat ion angle, modulus rat io , p late aspect rat io andradius of curvature on cr i t ical buckl ing temperature ares tu d i ed .

Hu an g an d Tau ch er t (1 9 9 1 ) s t u d y t h e rm al ly i n d u cedlarge def lect ions of lamina ted com posi te cy l indr ical anddoubly-c urved pane ls using the f in i te elem ent m ethod .Fi rs t o rder shear deformat ion theory i s used , and auniform temperature r i se i s assumed. Post -cr i t icalequi l ib r ium paths are t raced and the s t rength l imi ts oflaminates are pred icted using the Tsai -Wu cr i ter ion . Forcer tain laminate conf igurat ions, the equi l ib r ium path i scharacter ized by the ex is tence of l imi t -po in t and/orb i furcat ion insta b i l i t ies . Som e lamina tes are found toundergo two snap-fhroughs pr ior to fai lu re. Kossi ra andFfaupt (1991) s tudy the buckl ing of laminated p lates andcyl indr ical shel ls subjected to com bined thermal and mechanical loads using the f in i te element method . Thenonl inear pre- and post -buckl ing behavior of cy l indr icalcom posi te shel ls i s s tud ied for th ree d i f feren ttemperature d is t r ibu t ions. The effect o f rad ius ofcurvature i s s tud ied , and computat ions show that thebuckl ing temperature increases as the rad ius of curvaturedecreases . For large p ly angles , snap- through behavior i sp red i c t ed . Ch an d rash ek h ara (1 9 9 1 b ) s t u d i es t h e rm albuck l ing of an iso t ro pic cy l indr ica l ly curved panels usingSand ers ' shel l theory ex tended to laminated she l ls . F i rs torder t ransvers e shear deform at ion i s included .Numerical resu l ts are presented using the f in i te elementmethod , and the effect o f var ious parameters on thecr i t ical buck l ing temp eratu re of a curved graphi te-e poxypanel wi th s imple and clamped boundary condi t ions i sd em o n s t r a t ed .4 .2 Cy l in d r i ca l Sh e l l sEar ly work on cy l indr ical shel ls was in i t ia ted to s tudypoten t ial therm al buc kl ing of f rame-reinforced fuselagesfor ai rcraf t in h igh speed f l igh t . Due to aerodynamicheat ing , the th in cy l indr ical shel ls exper ience h ighertemperatures than the heavier reinforcing f rames. Thehotter shell tends to expand radially more than the

FIG 10 Circumferential mem brane force induced bytemperature distribution T(x) in cylindrical shell.

r es t r a in in g f r am e in d u c in g co m p ress iv e c i r cu m feren t i a lm em b ran e s t r es ses N e (x) , F ig 10 . In i t ia l research w asaimed at determinat ion of cr i t ical sk in temperature levelsfo r b u ck l in g d u e t o t h e co m p ress iv e m em b ran e s t r es ses .The f i rs t papers that s tud ied thermal buckl ing ofcy l indr ical shel ls so lved the l inear Donnel l s tab i l i tyequat ion s in uncou pled form:


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Appl Mech Rev vol 46, no 10, October 1993 Thornton: Thermal buckling of plates and shells 497uniform temperature r i se are no t l ikely to cause elast icbuckl ing . In a closely related paper , Anderson (1962a)developed a s imi lar theory to pred ict the buckl ingtemp eratu re of a un i formly heated , r ing-st if fened shel land included ax ial com press ion . He used a shel l theoryby Batdorf and considers bo th s imply supported andc l am p ed sh e l l en d co n d i t i o n s . Th e p ap er co n c lu d es ,consis ten t wi th Hoff, that buckl ing of a r ing-st i f fenedcyl inder due to un i form temperature alone wi l l no t occurfor small values of the radius to thickness ratio, a/h.However , the paper no tes that for large values of a/h (eg,a/h = 2 0 0 0 ) , su ch b u ck l in g can o ccu r . Su n ak awa (1 9 6 2 )also invest igated the deformat ion and buckl ing ofcy l indr ical shel ls due to aerodynamic heat ing . The bookby Johns (1965) g ives a s imple formula for the cr i t icalt em p era tu re fo r b u ck l in g b y c i r cu m feren t i a l m em b ran es t r es ses ,

where for a s imple shel l wi th s imply supported ends k =5.3 , for a s ing le or mul t ibay she l l wi th clamped ends k =3.76 , and for a mul t ibay shel l wi th s imply supporteden d s k= 2 . 1 .

This ear ly work on cy l inders wi th un i form temperatures was based on appl icat ions to fuselages wi thassu m ed ax i sy m m et r i c h ea t i n g . At ab o u t t h e sam e t im e ,wor k bega n on cy l inders nonun iformly heated aroundthei r ci rcumference. Prev ious research on f lat p lates wi thsp a t i a l t em p era tu re g rad i en t s , eg , Gossard and Seide(1952) , had estab l ished that thermal buckl ing couldoccur for relat ively smal l spat ial temperature grad ien ts .Abir and Na rdo (1959) invest igated therm al buckl ing ofci rcu lar cy l indr ical shel ls wi th ci rcumferen t ialtemperature grad ien ts . For a s imply supported shel l , theysolved Eqs 19 fo l lowing the approach employedpreviou sly by Hoff (1957b) , bu t they invest igatedbuckl ing due to the var iat ion of the ax ial membraneforce N around the shel l c i rcumfe renc e, F ig 11 . Thestudy concluded that the cr i t ical value of the ax ialthermal s t ress d is t r ibu t ion occurs at a value equal to thecr i t ical s t ress in un i form axial compression . The cr i t icals t ress for a cy l inder in un i form axial compression i s o v =[3{l-X^)]- in Eh/a, Tim o sh en k o an d G ere (1 9 6 1 ) . For afu l ly res t rained cy l inder the ax ial compression s t ress fora temperature increase AT is a v = EaAT. Thus thecr i t ical buckl ing temperature for un i form compression i s

Bi j laard and Gal lagher (1960) also s tud ied the elas t icinstab i l i ty of a cy l indr ical shel l under an arb i t raryci rcumferen t ial var iat ion of ax ial s t ress . Thei rconclusions were consis ten t wi th Abir and Nardo .

In a closely related s tudy , Hi l l (1959) analy t ical ly andexper imental ly s tud ied buckl ing of ci rcu lar cy l indr ical

shel ls heated along an ax ial s t r ip . In the analysis , theDonnel l shel l equat ions were used and were so lvedap p ro x im ate ly u s in g a R i t z m eth o d . A s im p l i fi ed ax i a lthermal s t ress d is t r ibu t ion was assumed where the s t ressis constan t in x bu t var ies around the ci rcumference. Inth e ex p er im en t s , a l u m in u m an d s t ee l cy l i n d er s werelocal ly heated by infrared lamps, and two types ofcy l in d er en d co n d i t i o n s were em p lo y ed . Tem p era tu reswere m easu red wi th t h e rm o co u p les , an d d i sp l acem en t swere m e asu red wi th L in ear Var i ab l e Di f f eren ti a lT ran s fo rm ers (LVDTs) . Cy l in d er b u ck l in g wasd em o n s t r a t ed wh e re t h e b u ck l in g d efo rm at io n s d ep en d edst rongly on the exper imental end condi t ions. Fora lu m in u m cy l in d er s wi th a/h = 430 , cr i t ical buckl ingtemp erature s were relat ively low rangi ng from AT =65F to 100F depending on end condi t ions.Exper imental resu l ts for the cr i t ical buckl ingt em p era tu re were i n r easo n ab le ag reem en t wi th t h etheoret ical calc u lat io ns. Th e s tudy showed the d i f ficu l tyof performing analy t ical p red ict ions for the thermalbuck l ing s t resses due to local ized tempe ratured is t r ibu t ions and demonst rated the pract ical p roblems ofc rea t i n g ex p er im en ta l en d co n d i t i o n s co n s i s t en t wi th am ath em at i ca l m o d e l .

An d er so n an d Card (1 9 6 2 ) d esc r ib ed a s t u d y wh ereseveral s tain less s teel r ing-st i f fened cy l inders weresubjected to a pure bend ing mom ent and heated rap id lyu n t i l b u ck l in g o ccu r red . Fo r m o s t o f t h e cy l i n d er s t h eh ea t i n g was n o n -u n i fo rm a ro u n d th e c i r cu m feren ce .Tem p era tu re an d d ef l ec t i o n s were m easu red a t sev era llocat ions, and in some tes ts s t rains were measured .St rains were successfu l ly measured below 175F, bu tdata was unrel iab le for h igher temperatures . Anelementary thermal s t ress theory was found to beinadequ ate for the pred ic t ion of the therm al s t resses . In arelated paper , Anderson (1962b) rev iews theoret ical andexper imental invest igat ions on buckl ing of cy l inders dueto bo th ci rcumferen t ial and ax ial thermal s t resses . Thepaper no tes that the sever i ty of the ci rcumferen t ialthermal s t ress i s s t rongly dependent on the boundarycondi t ions and suggested the need for addi t ionalexper imental resu l ts . For cy l inders that are heated non-

FIG11 Longitudinal membrane force N x induced bytemperature distribution T(6) in cylindrical shell.



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498 Appl Mech Rev vol 46, no 10, Oc t obe r 1993 Thorn t on : The rm a l buc k l ing of plates and shel lsuniformly, the paper discusses the difficulty ofpredicting axial thermal stresses and identified the needfor research in thermal buckling of longitudinallystiffened cylindrical shells for launch vehicles.In the mid-1960s, the Stanford University group of NJHoff continued to investigate thermal buckling ofcylinders. Three papers describe further research onbuckling of thin circular cylinders heated along an axialstrip. Hoff, Chao and Madsen (1964) analytically studythe problem assuming uniform compressive axial stressto occur in a narrow strip while the rest of the shell isstress free. The shell is assumed very long, and Donnell'sequations are used. The results of the analysis supportthe earlier conclusion of Abir and Nardo (1959) that thecritical stress of the heated strip is the same as thecritical stress of a complete cylindrical shell subjected touniform compression unless the heated strip is verysmall. With very narrow heated strips, the critical stressrises rapidly as the strip width decreases. Ross, Mayersand Jaworski (1965) continued and extended theexperimental studies initiated by Hill. A series ofstainless steel and cold-rolled steel cylinders (alh = 334and 291, respectively) were heated along an axial stripby infrared heat lamps. Ends supports were designed tosimulate fully-fixed conditions. Instrumentation includedthermocouples and a microphone to provide an audiosignal at the instant of buckling. The heated width wasvaried from 1.5% to 18% of the total shellcircumference. Tests were conducted by subjecting (hecylinders instantaneously to maximum heating from theinfrared lamps; axial and circumferential temperaturedistributions were then recorded until the shell buckled.Fo r the stainless-steel cylinders, buckling temperatureswere around 300F; for the thicker, cold-rolled steelcylinders buckling occurred around 500F. There wassignificant scatter in the data. For tests conducted oncylinders having narrow heated strip widths, bucklingtook place as a sudden localized "snap-through"accompanied by a distinctly audible sound. Al largervalues of the heated strip width, localized bucklingpatterns appeared simultaneously within the heated stripat several locations. For tests conducted on the thickercold-rolled sheet steel cylinders with larger values of thestrip width, a different failure mode occurred with abarrel-shaped deformation leading to yielding failure atthe clamped support. Computations were performed withthe Donnell theory using measured circumferentialtemperature distributions. Agreement of analysis withexperiment was fair, but it was judged good enough forthe authors to confirm previous conclusions of severalinvestigators that the buckling temperature corresponding to uniform axial compression provides alower limit of stability for thin cylindrical shellssubjected to highly nonuniform circumferential healing.Hoff and Ross (1967) performed further analysis of theproblem using the experimental temperature distributionsand obtained reasonable agreement with theexperimental data. The consistent trend demonstrated by

these analyses was that experimental critical bucklingtemperatures were consistently higher than valuespredicted by analyse s. This trend was in directcontradiction to the well-established trend that formechanical loads experimental critical buckling valuesare consistently lower than predicted v alues.Ross, Hoff, and Horton (1966) conducted additionalexperiments to address the thermal buckling anomaly.Six stainless-steel cylinders (a/h = 344) and five cold-rolled steel cylinders (a/h = 291) were heatedaxisymmetrically by an array of twelve infrared heatlamps. The cylinders' ends were longitudinally restrainedand clamped. The test procedure was essentially thesame as used by Ross et al (1965). Upon heating, a testcylinder experienced thermal expansion in the radialdirection, and the thin cylindrical shell became barrel-shaped. Ultimately, the shell buckled near the clampedend supports. Comparison of experimental results withthe predictions of available shell-buckling theoriesshowed poor agreement. The authors concluded thatbecause the cylinder "barrels out" during heating, theaxial compressive stress is reduced resulting in highercritical temperatures than is predicted by linear bucklingtheory.

Hoff (1965) and Ross (1966) also investigated the useof simple models to explain thermal buckling of shells.Hoff uses a two bar mechanism with a nonlinear spring,and Ross uses a beam-column with a nonlinear spring.Gellatly, Bijlaard, and Gallagher (1965) investigatedthermal buckling of sandwich cylindrical shells forsimply and clamped supports. Using the approach ofHoff (1957b), critical buckling temperatures weredetermined for circumferential stresses varying with x.Numerical results are presented for isotropic andsandwich cylinders.By the late 1960s and early 1970s, digital computershad become widely used in engineering research, andnumerical models were being developed to solvecomplex problems. Chang and Card (1970,1971) atNASA Langley developed programs for analyzingthermal buckling of orthotropic, multilayered, cylindricalshells stiffened by uniform, equally spaced rings andstringers. Thermal effects are accounted for byspecifying axisymmetric temperature distributions in theshell and stiffeners. The theory was developed fromstrain energy expressions corresponding to nonlinear

Donnell-von Karman displacements, and equilibriumequations were derived by the principle of minimumpotential energy. These equations were separated intoequations governing axisymmetric prebuckling behaviorand equations governing behavior at buckling. The endsof the stiffened cylinder were assumed to be free toexpand longitudinally. In the analyses thermal bucklingoccurs as a consequence of circumferential compressivestresses (Fig 10) introduced by radial restraint at theboundaries (w = 0) and/or from restraints resulting fromdifferences in expansion between the stiffeners and shell.The prebuckling and buckling solutions were based on



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ApplMech Rev vol 46, no 10, October 1993 Thornton: Thermal buckling of plates and shells 499finite difference discritizations of the shell equations.Thermal bu ckling behavior of unstiffened, ring-stiffened,stringer-stiffened and ring/stringer-stiffened shells wereinvestigated. For unstiffened cylinders the numericalsolutions were validated by comparisons with theanalytical solutions of Hoff (1957b) and Sunakawa(1962). Recall that the Hoff analysis (1957b) of a simplysupported cylinder under hoop stresses varying in the xdirection predicted buckling at a very high temperature.Chang and Card's computer analysis of the same cylinderwas unable to determine this critical bucklingtemperature; no buckling temperature could be foundwithin a practical temperature range. The differencebetween the two analyses was determined to lie in theassumptions made in the linearized Donnell prebucklingequations employed by Hoff where rotations and derivatives of rotations are neglected. These prebucklingrotations were included in the Chang and Card formulation, but when these terms were suppressed theChang and Card predictions were in agreement withHoff. The conclusion was that the prebuckling rotationsare required, and for simple supports the bucklingtemperature is beyond the elastic range. Chang andCard's predictions were in good agreement withSunakawa's analysis for a clamped cylinder sinceSunakawa's shell theory included prebuckling rotations.For ring-stiffened cylinders compa risons were madesuccessfully with Anderson's analysis (1962b). Thermalbuckling studies of an aluminum large diameter,longitudinally stiffened cylinder and a titanium ring- andstringer-stiffened cylinder were also conduc ted.

At about the same time, Bushnell of the LockheedPalo Alto Research L aboratory was de veloping acomputer program called BOSOR for the generalanalysis of shells of revolution. Applications of BOSORto thermal buckling are described in a book (1989) andseveral papers (1971a, b, 1973). The BOSOR program isbased on an energy formulation with the method of finitedifferences. Axisymmetric ring-stiffened shells are assumed. The program has been used to solve several ofthe thermal buckling problems discussed previously andother, more complex, thermal buckling problems.Bushnell (1971a) solves the Hoff (1957b) problem of thesimply supported cylinder subjected to a uniformtemperature rise. Buckling of a ring-stiffened aluminumcylinder subjected to axial compression, externalpressure and various temperature distributions is alsostudied in detail. The effect of ring-oul-of-plane bendingstiffness on thermal b uckling of ring-stiffened cylindersis discussed by Bushnell (1971b) and demonstrated to beimportant.Bushnell and Smith (1971) describe calculations ofthermal stresses and buckling of nonuniformly heatedcylinders and cones. The BOSOR program was used toanalyze several cylinder tests including the studies ofHill (1959), Anderson and Card (1962), Ross, Mayersand Jaworski (1965), as well as Ross, Hoff, and Horton(1966). These analyses arc valuable because they

represent the first systematic comparison ofcomputational results with experimental data. Inaddition, the computational approach permitted issues tobe addressed that previously had been intractablebecause of limitations of analytical methods. Particularattention was given to the effect of boundary conditionson predicted stress and critical temperatures. Forexam ple, the anom aly raised in the experimental studyof Ross et al (1966) is attributed to undesired flexibilityin the experim ental boundary con dition. The paper alsoconcluded tha t for shells which a re long compared to a"boundary layer (ah)112", critical temperatures ofuniformly heated monocoque cylinders and cones are assensitive to initial imperfections as are critical axialloads.

In a later paper, Bushnell (1973) uses BOSOR tostudy thermal buckling of cylinders with axisymmetricthermal discontinuities. The first problem consideredwas buckling of a cylinder heated halfway along itslength. The problem was considered because ofquestions raised by the Hoff (1957b) problem of thermalbuckling due to circumferential stresses. Bushnellreaches the same conclusions that Chang and Card;prebuckling rotations should be included in the analysis,and elastic buckling will not occur for the simplysupported cylinder with uniform temperature. Thesecond problem considered was a clamped cylindricalshell (a/h = 2540) analyzed and tested by Johns (1962).A BOSOR analysis (with prebuckling rotations) of theJohns' cylinder predicted a critical temperature of 150Cwhereas the experimental value was 324C. Subsequentinvestigation showed that the large discrepancy betweenthe computation and test could be explained by thepresence of temperature gradients near the boundaries ofthe cylinder. The paper concludes that critical bucklingtemperature calculations are sensitive to the shape of thetemperature distribution. To obtain good correlationbetween predictions and tests, temperature distributionshave to be measured carefully and the spatial variationsincluded in the analysis.

Two papers in the mid-1970s analyzed thermalbuckling of orthotropic cylindrical shells. Gupta andWang (1973) use a Rayleigh-Ritz approach to analyze asimply-supported orthotropic shell with uniformtemperature. Prebuckling rotations are not considered.Parametric studies illustrate the effects of the axial andcircumferential coefficients of thermal expansion.Radhamohan and Venkataramana (1975) analyze anorthotropic clamped cylindrical shell using an approachbased on Sanders' nonlinear shell theory. Effects ofprebuckling rotations, different forms of clampedboundary conditions and other parameters are studied.

A brief paper by Belov (1978) describes an experimental study of stability of cylindrical shellspartially filled with liquid. The study was motivated byaerodynamic heating of fuel tanks. In the experimentalprogram, cylindrical shells were subjected to heating andwere loaded by axial compression and internal pressure.



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500 - Appl Mech Rev vol 46, no 10, October 1993 Thornton: Thermal buckling of plates and shellsAlthough good agreement between calculations andexperiment data is stated, details of the analysis and testsare not given.In the late 1970s, two test programs on buckling ofcylinders with combined mechanical and thermal loadswere conducted at the Technion in Israel. Frum andBaruch (1976) describe buckling of cylindrical shellsheated along two opposite generators. A series of forty-six tests of aluminum cylinders with a/li = 301 wereconducted. End supports were designed to be fullyrestrained. Axial loads were applied by a hydraulic jack.Two infrared line heaters were installed above and belowthe shell. The temperature distribution was measuredwith thermocouples, and displacements were measuredwith LVDTs. The instant of buckling was detected witha microphone. The effects of the u displacementboundary conditions were studied, and the authorsconclude that previous investigators had not treated thecondition with enough care. They conclude that the ndisplacement has a dominant influence on the bucklingresults. Comparisons of experimental data withcomputations are only fair. The experimental data wasused to construct an axial load-temperature interactioncurve. Ari-Gur, Baruch and Singer (1979) describebuckling of cylindrical shells under combined axialpreload, nonuniform heating and torque. Similar testcylinders and the test rig employed in the previous studywere used after modifications to allow torsion. A seriesof 35 tests were conducted. A temperature-torque interaction curve was developed from the experimental data.

Studies of thermal buckling of laminated compositecircular cylinders began in the 1980s and continue to thepresent time. So far, the studies have been analytical;experimental studies have not yet been conducted.Wilcox and Ma (1989) use an energy approach to derivea set of equilibrium equations based on classical thinshell theory with Donnell's assumptions. Galerkin'smethod with an assumed trigonometric variation for wfor a simply-supported cylinder leads to a matrixeigenvalue problem. Numerical results are presented forcritical buckling temperatures for various composite parameters such as lamination angle. Thangaratnam et al(1990) uses the finite element method to conductparameter studies of a simply supported cylinder withuniform temperature.Birman (1991) studied the thermally induced dynamic

response of reinforced composite cylinders. The study isbased upon Donnell's theory of geometrically nonlinearshells and includes axial and ring stil'fcncrs. Solutionsare developed for a simply supported cylinder subjectedto a uniform rise in temperature. The paper concludesthat if a shell is subject to an instantaneous rise intemperature it exhibits stable steady-state oscillations butif the temperature exceeds a critical buckling level, thecharacter of the response changes and the deflections canincrease dramatically.

4.3 Conical ShellsWork on thermal buckling of conical shells began in themid-1960s and was motivated by aerodynamic heatingeffects. The analyses typically were based on Donnell-type equations written in terms of a stress function F andthe w displacement. A truncated cone with vertex angle2a is considered. The non-zero stresses are defined interms of the Airy stress function F(s, Id 1. + _ +s ds s

2 dip2

(25)

(26)In the above, the temperature is assumed uniformthrough the thickness of the shell, and the thermalmoment is zero. For a = 0, Eqs 24 and 25 reduce to thevon Karman equations for a flat circular plate.Some of the earliest analytical and experimental workon conical shells was done by Bendavid and Singer(1967) who studied buckling of truncated conical shellsheated along a generator. The work is closely related toHill's (1959) study of buckling of thin cylindrical shellsheated along a strip. Critical buckling temperatures areinvestigated for axial compressive stresses induced bycircumferential temperature distributions. A Rayleigh-Ritz analysis is employed, and numerical results arecompared with an experimental study. In theexperimental study a steel cone with vertex angle a =

12.4, thickness h = 0.0155 in and a small opening radiusof 6 in was heated along a generator with a quartz lamp.



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Appl Mech Rev vol 46, no 10, October 1993 Thorn ton; Thermal buckling of plates and shells 501The experimental temperature distribution varied both inthe circumferential and axial directions. Boundaryconditions are assumed simply supported, and the endsare assumed restrained in the direction of the generator.The experimental buckling temperature was 103C andthe calculated value was 137C.

Lu and Chang (1967) calculated thermal stresses andbifurcation buckling temperatures of axisymmetrical andnonsymmetrically heated simply-supported cones. In theaxisymmetric case, the temperature varies along thegenerator which is restrained at the ends. In the othercase, the temperature varies circumferentially as well,but the shell is assumed free of axial forces at the ends.A formulation similar to Eqs 24-25 is used, but thenonlinear von Karma n strains are neglected. A Galerkinapproach with assumed deflection shapes is used toobtain critical temperatures. The variation of criticaltemperatures with various shell parameters such as theaverage radius-thickness ratio, the average radius-heightratio and the cone's vertex angle are evaluated. Theradius-thickness ratio has the most significant effect onthe critical temperature. In further work, Chang and Lu(1968) included the nonlinear strain terms neglected intheir previous paper. The analysis showed that thenonlinear theory predicted lower crilical temperaturesthan the linear theory. A brief discussion is given of anexperimental study of a brass cone heated uniformly byheat lamps. The experimental cone had vertex angle a =10, an average radius to thickness ratio of 500 and aheight to average radius ratio of 2. The ends of the conewere assumed simply-supported but restrained againstaxial expansion. The average experimental bucklingtemperature for three lesl.s was 128'F, and the predictedvalue was 9SF.

Bushnell and Smith (1971) used the computerprogram BOSOR to analyze thermal buckling of conicalshell experiments performed by Smith (1964) at StanfordUniversity. Truncated steel conical shells with clampedends restrained against axial displacement were heatedalong axial strips of various widths by quartz lamps. Testcones have a wall thickness of 0.0155 in, a smallopening radius of 6 in, a large opening radius of 12 inand a slant length of 28 in. Circumfe rential-and axialtemperature variations were induced and measured withthermocouples. The instant of buckling was determinedby a microphone; buckling occurred with a loud noise.Critical buckling temperatures for eight test specimensvaried from about 55C to over 100C; the scatter in thedata was attributed lo initial imperfections in the testspecimens. In the analyses, a test was modeled in threeways: as a fully clamped shell, as a simply supportedshell, and as a portion of a larger structure whichincludes the test rig. Using measured critical temperaturedistributions, axial membrane stress distributions werecomputed for the three models. Treatment of the test rigas a structure of finite flexibility led to results that were

with 5% of those obtained with the assumption that thetest rig is rigid. In the stability analysis, the meridionalprebuckling membrane stress distribution was assumedto be axisymmetric. Depending on the experimentaltemperature distribution used, the discrepancy betweentest and theory for critical temperatures was from 11 % to52%.Tani (1978) studied the effect of axisymmetric initial

deflections on the thermal buckling of shallow, truncatedconical shells under uniform heating. The shell isanalyzed for two cases where the supports are assumedclamped but may be either constrained or free in theaxial direction. A uniform temperature distribution isassumed. Nonlinear buckling equations based on Eqs 17-18 are modified to include initial shell deformations. Acentral finite difference scheme is used to discritize theequations in the s direction with a cos nq expansion inthe circumferential coordinate. Numerical results showthat the buckling temperature and number n of circumferential buckling waves of shallow cones with axialconstraint vary significantly with the amplitude ofaxisymmetric initial deflections. Without axial restraint,the initial imperfections have only small effects on thebuckling mode. Initial deflection amplitudes equal toplus or minus the shell thickness were studied. Inshallow shells, local buckling occurred near the coneedges due to hoop compression. In a further study Tani(1984) analyzed instability of truncated conical shellsunder combined uniform pressure and uniformtemperature. Following the approach of the previouspaper, the Donnell-type shell equations with nonlinearprebuckling deflections are solved using a finitedifference method. Numerical results show that forshallow conical shells, the critical combination ofbuckling loads changes with the loading order; for deepconical shells, the critical combination of buckling loadsdoes not change with loading order. Generally, thecritical buckling temperature increases with internalpressure.

4.4 AssessmentIn the 1950s and 1960s, there was significant research onthermal buckling of metallic shells as aerodynamicheating became important in the design of supersonicaircraft and missiles. Since then, research in thermalbuckling of metallic shells has diminished. Beginning inthe 1970s, there has been significant growth in theresearch efforts devoted to thermal buckling ofcomposites. Figure 12 shows a bar graph illustrating therelative research efforts devoted to thermal buckling ofme tallies and comp osites as measured by publishedpapers. In the 1990s thermal buckling research so far hasbeen devoted exclusively to composites.



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502 . Appl Mech Rev vo l 46, no 10, Oc tober 1993 Tho rnton: Therm al buck l ing of p la tes and shel ls

Exper imeCy l indr i ca l Exper iments1. Hill2. Johns3. Anderson and Card4. Ross, Mayers and Jaworski5. Ross, Hoff and Horton6. Frum and Baruch7. Ari-Gur, Baruch and SingerC one Ex pe r i men t s8. Smith9. Bendavid and Singer

Table I Inta l Studies of Thermal Buckl ing of She

Ye a r1959196219621965196619761979

19641967

Mater ia lsAluminumand steelSteelStainless steelStainless andcold-rolled steelStainless andcold-rolled steelAluminumAluminum

SteelSteel

illsLoadingHeating Along an Axial StripAxisymmetric HeatingBending Moment andNon-uniform HeatingHeating Along Axial StripHeating Along Axial StripHeated Along Two OppositeGenerators and Axial LoadsCombined A xial Preload,Nonuniform Heating and Torque

Heating Along Axial StripHeating Along Axial Strip

post-buckling damage is available. Virtually no straindata was taken in the experiments because of strain gagelimitations at elevated temperatures. Most experimentswere conducted on steel or stainless steel. There islimited data available for aluminum, and no experimentshave studied the buckling of titanium shells.Experimental studies of composite shells have not yetbeen published.Details of shell temperature distributions, particularlyin stiffened shells, have an important role in the bucklingresponse. The assumption of uniform temperature

between stiffeners is usually not a good approximation.Since material properties such as thermal conductivityhave significant effects upon the shape of temperaturedistributions, thermal buckling tests should be conductedusing the materials that will be used in actual structures.With one notable exception, there have been novalidation studies of computer analyses of thermalbuckling of shells. The exception is the very thoroughanalyses of several experiments by Bushnell using thecomputer code BOSOR. Generally, agreement betweenanalytical solutions, computer solutions andexperimental data has been only fair to good. A typicalreason cited for discrepancies has been uncertainties inexperimental boundary conditions. There have beenrelatively few tests with combined mechanical andthermal loads. For exam ple, no tests of heated cy linderswith internal pressures have been reported. Th ere is aneed for further experimental studies of thermallyinduced buckling of shells for both isotropic materialsand laminated composites. Applications to aerospacestructures strongly suggest the need for experimentalprograms for unstiffened and stiffened shells withthermal loads as well as combinations of thermal andmechanical loads.

Since the original investigations of Hoff (1957),numerous investigators have studied thermal buckling ofshells. Many of the studies have been analyticalparticularly through the 1950s-1960s, but since 1970there has been increasing reliance on computationalmethods. Many of the computer programs such asBOSOR have used finite difference methods, butanalyses of composite shells are using the finite elementmethod. There was also an effort made through the1950s-1970s to investigate thermal buckling of shellsexperimentally. Table II summarizes experimentalstudies for cylinders and cones. The vast majority of theexperimental studies considered monocoque shells. Withthe exception of the study by Anderson and Card (1962),there have been no further investigations of stiffenedshells. Experimental measurements were typicallylimited to a modest number of transducers. No data arcavailable for initial imperfections, details of temperaturedistributions are lacking, and little documentation of

THERMAL BUCKLING OF SHELLS

TotalNumber loot Papers

FIG 12 Re la t ive num ber o f pape rs on the rma l buck l ing o frrmrnllin and mm nn si te shn lls



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Appl Mech Rev vo l 46, no 10, Oc tober 1993 Tho rnton : The rmal buck l ing of p la tes and shel ls 5035.0 C O N C L U D I N G R E M A R K SThermal buckl ing research for p lates and shel ls has beendescr ibed . The ro le of mater ial thermal parameters onth ickness and spat ial temperature grad ien ts wasdemonst rated f i rs t . For metal l ic and metal matr ixcomp osi tes subjected to surface he at ing , th icknesstemperature grad ien ts are smal l and may be neglected inappl icat ions. Indeed , th is assumpt ion has been made inmost analy t ical and computat ional research s tud ies , andi t has been val idated exper imental ly . Of the mater ialsconsidered , on ly graphi te-cpoxy which has very lowthermal conduct iv i ty exhib i ted an appreciab le th icknesst em p era tu re g rad i en t . Ho wev er , m o s t m ate r i a l s ,par t icu lar ly in aerospace appl icat ions, wi l l exper iencespat ial ( in -p lane) temperature grad ien ts . Due to local izedheat ing , metals wi th relat ively low thermal conduct iv i ty ,eg , s tain less s teels , wi l l exhib i t h igh local tem peraturegrad ien ts and are l ikely to exper ience inelast ic buckl ing .Metal panels wi th h igh thermal conduct iv i ty , eg ,aluminum, tend to have much smal ler temperaturegrad ien ts as they conduct thermal energy to support ings t ru c tu ra l m em b ers .

The rev iew of research on thermal buckl ing of p latesshowed research in the 1950s-1960s was l imi ted tometals . Composi te p lates were considered f i rs t in the1970s, and since then the majority of research has beenfor com pos ite plat es. Th e majority of resear ch in the lasttwo decades has been analy t ical o r computat ional wi thvery few exper imental s tud ies . There i s a need forfur ther exper imental s tud ies of thermal ly inducedbuckl ing for bo th i so t ropic mater ials and laminatedcomposi tes . Appl icat ions to aerospace s t ructures s t ronglysuggest that exper imental p rograms be conducted forp lates wi th spat ial temperature var iat ions.The rev iew of thermal buckl ing research for shel lsshowed t rends s imi lar to thermal p late buckl ing .Research in the 1950s and 1960s considered onlymetal l ic shel ls . Research on composi tes began in the1970s and has become the predominant effor t s ince then .However , the curren t level o f research on thermalbuck l ing of compos i tes i s s t i ll relat ively smal l com paredto the efforts devoted to metallic shells in the 1960s.Many of the ear ly shel l thermal buckl ing s tud ies wereanaly t ic al , bu t s ince the 1970s there has been increasingre l i an ce o n co m p u ta t i o n a l m eth o d s . Th ere was asignifica nt early effort to stud y thermal shell buc klin gexper imental ly al though there are def iciencies in thee x p e r i m e n t s , eg , the lack of s t rain data. Anotherdef iciency has been uncer tain t ies in exper imentalboundary condi t ions. So far , there have been nopubl ished exper imental s tud ies of thermal buckl ing ofcom posi te sh el ls . The re i s a clear need for such s tud iesfor h igh temperature metal l ics such as t i tan ium as wel las for composi te shel ls . There i s also a need for exper i mental s tud ies of heated cy l inders (wi th and wi thouts t i f feners) subjected to mechanical loads includingin ternal p ressure. As wi th p lates , the . ex p er im en ta l

p ro g ram s sh o u ld b e co n d u c t ed wi th sp a t i a l t em p era tu red is t r ibu t ions.

For bo th thermal buckl ing of p lates and shel ls ,r en ewed em p h as i s sh o u ld b e p l aced o n v a l i d a t i o n o fan a ly t i ca l / co m p u ta t i o n a l s t u d i es wi th ex p er im en ta l d a t a .Co m p u ta t i o n a l an a ly ses o f sev era l t h e rm al -b u ck l in gexper iments by Bushnel l are descr ibed in the rev iew.The lessons learned f rom these analyses include theunce r tain t ies in boun dar y cond i t ions prev iouslyment ioned as wel l as l imi tat ions and/or gaps inex p er im en ta l d a t a . On th e o th er h an d , su ch co m p ar i so n sin the past have also ident i f ied def iciencies inco m p u ta t i o n a l m o d e l s .

An in t e res t i n g co n t r as t b e tween b u ck l in g d u e t omechanical and thermal loads i s the respect ive ro les ofelast ic mater ial p roper t ies . Cr i t ical buckl ing forces forbea ms , p lates and shel ls made f rom iso t ro pic mate r ialsare d i rect ly proport ional to the modulus of elas t ici ty .Cri t ical buckl ing temperatures for i so t ropic mater ials aretyp ical ly independent of the modulus of elas t ici ty andare inversely proport ional to the coeff icien t o f thermalex p an s io n . Th u s fo r o p t im u m p er fo rm an ce u n d erco m b in ed m ech an ica l an d t h e rm al l o ad s , d es i r ab l emater ial character is t ics are h igh elast ic modulus and lowcoeff icien t o f thermal expansion . Of mater ials avai lab le,on ly advanced composi tes provide the po ten t ial forach i ev in g su ch b en ef ic i a l ch arac t e r i s t i c s .

A C K N O W L E D G E M E N T SThe research effor ts o f the au thor are supported in par tby the Aero thermal loads Branch and the Aircraf tS t ru c tu res Bran ch a t NASA Lan g ley an d t h e L ig h tThermal St ructures Center at the Universi ty of Virg in ia .The au thor i s most appreciat ive of the encouragement ofh i s NASA t ech n ica l m o n i to r s Drs Al l an R Wie t in g an dJames H Starnes , Jr . The au thor i s also gratefu l for veryhelpfu l d iscussions wi th Dr Michae l P Ne meth of theAircraf t S t ructures Branch . Max L Blosser of theAircraf t S t ructures Branch provided importan t ass is tancein ident i fy ing and obtain ing references and valuabled iscussion of past exper imental s tud ies .



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