Thermal buckling of bimodular sandwich beams

Composite Structures 25 (1993) 345-352

Thermal buckling of bimodular sandwich beams Tin Lan, Psang Dain Lin & Lien Wen Chen* Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, 70101 This paper presents a model that explores the thermal buckling of three-layer sandwich beams possessing thick facings and moderately stiff cores. Bimodular facings and core material are used. In contrast to conventional theory, the effects of transverse shear deformation in the facings as well as the effect of the stretching and bending action in the core on thermal buckling are considered. The governing equations are derived using the principle of minimum total potential energy and the fact that its second derivative is zero. The finite-element results are presented in order to investigate the effects of important parameters such as thickness, thermal expansion coefficients and moduli ratio on critical buckling temperatures.

1 INTRODUCTION One of the most important and widely popular composite structures is that of the sandwich. As the core maintains a definite separation of two facings, the bending resistance of a sandwich structure is larger than a single facing. Although the volume of a sandwich structure is larger, it has the benefits of light weight and high bending resistance which has made the sandwich structure an important component in widespread applications. The study of the sandwich structure was first addressed by Hoff and Mautner in 1945.1 Many subsequent authors, such as March, 2 Hoff, 3,4 Libove and Batdorf, 5 Hemp 6 and Eringer 7 successively investigated its bending problems. Gordaninejad and Bert developed a new model for the sandwich beam, 8 and applied it in the study of vibration and bending problems. 9 The thermal stress of the sandwich structure was investigated using the theory of elasticity by Bijlaard 1° and Pogorzeiski, 11 who subjected sandwich plates to thermal gradients. Ebcioglu, 12,13 using the principle of minimum total potential energy, determined the stress induced on the sandwich structure by transverse loadings, boundary forces and temperature gradients. Hussein et al. 14 considered the effects of inter-layer shear stresses on thermal stresses. The concept of a bimodular material was introduced by Timoshenko,15 who in 1941 considered *To whom correspondence should be addressed.

the flexural stresses of such a material undergoing pure bending. Tran and B e r t , 16 u s i n g the transfer matrix method, studied the small bending deflection of homogeneous bimodular beams. Column buckling of general bimodular materials was considered by Rigbi 17 and Rigbi and Idan, 18 while the bifurcation buckling of a uniform, slender, cantilever column constructed of bimodular material was studied by Bert and K o . 19 Dynamic analyses of bimodular beams known at present were investigated by Khachatryan, 2° Galoyan and Khachatryan, 21 Bert and T r a n 22 and Rebello et al. 23 A l l of these works involved either free or forced vibrations. This paper studies the thermal buckling of bimodular sandwich beams possessing thick facings and moderately stiff cores under uniform or tent-like temperature fields. The governing equations are derived using the principle of minimum total potential energy. The finite-element results are presented in order to study the effects of a number of important parameters on the critical buckling temperature.

2 FINITE ELEMENT FORMULATION Consider a three-layer bimodular sandwich beam with total thickness h, and length /, as shown in Fig. 1. h m is the thickness of the mth layer, x and z are respectively the Cartesian position coordinates along the length of the beam and normal to it, measured from the midplane of the beam. Based on the Duhamel-Neumann form of

345

Composite Structures 0263-8223/93/S06.00 © 1993 Elsevier Science Publishers Ltd, England. Printed in Great Britain

Tin Lan, PsangDain Lin, Lien Wen Chen

346

Hooke's law, the stress-strain relations of the mth layer are:

[TO']=[E'm ;kin] [£--: AT]

(2a) (2b)

uo and wo are associated midplane displacements and W is the bending slope. The displacement, u, is required in order to account for the lack of symmetry about the midplane, due to different facing thicknesses or bimodular material behaviour. With the assumptions that the material is not extensible in the thickness direction and that w(x,z) is independent of z. 8 Consequently, the strain-displacement equations of linear elasticity are:

Ou OU°+z Oql '=

= 0-;

G-'° -

+z~c

topfacing(layer1)

(3a)

'>/ I ha I~,\\\\\\'X~X-'X,~.'~'X\\\~ core(layer2) 1L

C

0w 0u

0Wo+ttt 0x

(3b)

As usual, the normal stress, moment, and transverse-shear stress resultants, each per unit width, defined respectively by: =

dz

(4)

where extensional stiffness A, flexural-extensional coupling stiffness B, flexural stiffness D, and transverse shear stiffness S (per unit width) for a laminate beam are defined as:

(A,B,D,S)=[hi2 ( Ekm,ZEkm,Z2Ekm,

K 2 Gk m )

dz

J-h/2

(6)

The shear correction factor, r 2, is obtained from the equilibrium equation of elasticity in the x-z plane and the concept of equivalent strain energy, 24 and is expressed as:

2

( A B - B2)2 =

[ dz] rr Gkm

LJ - h / 2

dz]

(7)

(Ab- Ba2)2

where a and b are the extensional and flexural partial stiffnesses defined as

(z,b)=l

O- hi2

(1,z*)&mdz*

(8)

The thermal stress resultant, N T, and the thermal moment resultant, MT, are defined as:

J - h/2

h/2

can be obtained by integration of stresses over the thickness h with eqn (1):

[Q]N = [ 0BA B OoDOs]I/~i]-[oMt ]Nt

bottomfacing(layer3)

Fig. 1. Three-layer sandwich beam with composite facings.

J-h/2

7=Ox~ 0z =

/

(1)

where e and 7 represent the axial normal and transverse shear strain respectively and o and r are their corresponding stresses; a is the thermal expansion coefficient, and A T is the temperature rise. Note that the subscript k of Ek,n and Gk,~ refers to the sign of strain (k = t for tension and k = c for compression), while the subscript m denotes the layer number (see Fig. 1 ). In order to account for the transverse shear deformation effects, the Timoshenko displacement field is used:

u(x,z) = uo(x ) + zW(x) w(x,z)=Wo(X)

x

(5)

(NT,Mw)=

I

EkmaAT(1,Z) dz

(9)

J - hi2

The integration of eqns (6)-(9) were carried out in a piecewise fashion from one layer to the next, with account taken of the tension or compression within appropriate portions. The problem was solved by a number of quadratic Lagrangian elements with three nodes and

Thermal buckling of bimodular sandwich beams

1

nine degrees of freedom elements. The displacement components were approximated by the product of the shape function matrix, [Nil, and the nodal displacement vector, {qe}= [Uoi Woi ~ i]r, i.e. {q~}= 2 [Ni]{q e}

'dx dZ= e=1

1

The superscript 'e' indicates that these variables are defined on the element and are to be determined. The shape functions are given by: 1

N Ox

(10)

i=1

Nl=~

2 {q}T[KsJ{q}--{q}T[FT] ~z2= ~

3

347

~ ( ~ - - 1)

(lla)

2 {q}T[KG]{q} with element structural stiffness matrix [ge] =

I'

{[NtlTA[Nt]+[Bt]TB[Bb]

-1

N2=l-~ 2

(llb)

1 N3=~ ~(~+1)

(llc)

The midplane strains and curvatures on the element, e, can be rewritten as: 3

{ee} = ~ [B.]{qT}

(12)

+[Bb]'rD[Bb]+[B~]TS[Bs]}[JI d~

(18)

element geometric stiffness matrix [g~l =

I1 [Bg]TN[Bg]I J I

at

(19)

-1

element thermal load vector

i=1

{re} = Z [Bb,]lqT}

(13)

i=1 3 {~e}= ~ [BseJ{qT} i=l e

(14)

3

0w°= ~ [Bgi]{q~i} 0X

(15)

i= 1

The strain matrices of [Bti], [Bbi], [B~i] and [Bgi] are given by:

[B.]=[ON]OxO 0] [Bbi]=[OO ON/Ox] [Bsi]=[00N /Ox Ni] [Bgi]=[OONi/OxO]

(16a) (16b) (16c) (16d)

The total potential energy, at, is the sum of the strain energy, hi, and the work done by the applied distributed normal loading, n2, i.e. n=~l

J-n2

where

.rr,1= ~

{[Bt]TNT+{BblTMTIIJI d~

[f~-] =

3

[Ne + QT] dx dz

(17)

(20)

-1

where IJI is the determinant of the Jacobian transformation matrrix. For the displacement field of prebuckling, minimization of :r I with respect to the generalized displacement vector {q} gives the following set of equations:

[Ks]{q}=[FT]

(21)

For the critical buckling state corresponding to the neutral equilibrium condition, the second variation of total potential energy, n, must be set to zero which gives the following standard eigenvalue problem: [ [K~]+2[KG][ = 0

(22)

The product of ;t and the initial guessed value A T is the critical buckling temperature, Tcr. The different properties of the tension and compression of bimodular materials cause the neutral surface position to shift away from its geometric midplane. In the calculations, as the actual neutral surface positions were not known, an iterative technique was employed in order to compute their locations. The iteration procedure began with an assumed value of the neutral surface position. Dummy stiffness and thermal load were then computed from eqns (6) and (9) using these assumed values. After the eigenvalues and

348

Tin Lan, Psang Dain Lin, Lien Wen Chen

eigenvectors were obtained, the new neutral surface locations were given by: Z , = - eo*/r* = - ( c o - a A T ) / r

(23)

Using these new values, the stiffnesses for the next iteration are computed. This procedure is repeated until the differences between any two consecutive values of the neutral surface at all nodal points are smaller than one tenth of the thickness.

3 NUMERICAL RESULTS AND DISCUSSION In this section, two sets of boundary conditions for beams subjected to either a uniform or tentlike temperature field are studied. For the sake of simplicity, the thermal expansion coefficient of the core is assumed to be zero since its value is in general, very small. 3.1 A beam hinged at both ends (denoted as H-H)

10

8

E, /E¢ - 1000

6

4 ~

0

'~

'

~

'

E, /E, - 100

. . . .

'

. . . .

I'5

.

.

.

.

10 e{~ / ct 3

20

Fig. 2. Effect of transverse expansion ratio al/ct 3 on thermal buckling load T~r for various bimodulus ratios (E, > Ec) for H-H beam under uniform temperature field ( E c t = E o = l x l 0 5 , E z = l x l 0 2 , a2=0, a 3 = l x l 0 -4, GkffEkl = 0"5, L = 600, hi = 3).

0.5 0.4

In this case, the midplane displacements, Uo and wo, at both ends (i.e. x = + l/2) are zero.

3.1.1 A H-H beam under a uniform temperature field The neutral surface of the core is located exactly at the midplane of the beam since strain eo is zero. Moreover, the strain of the facing having the smaller thermal coefficient is always negative. Consequently, the stiffness of the beam depends only upon the neutral surface of the other facing possessing the higher thermal coefficient. The influence of the bimodular elastic moduli ratio (Et/Ec) of the facing and the thermal expansion coefficient ratio, a~/a3, on the critical buckling temperature, Tcr, is shown in Figs 2 and 3. Note that by increasing al/a 3, the beam deflection also increases, and the section bending moment, the neutral position location, and the beam stiffness all change simultaneously. Note in Fig. 2 where E t > Ec, the increase in the thermal expansion coefficient ratio and the shift in the neutral position, sharply changes the stiffness and the buckling temperature. The effect of a l/a3 on the critical buckling temperature for the case of E~ > E t is illustrated in Fig. 3. Note that a decrease in stiffness corresponds to a shift in the neutral position of the upper facing. Since E t has more of

0.3

I E , = 0.2

0.2

=

.

0.1 Et / E

~

0.0

I

50

(A)

~ l

I

I00

1

~

I 0

i

200

~1 / Ct3

Fig. 3. Effect of al/a 3 on Tcr for various bimodulus ratios (Ec>Et) for H-H beam under uniform temperature field (Ecl=Ec3= l x 105, E 2 = l x 102, a2=0, a 3 = l x 10 -4, Gkl/Ekl = 0"5, L = 600, h I = 3).

an influence on a decrease in thermal stress than on a decrease in stiffness, a lower E t / E c value corresponds to a higher critical buckling temperature. The influence of the core's rigidity on the critical buckling temperature is presented in Fig. 4. Higher core rigidity is accompanied by a higher buckling temperature and a larger change in the neutral position. The influence of the upper and lower facings' modulus ratio (E3/E1) and the thermal expansion ratio on the critical buckling temperature is dis-

349

Thermal buckling of bimodular sandwich beams 4

3

/~E:/

Et - 5

4

0

12

(n)

16

a, / ct3

20

Fig. 4. Effect of ctl/ct 3 and core moduli on T~r for H - H beam under uniform temperature field (Ec~ = E¢3-- 1 x 105, E,]=Et3=Ix106, a 2 = 0 , a 3 = l × 1 0 -3, Gkl/Ekl=0"5, L - - 600, h I = 3).

2"

Tl / T0 =" 3

1" ~ , ~ Ts/To m

T t / T o -- 5 ,

~

3.0 i

2.0 v

~E3 0.5

I

10

i

I

20

i

30

Oq / 0~3

/Ez ,=

Fig. 6. Effect of al/a3 and TI/To on Tcr for H-H beam under tent-like temperature field (Ecl -- E¢3 = 1 × 10 5, Etl = E t 3 - 5 x 105, E2 = 1 x 10 2, a2 = 0, 6t3 = 1 x 10 -4, Gkl/Ekl = 0"5, L = 600, hi = 3.

1.0

0.0

more of an effect on the critical buckling temperature. I

10 (A)

'

I

20

'

I

,30

'

I

40

'

50

a~/o~3

Fig. 5. Effect of al/Ct 3 and facing moduli on Tcr for H-H beam under uniform temperature field ( E c t = l x l 0 4 , Etl/E~l= 10, a 2 = 0 , a3 = 1 x 10 -4 , Gkl/Ekl=0.5, L = 6 0 0 , h1=3).

played in Fig. 5 where E t > E c. Note that a higher value of E3/E 1 is accompanied by a lower buckling temperature. This phenomenon can be explained in terms of the lower facing's stiffness, an increase of which affects the location of the upper facings' neutral position, reducing the upper facing stiffness and that of the entire beam as well. Although the thermal stress is also reduced, it only decreases linearly with a change in the neutral position whereas the stiffness decreases by a power of three. A comparison of the effects of a change in the thermal stress and that of the stiffness reveals that the stiffness has

3.1.2 A H-H beam under a tent-like temperature field The tent-like temperature field is given by:

r(x)--

ITo+T (1-2x/l) [ Zo + TI(1 + 2x/Z)

where TO and T1 are the uniform and gradient temperatures, respectively. The influence of temperature distribution (7"1/ To) and the thermal expansion ratio on the buckling behaviour of the bimodular sandwich beam under a nonuniform temperature is shown in Fig. 6. Note that a steep temperature gradient corresponds to a large change in the deflection and in the location of the neutral position. Likewise, a steep buckling temperature gradient corresponds to a large change in the stiffness. The effect of the H-H beam thickness on the critical buckling temperature gradient is shown in

Tin Lan, Psang Dain Lin, Lien Wen Chen

350

Fig. 7. In comparison with a homogeneous beam of the same facing material, the sandwich beam possesses lower stiffness and a highly buckling temperature gradient due to the low thermal expansion coefficient of its core. Note that in general, increasing the thickness of the core is an effective way of increasing the buckling temperature gradient.

perature distribution on the unimodular beam's critical buckling temperature gradient (Tier) is shown in Fig. 8, while the influence of temperature distribution and the thermal expansion co-

4.0 ax / ~3

3.2 A beam clamped at both ends (denoted as

=" 5

3.5

C -C) Note that the displacements, Uo, Wo, and the bending slope qJ are zero at both boundary ends. Also note that, in a uniform temperature field, the C-C beam will not deflect before buckling. Its curvature is zero and the whole beam is subjected to compression. Consequently, the extension modulus, Etm, has no influence on its stiffness before buckling. The critical buckling temperature of a C-C beam subjected to a nonuniform temperature field is illustrated in Figs 8-10. The influence of tern-

3.0

b-

2.0 1.5 ~

1.0

I.o

,

,

,

0.8

!

,

I'o

T1 / To Fig. 8. Effect of cq/a3 and TI/To on Tl~r for u n i m o d u l a r C-C beam u n d e r tent-like temperature field (E = 1 × 105, a = l × 10 -4 , G/E=O.5, L=600, h = 9 ) .

0.8

0.4

h,=6

4

0.2

2

h=*3

0.0

'

!

,

j,

,

~

,

| 12

'

i 18

, 2O

TI / To Fig. 7. Effect of thickness and T l / TOo n T,~r for H - H b e a m u n d e r tent-like temperature field (E~l = Ec~ = l x 105, EtI=Et3=5x105, E 2 = l x l 0 z, a i = l x 1 0 -, %=0, a 3 = 1 x 10 -4, Gkt/Ekt----0"5, L = 600, hl = 3).

Fig. 9. Effect of al/a 3 and TllTo on T, cr for C-C b e a m u n d e r tent-like temperature field (Ec = 1 x 104, E t = 2 x 104, a = 1 x 10 -4, a2=O , ct3--- 1 × 10 -4, Gkl/Ekl =0"5, L = 600, h i - - 3 ).

Thermal buckling of bimodular sandwich beams

I }E' 3

to differences in the thermal expansion coefficients of the facings. When this difference is large, the effects on stiffness, critical buckling temperature, and buckling temperature gradient are more pronounced. (2) In a uniform temperature field, a H-H bimodular beam facing possessing a lower thermal expansion coefficient always undergoes compression, whereas the facing of a C-C bimodular beam is subjected to axial compressive strain and its stiffness remains unchanged until buckling. (3) In a nonuniform temperature field, strain of a C-C beam varies in the axial direction. The thermal strain is nonsymmetric in the direction of thickness if the thermal expansion coefficients of the layers differ. (4) Provided the core's thermal expansion coefficient is low, the best method for increasing sandwich stiffness and the critical buckling temperature gradient is to increase the core thickness.

7.5

E, I E~ "= 5 Et / E c ~' 2

o.o E~ / E~ "= 0.2 Et / E c " 0.13

-2

(() (6) 12

351

16

20

Oh/(X3 Fig. 10. Effect of

a l i a 3 and E t / E c o n Ticr for C-C beam under tent-like temperature field (To=0 , E t ~ = l x l 0 4 , a 2 = 0, a 3= 5 x 10 -5, Gkl/Ekl = 0"5, L = 600, h I = 3).

efficient ratio on the bimodular sandwich beam's critical buckling temperature are shown in Fig. 9. Figure 10 reveals that a C-C beam possessing a high modulus ratio and a lower thermal expansion coefficient ratio has higher critical buckling temperature gradient, Tier.

4 CONCLUSIONS This paper investigated the thermal buckling of three-layer bimodular sandwich beams possessing thick facings and moderately stiff cores. The effects of transverse shear deformation in the facings as well as the effects of the stretching and bending action in the core on thermal buckling are considered. Two sets of boundary conditions, hinged or clamped at both ends, are studied under uniform or tent-like temperature fields. The following important conclusions were obtained: (1) The stiffness of the bimodular sandwich beam varies with the temperature rise due

ACKNOWLEDGEMENT The authors thank to the National Science Council of the Republic of China for supporting this research under grant NSC 81-0401-E-006-05.

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Tin Lan, Psang Dain Lin, Lien Wen Chen

11. Pogorzeiski, J., Thermal deformations and stresses in rectangular sandwich panels with non-rigid cores. Build. Sci., 4 (1969) 79-92. 12. Chang, C. C. & Ebcioglu, I. K., Thermoelastic behaviour of a simply supported sandwich plane under large temperature gradient and edge compression. J. Aerosp. Sci., 28 (6)(1961)480-92. 13. Ebcioglu, I. K., Thermo-elastic equation for a sandwich panel under arbitrary temperature distribution, transverse load and edge compression. Proc. 4th US Nat. Cong. of Applied Mechanics, ASME, New York, 1962, pp. 537-46. 14. Hussein, R., Ha, K. & Frazio, P., Thermal stresses in sandwich plates. J. Therm. Stresses, 12 (1989) 333-49. 15. Timoshenko, S., Strength of Materials, Part 2, Advanced Theory and Problems. Van Nostrand, Princeton, NJ, 1941. 16. Tran, A. D. & Bert, C. W., Bending of thick beams of bimodulus materials. Comput. Struct., 15 (6) (1982) 627-42. 17. Rigbi, Z., The buckling of bimodular columns. Acta Mech., 18 (1973) 317-32.

18. Rigbi, Z. & Idan, S., Buckling and immediate postbuckling behavior of bimodular columns. J. Struct. Mech., 6 (1978) 145-64. 19. Bert, C. W. & Ko, C. L., Buckling of columns constructed of bimodular materials. Int. J. Struct. Dynam., 10(1982) 551-60. 20. Khachatryan, A. A., Longitudinal vibrations of prismatic bars made of different-modulus material. Mech. Solids, 2 (1967) 94-7. 21. Galoyan, A. G. & Khachatryan, A. A., On transversal vibration of beams made from different modulu material (in Russian). Dokl. Akad. Nauk Armyanskoi SSr, 66 (1967) 2-26. 22. Bert, C. W. & Tran, A. D., Transient response of a thick beam of bimodular material. Earthqu. Engng Struct. Dynam., 10 (1982) 551-60. 23. Rebello, C. A., Bert, C. W. & Gordaninejad, E, Vibration of bimodular sandwich beams with thick facings: A new theory and experimental results. J. Sound Vib., 90 (1983) 381-97. 24. Sun, L. X. & Hsu, T. R., Thermal buckling of laminated composite plates with transverse shear deformation. Comput. Struct., 365 (5)(1988) 883-9.