This article was downloaded by: [Universitaetsbibliothek Dortmund]On: 17 October 2014, At: 07:51Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Journal of Thermal StressesPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/uths20

TEMPERATURE-DEPENDENT CREEP BUCKLING OF PLATESDavid A. Ross a & Laslo Berke ba University of Akron , Akron, OHb NASA Lewis Research Center , Cleveland, OHPublished online: 23 Feb 2007.

To cite this article: David A. Ross & Laslo Berke (1981) TEMPERATURE-DEPENDENT CREEP BUCKLING OF PLATES, Journal ofThermal Stresses, 4:2, 237-247, DOI: 10.1080/01495738108909965

To link to this article: http://dx.doi.org/10.1080/01495738108909965

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

TEMPERATURE-DEPENDENT CREEP BUCKLING OF PLATES

David A. Ross University o f Akron. Akron, OH

Laslo Berke NASA Lewis Research Center, Cleveland, OH

The Norton-Bailey power law for material creep is used to predict the time- dependent lateral deflection of pat rectangular plates. In particular, the plates considered have a through-thickness steady-state temperature distribution. This effect is considered by using Maxwell's law to modify the power creep law. The plate lateral-deflection equations are derived for creep exponents of 3 and 5, using the "sandwich plate element" frequently used to predict creep buckling of plates. It is shown that, for one engineering material, the predictions have approximately the same degree of agreement with experimental data as have the respective creep laws.

INTRODUCTION

The problem of creep buckling of structures with a nonconstant temperature distribution has long been recognized. Hayashi [5] investigated this problem for a column with a temperature variation across its cross section,and he and others [4,9,10] have investigated a column with a nonconstant temperature distribution along its length. This problem has, however, become more important with the development of high-performance jet-aircraft engines. In particular, the combustor liners of such engines are required to act both as structural members and as cooling fins. The situation is thus one of a shell subjected to both axial stress (due to bending stresses) and a through-thickness temperature variation. As an approach to a solution of this problem, this paper presents two solutions to the temperature-dependent creep buckling behavior of a flat, simply supported plate.

To analyze creep buckling behavior, it is usual to assume a Norton-Bailey or power creep law [ 2 , l l ] of the form

where P = creep strain rate, k = creep constant, a = stress, and n = exponent (constant).

The work reported herein was commenced while the senior author was participating in a NASA/ ASEE Summer Faculty Fellowship at NASA Lewis Research Center, Cleveland, Ohio, during summer 1980. Among the many helpful colleagues, the authors would like to express their special appreciation for the encouragement and advice of Dr. C. C. Chamis.

Journal of Thermal Stresses. 4:237-247.1981 Copyright O I981 by Hemisphere publishing ~ o r ~ o k a t i o n

0149-5739/81/020237-11$2.75

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek D

ortm

und]

at 0

7:51

17

Oct

ober

201

4

238 D. A. ROSS AND L. BERKE

Figure I Plate element properties and forces.

T o consider temperaturevariationsit is usual t o modify Eq. (1)so that i t attains the form, given by Maxwell's law [ I 31 .

where H = creep activation energy, R = gas constant, and T = temperature (absolute). For Eq. (2) to be used for investigations of creep buckling of plates, it is necessary

to consider the two-dimensional equations given by

where the subscripts i and j are x and y in cyclic substitution, a,, oy = inplane stresses in the x and y directions, respectively, and

J2 = stress invariant = u + o 2 - u u

X Y X Y

However, to apply the plate bending equations to the creep expressions of Eq. (3), "plate sandwich element" [6, 121 of the type shown in Fig. 1 is usually assumed. Under this approximation, a cross section is considered t o be composed of two flanges rigidly connected by a web of zero bending rigidity. Thus it is no longer necessary to derive a stress/strain distribution across a cross section (with its attendant integration difficulties), but it is assumed that the stresslstrain state is considered to be adequately represented by the stresslstrain state of the centroid of each flange. When steady-state temperature distributions are introduced through the thickness of a plate, it may be shown that the sandwich element may adequately consider temperature distributions of parabolic or lower order.

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek D

ortm

und]

at 0

7:51

17

Oct

ober

201

4

TEMPERATURE-DEPENDENT CREEP BUCKLING OF PLATES 239

The adoption of a suitable exponent of the creep power law, n, in Eq. (I), has also been the subject of investigation for the case of creep buckling of plates of constant temperature [7]. This problem has not, however, been adequately addressed for the case where a flat plate has a through-thickness temperature differential. It is the purpose of this paper, therefore, t o consider this topic, presenting a comparison of creep buckling predictions for the cases where the creep exponents are 3 and 5.

THEORETICAL DEVELOPMENT

If it is assumed that the deflected shape of a simpy supported flat plate may be adequately represented by a half-wave cosine function in each of two perpendicular directions at all times (which is reasonable for approximately square plates), then the behavior of that plate is determined by the plate deflection at one point, say the center, of the plate. If the plate has an initial manufacturing imperfection given by its value at the plate center, w,,i, and a temperature differential is applied between sides of the plate, then the central deflection due to that temperature differential, WOT [IS] ,may be approximated by

where a, b = plate dimensions (see Fig. 2), h = plate thickness, Ti, To = inner and outer plate temperatures respectively (see Fig. 2), a = material linear coefficient of thermal expansion, and v = Poisson's ratio for the material. The central deflection w,, after the application of a through-thickness temperature differential, but before the application of

Figure 2 Plate dimensions, applied stresses, and deflections.

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek D

ortm

und]

at 0

7:51

17

Oct

ober

201

4

240 D. A. ROSS AND L. BERKE

inplane stresses, is thus taken to be given by

At time t = 0, an axial force is added t o the plate, as indicated by Fig. 2. It has been shown 181 that the immediate deflection upon adding the axial load t o the system, w,,,, is given by

where a = applied axial stress, and o~ = Euler buckling stress. For a plate subject t o an inplane compressive load. it can be shown that the critical

Euler buckling stress is given by

Since it is not reasonable that a legitimate value of wOp would be obtained if the applied stress a was greater than the material yield stress, this investigation inserted the material yield stress f, in place of the Euler buckling stress in Eq. (6 ) wherever the yield stress was less than as. While this is only an approximation t o a plate buckling criterion, it is certainly better than using an without qualification in Eq. (6) .

The stresses and lateral displacement are now assumed to have the form

ax u = A3 + A4 C O S C O S % y i

0 - - TY

xo COS B1 cos - a

71X B2 cos -- a cos 2 b

ax w ( t ) = w o w cos < cos zi

where A1 to A4 and BI , B2 are constants, oko, ski = stress in direction k on outside and inside faces of the sandwich element, respectively, and w,,(t) = central deflection of the plate at time t. By applying plate equilibrium and continuity conditions t o Eq. (8) it is possible to evaluate all the unknown constants. However, for algebraic convenience, it is usual [3] t o assume that the solution for the lateral deflection of the plate at time 1, w,(t), is given by the sum of a small deflection w,(t) and a large deflection wp(t). It

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek D

ortm

und]

at 0

7:51

17

Oct

ober

201

4

TEMPERATURE-DEPENDENT CREEP BUCKLING OF PLATES 241

was shown [I41 that when the creep exponent is 3, the expressions for these components of the plate deflection are given by

and

where = b2/a2, and k j = material creep constant in Eq. (3) for n = 3. It may also be shown that , when the creep exponent is 5 , Eq. (9) and (1 0) become

and

where k5 = material creep constant in Eq. (3) for n = 5.

NUMERICAL EXAMPLE

To determine the reliability of these predictions the two sets of developed expressions, represented by Eq. (9) through (12), were used t o make predictions on a flat plate of

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek D

ortm

und]

at 0

7:51

17

Oct

ober

201

4

242 D. A. ROSS AND L. BERKE

material likely t o be used in jet-engine combustor liners. To simplify the comparison it was assumed that the material properties of the plate could be taken as those a t the mean temperature T, of the two sides of the plate. For the purpose of the example, the material chosen was Incoloy Alloy 800, which, at a temperature of 1000 K (1341 O F ) has the following material properties

f = 103.4 MPa (15 k s i ) Y

E = 159 GPa (23000 k s i )

The plate dimensions and the initial imperfection were chosen t o be

a = b = 0.305 m (12 i n )

h = 5.08 x 1 0 - ~ m (0.2 i n )

' 0 i = 3.05 x I O - ~ ~ (0.012 i n )

with the gas constant R given by

The other values required are the appropriate creep constants k, and the energy of actha- tion H. Determination of these values proved somewhat problematic, owing to a significant scatter in experimental data available. [t seems reasonable that the energy of activation for the creep process should not be a function of the creep expression used. The value adopted was that o f

which gave the best fit t o available experimental data [ I ] . Using the same data the creep constants were derived as

3 = 3.05 x 1 0 8 / ~ p a 3 . sec (1.0 x 1011/ksi3 . sec)

= 5.71 x 1 0 5 / ~ p a 5 . sec (8.2 x 109/ks i5 . sec)

It is important to recognize that these values are mean values derived for temperatures ranging from approximately 900 t o 1240K and for stress values in the range 6.90 to 69.0MPa (1 to 10ksi). Within this range of temperature and stress a large range in the

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek D

ortm

und]

at 0

7:51

17

Oct

ober

201

4

TEMPERATURE-DEPENDENT CREEP BUCKLING OF PLATES 243

values of k, may be justified. Certainly any one experimental data point may produce a k, value that varies from that assumed by a factor up t o one order of magnitude. This variation may be due to either experimental scatter or t o dubious assumptions in the derivations of the creep buckling expressions. Whatever its source, these variations in possible k, values do give an indication of the permissible variations in analytical predictions made.

Figures 3 , 4 , and 5 present predictions of lateral deflection as functions of time for applied axialstressesof 6.9, 13.8, and 34.5 MPa (1 ,2 , and 5 ksi), respectively. A particular value of AT represents a temperature variation between inner and outer plate surfaces such that

For given values of AT and o, Figs. 3 , 4 , and 5 compare the predictionsof lateral deflection made by the creep power law of Eq. (3) with the creep exponent n taken as 3 and 5. Although all the curves show the familiar increasing strain rate with time, the predictions clearly have a degree of consistency which varies with both applied stress and temperature variation between plate surfaces. For example, the curves of Fig. 3 seem to show that, for an applied axial stress of 6.9 MPa ( I ksi), the agreement between the predictions gets better as the temperature variation increases. However, this trend is not supported, by Figs. 4 and 5 , where the reverse appears to be true.

a = 1 k r i

I I I I I I I

lo5 I

lo6 I

1 o8 Time (sec)

Figure 3 Creep deflection as a function of temperature differential and creep exponent for applied stress o f I ksi (6.9 MPa).

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek D

ortm

und]

at 0

7:51

17

Oct

ober

201

4

o = 2 ksi

Figure 4 Creep deflections as a function of temperature differential and creep exponent for applied stress o f 2 ksi (13.8 MPa).

Figure 5 Creep deflections as a function of temperature differential and creep exponent for applied stress o f 5 ksi (34.5 MPa).

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek D

ortm

und]

at 0

7:51

17

Oct

ober

201

4

TEMPERATURE-DEPENDENT CREEP BUCKLING 01: PLATES 245

Figure 4 also illustrates that there is no consistency in which creep exponent leads t o greater collapse times. For an applied axial stress of 13.8 MPa (2 ksi) there is a transition at a AT value of approximately 60K. It would appear that for temperature differentials with a AT value greater than about 6 0 K, the creep exponent of 3 in the creep power law leads to longer predicted time values until creep buckling occurs, while for temperature differentials less than about 60 K, the creep exponent of 5 leads t o longer predicted time values until creep buckling occurs.

Figure 5 provides further insight into this consideration of the consistency of predic- tions of time until creep buckling occurs. A comparison of thecurves derived for ATvalues of 10 and OK (the latter representing a constant temperature across the section) shows that there is greater variation between the predictions using the two different creep exponents than between curves using the same creep exponent and a temperature variation corresponding to a AT value of 10 K. It is worthwhile remembering that a AT value of 10 K corresponds to a temperature difference between the faces of the plate of 2 0 K (36OF).

Another method of considering the consistency between the predictions using the different creep exponents is to consider the time required for the lateral deflection of the center of the plate due t o creep t o reach a certain arbitrary value. If this arbitrary value is chosen t o be equal to the thickness of the plate, 5.08 X m (0.2 in), then a comparison between the predictions can be made by observing changes in the relationship

where t3 is the time required for lateral deflection of the plate center due t o creep t o reach a value equal t o the plate thickness with the creep exponent n = 3 , and t5 is the time required for lateral deflection of the plate center due t o creep to reach a value equal t o the plate thickness with creep exponent n = 5 . This comparison is shown in Fig. 6 , where the variation of this function is presented as a function of applied stress and temperature variation. The comparison of the predictions is represented by the ordinate of the graph in its logarithmic form, in order that values of t3/tS may be adequately represented when the function is both greater than and less than unity. If all the predictions agreed, therefore, the curves of Fig. 6 would become a straight line along the abscissa. Also represented in Fig. 6 is a discontinuous vertical line corresponding t o a temperature variation at which the equations all predict that there will be no creep buckling at any time. For this example it can be shown that this temperature variation corresponds to a AT value of -4.7 K, which represents the temperature distribution that exactly negates the initial manufacturing imperfection and at which the plate is exactly flat when the axial load is applied.

The range of applicability of these expressions also needs some consideration. It has already been mentioned that the material properties have been adopted for a constant mean temperature of the plate. Experimental data suggest that these properties may be strongly dependent on temperature, and that this dependence is unlikely t o be linear. For this reason it is recommended that the range of AT considered be restricted. For this particular example a maximum AT value of IOOK, corresponding to a temperature difference between the forces of the plate of 200 K (360°F), might be a reasonable value.

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek D

ortm

und]

at 0

7:51

17

Oct

ober

201

4

246 D. A. ROSS AND L. BERKE

Figure 6 Cornparison o f predictions for time until creep deflection reaches specified value for creep esponenls of 3 and 5 .

At a temperature of 1000 K the modulus of elasticity (which affects only the buckling stress) is changing rapidly with temperature.

CONCLUSIONS

The predictions of creep buckling behavior of a flat plate with a temperature variation between its surfaces appears to be somewhat dependent on the creep buckling relationship assumed. When significant scatter in available experimental creep data such that a variation in the creep constants up t o an order of magnitude might be justified, the discrepancy in predictions using the two creep exponents is not unreasonable. For a relatively restricted range of temperature differences between inner and outer plate surfaces the predictions made are at least as good as the experimental data.

This investigation also highlights a continuing problem in creep buckling investigations. It has been common practice to adopt a power-law creep buckling expression, modified if necessary by Maxwell's law and the "plate sandwich element," t o make the differential equations amenable to solution. With these assumptions it is difficult to make the available experimental data produce creep "constants," which indicates either some large scatter in experimental data or some inaccuracy in the assumed relationships.

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek D

ortm

und]

at 0

7:51

17

Oct

ober

201

4

TEMPERATURE-DEPENDENT CREEP BUCKLING O F PLATES 247

REFERENCES

1. Aerospace Stmcturol Metals Hondbook, DOD, Army Materials and Mechanics Research Center, Watertown, Ma., 1975.

2. R. W. Bailey, The Utilization of Creep Test Data in Engineering Design, Proc. Insf. Mech. Eng.. vol. 131 ,p . 131,Nov.-Dec. 1935.

3. L. Berke, Two Problems in Structural Stability, Ph.D. Dissertation, Standord Univ., Stanford, Calif., 1968.

4. T. Hayashi and A. Kuuchi, Creep Buckling of Columns under Axially Non-uniform Temperature Distributions, Trons. Jpn. Soc. Aeronouf. Space Sci.. vol. 8 , no. 12, pp. 15-22, 1965.

5. T. Hayashi, Creep Buckling of a Column Under Non-Uniform Temperature Distribution Over Its Sections, Dons.. Jpn. Soc. Aeronavf. Spoce Sci., vol. 10,no. 17, pp. 49-53. 1967.

6. N. J. Hoff, Creep Buckling of Plates and Shells, in E. Becher and G. K Mikhailov (eds.), Proc. Mech., Moscow University, pp. 124-140, August 1962.

7. N. J . Hoff e t a]., Creep Buckling of Flat Rectangular Plates when the Creep Exponent Ranges from 3 to 7, in A. I . Smith and A. M. Nicholson (eds.),Advances in Creep Design. The A. E. Johns011 Memorial Volume, pp. 4 2 1 4 4 1 , Applied Sciences Ltd., London, 1971.

8. N. .I. Hoff, and I. M. Levi, Short Cuts in Creep Buckling Analysis. Stanford University, Stanford, Calif., Sudaer Report No. 425, June 1971; also in Int. J. Solids Sfruc., vol. 8, 1103 ff, 1972.

9. W. N. Huang, Non-Linear Creep Buckling with Random Temperature Variations, Nuc. Eng. Des., vol. 25, pp. 4 3 2 4 3 7 , 1973.

10. A. Kobayashi, Non-Uniform Temperature Distribution Effects on a Viscoelastic Column Creep Buckling,Proc. Sixth Int. Space Technol. Sci., Tokyo, pp. 331-338, 1965.

I I . R. N. Norton, Creep in Steel at High Temperatures, McGraw-Hill, New York, 1929. 12. F. K. G. Odquist, Mathemoticol Theory of Creep and Creep Rupture. 2d ed., Oxford University

Press, New York-London, 1974. 1 3. Yu N. Rabotnov, Creep Problems in Structural Members. North Holland Publishing, Amsterdam

(Wiley, New York), 1969. (English trans. by F. A. Leckie.) 14. D. A. Ross, Creep Buckling of a Flat Plate with Through-Thickness Temperature Variation,

NASA Technical Memorandum 81627, Nov. 1980. 15. S. Timoshenko, and S. Woinowsky-Kreiger, 771eory of Plates and Shells. 2d ed., McGraw-Hill,

New York, 1959.

Received July 23, 1980

Request reprints from D. A. Ross.

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek D

ortm

und]

at 0

7:51

17

Oct

ober

201

4