Thermal buckling behavior of thick composite laminated plates under nonuniform temperature distribution

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Compvrers & Stntcnws Vol. 41, No. 4, pp. 637-645, 1991 Printed in Great Britain.

THERMAL BUCKLING BEHAVIOR OF THICK COMPOSITE LAMINATED PLATES UNDER NONUNIFORM TEMPERATURE DISTRIBUTION W. J. &EN, F’. D. LIN and L. W. CHENf Department of Mechanical Engineering, Cheng Kung University, Tainan, Taiwan, 70101, Republic of China (Received 25 August 1990)

thermal buckling behavior of composite laminated plates subjected to uniform or nonuniform temperature fields is analyzed with the aid of the finite element method. To account for transverse shear deformation and rotatory inertia, the thermal-elastic Mindlin plate theory is used. It is found that the effects of lamination angle, modulus ratio, plate aspect ratio, and boundary constraints upon the critical buckling temperature are significant. Abstrac-The

INTRODUCTION

Fiber-reinforced composite laminates, which have high strength-to-weight and stiffness-to-weight ratios, are becoming important in weight-sensitive applications such as aircraft and space vehicles. As a result of this thermal buckling analysis of composite laminates is very important, especially in thin-walled members, since structural components of these highspeed machines are usually subjected to nonuniform temperature distribution due to aerodynamic and solar radiation heating. Considerable work has been done on thermal problems of plates. Gossard et al. [l] used the RayleighRitz method to find the deflections of flat or initially imperfect plates subjected to thermal buckling. Hoff [2] established a buckling criterion for the panels of cover plates subjected to thermal stresses. Klosner and Forray [3] analyzed the thermal buckling of simply supported plates under an arbitrary symmetrical temperature distribution. A conformal mapping technique was used to obtain the critical buckling temperature of an irregularly shaped plate by Biswas [4]. Whitney and Ashton [5] studied the thermal buckling of symmetric composite laminated plates with simply supported edges. Prabhu and Durvasula [6] analyzed skew plates under arbitrarily varying temperature distribution using the Galerkin method. Bargmann [7] obtained explicit closed-form approximate solutions for the stability boundary of an initially stress-free plate subjected to a nonuniform temperature field. Bednarczyk and Richter [8] investigated the influence of temperature distribution on buckling modes. They found that the temperatureshock type is most critical and leads to the lowest buckling temperature. Tauchert [9] analyzed thermal buckling of simply supported antisymmetric laminated plates subjected

to a uniform temperature increase. By using the finite element method, Thangaratnam and Ramachandran[lO] studied thermal buckling of composite laminated plates. Chen and his colleagues [12-141 also used the &rite element method to study the problem of thermal buckling and postbuckling behavior of composite laminated plates. To account for the effect of transverse normal strain on the thermal buckling load of laminated plates, Chang and Leu [l l] expanded the z displacement as a thickness quadratic function. In this paper, the finite element method is used to study the thermal buckling of moderately thick antisymmetric angle-ply laminates under a uniform or nonuniform temperature rise. The effects of various important parameters, such as lamination angle, modulus ratio, plate aspect ratio, and boundary constraints, on the thermal buckling loads are presented. PROBLEM

FORMULATION

Consider a laminated plate of N layers, with thickness h, length a, and width b, as shown in Fig. 1. Each layer is taken to be macroscopically homogeneous \ a

‘N-1’N

/

/V

t To whom all correspondence to be addressed.

I

2

Fig. 1. Composite laminated plate.

637

W. J. CIEN et al.

638

and orthotropic. Based on the Duhamel-Neumann law, the stress-strain relations of the mth layer are

Qu QIZ QH

0

0

0

0

0

0

0

Q22

Qn Qz

Qa

0

0 0 0

0 0 0

QM 0 0

QIZ

=

0

Q2)

0 0 0

0 0 QJ5 0 0 QM

+

X

(1)

where al,, az2, and aj3 are thermal expansion coefficients in the principal directions. AT is the temperature rise. In the de~vation of eqn (l), the stresses and strains are defined in the principal material directions (1, 2, and 3) for that orthotropic lamina. However, in angle-ply laminated plates the principal directions of orthotropy of each individual lamina do not coincide with the geometrical coordinate frame. It is necessary then to use the transformed reduced stiffness r

-

=

a,

a2 a3

a*

Q22

023

a3

Q23

033

0

0

0

0

e44

CL

0

0

0

0

cl45

Ps

0

L Q16Q26B6

L *, 1

0

0

&f6

0

0

026

0

036

0

0

xz and yz planes. The coordinate frame is chosen in such a way that the xy plane coincides with the midplane of the plate. Consequently, the strain-displacement equations of linear elasticity are

av -av,+z -J=t;+zlCy ay ay

7.W = %J+ ax

~+z(~+~)=r~y+zKxy (4d)

-d”“+Y$.

YYZ -

ay

Substituting eqn stresses expressed (e”f = [& $, c&]r [ic,, rcY,rc,,]r, [r’]=

(4) into (2), one has the in terms of midplane strains and midplane curvatures (K j = [yXz r,,]r. The stress resultants

Pa)

and moment resultants

i766 M=[;]=/~;z[~]r~

(5~1

.zX- ~1,AT cY

-a,,AT

(2)

can be obtained by integmtion thickness h with eqn (2) ’

The thirteen constants Q, are related to the nine Q, through the usual transformation law. In order to account for transverse shear deformation and rotary inertia effects in the plate, the displacement components are of the form ~(x,Y,z)=uo~~,Y)+z~*,(~,Y)

W

w,

WI

wx,

Ya z) = UokY)

+ Z@y,(&Y)

Y9 z) = web, Yh

(3c)

where a,,, uo, and w, are associated midplane displacements, and eX and eY are the bending slopes in the

of stresses over the

AII

A2

46

&

42

46

A12

A22

A26

42

B22

826

A,6

4426

46

46

826

8%

42

&6

-

I

-

41

42

46

&I

42

42

B26

&2

D22

026

46

B26

46

46

D26

4

-

X

_

0

6,

0 fY 0 6xY K.x

KY - KXY

W

639

Buckling behavior of laminated plates

or compactly

w

where extensional stiffness A,, flexural-extensional coupling stiffness BV, and flexural stiffness Do of the plate are defined as

Strain in the The energy i.e.

matrices of [B,,], [&,I, [&], and [B,] are listed Appendix. total potential energy rc is the sum of strain rt, and work done by the membrane force II*, II =x,+x*,

where [N, 6x+ Ny67+ Nxyv,y + Qxrxz

Mj,&D,)=

5 m-l

Qyryzlh dy

+

%I+1

X

I

&‘(l,Z,Z2)dz,

i,j= 1,2,6

(7a)

Gl

a,, aj are shear correction factors. @ is the transformed reduced stiffness of the mth layer with ply thickness from z,,, to z,,,, , . Thermal stress resultants, [N,], and thermal moment resultants, [M,], are defined as

~N,l=[~]=~*~“~[~~A~dz

@a)

~~,l=[~]=~,~~+‘~[~~A~zdz.

@W

3=3-X(3+%$ 2

=e~~~~~!~~hdy = fwr&lbd with element structural stiffness matrix

The problem is solved by n number of quadrilateral finite elements with eight nodes, 40 degree of freedom elements. The displacement components are approximated by the product of shape function matrix [NJ and the nodal displacement vector

14;) = hi

OOi wOi Yx~

rgYil’9 i.e.

{QJ=,i, [Nil{qFI* The superscript e of {q;} denotes these variables are defmed on the element and need to be determined. The midplane strains and curvatures can be rewritten as

+

PG1VW,1+ b%17tDIP,l

+ P,l%fl[411 dx dy, element geometric stiffness matrix

PGI =

[~,l~NolP,l dr dy,

and element thermal load vector

El = [No] =

[

2 : . Y

1

(9)

640

W. 3. CHW el al.

For the displacement field of prebuckling, the rn~~rn~t~on of nt with respect to the generalized displacement vector (q) gives the following set of equations

Wb >= VII.

w-3

For the critical buckling state corresponding to the neutral equilibrium condition, the second variation of total potential energy n must be set to zero and the following standard eigenvalue problem is obtained II4l+4&ll=O.

(11)

of d and the initial guessed value AT is T,.

The product

the critical buckling temperature

Finite element results are presented for antisymmetric angle-ply laminated plates, which are made from an equal number of layers oriented at +0 and -8 to the midplane. Each layer has equal thickness with the same foilowing material properties

4

G2

4

xy=z

v,* =

v,3 =

0.25,

G3

=

0.6,

Nondimensional critical buckling temperature for a simz supported laminated plate EJE,, = 181.0, E,/E, = 10.3, G,,/E,, = 7.17 G,,& = 2.39, G,,/E,, = 5.98, v12= 0.28, 01,/a,, = 0.02, aria0 = 22.5, 6 = 45” No. of layers 0

Ref. [9]

Present

4 8

2.032 2.281

2.038 2.278

The 3 x 3 Gaussian quadrature rule is used in numerical inaction. Two cases are considered: with uniform temperature rise and with nonunifo~ temperature rise. Part (a): Uniform temperature rise

RESULTSAND DISCUSSION

-=40,

Table 2. Comparison between the present solution and that of ref. [9] for ~ti~et~c a&e-ply laminate with a different number of layers

G23

E

= 0.5

The finite element mesh is a 3 x 3 mesh. Since the plate is subjected to uniform distribution stresses, there is no prebuckling deformation and work done by in-plane stresses is zero. It is not necessary to determine the stresses in advance. To validate the derived equations, the obtained critical buckling temperatures of simply supported isotropic plates and simply supported orthotropic

2

a, = 1 x 10-6/OC,

cr, -=

2

4

aI

(8)

5 K;=K&=-.

The boundary

conditions

6

are:

1. Simply supported edge x = 0, a,

,=w,=Q),=o

y=O,b,

u,=w,=@,=O.

2. Clamped

I

I 75

I 30

edge u,=uO=wO=QI~=Qiy=O

x=O,a, y =

0

0, b,

110 =

I 45 e

I 60

I 76

90

N=8

1

75

90

61

M

u, = wg = Qi, = By = 0.

Table 1. Comparison between the present solutions and results of ref. [IS] for an isotropic plate Nondim~sional

critical buckling temperature for a simply supported isotropic plate E = 1, a/t = 100, Y = 0.3, 01= 1.0 x 1O-6

0 0.25 0.5 1.0 1.5 2.0 2.5 3.0

Ref. [15]

Present

0.686 0.808 1.283 2.073 3.179 4.599 6.332

0.691 0.814 1.319 2.101 3.191 4.601 6.330

,0

15

30

45

60

e Fig. 2. Effect of ply orientation on tbe critical buckling temperature of laminates (u/b = 1, a/t = 20); (a) simply supported, (b) clamped.

Buckling behavior of laminated plates

laminated plates subjected to a uniform temperature increase are compared with these of Boley and Weiner [15] in Table 1 and the results of Tauchert [9] in Table 2. They are in excellent agreement. The critical buckling temperature T, versus lamination angle 6 is shown in Fig. 2 for both simply supported and clamped plates. It can be seen that variations in lamination angle 0 may result in large changes of T,. Also it can be observed that the critical temperature for a given thickness of laminated plates increases as the number of layers, N, increases. The maximum value of T,, occurs at 0 = 45” for simply supported N = 4 and N = 8 plates. However, the reverse phenomenon is observed in twolayer laminates. This is because bending-stretching coupling stiffness reach their maximum values at stacking layers N = 2 and decrease rapidly as N increases. Figure 3 shows the effect of plate thickness ratio a/t on the critical temperature for square laminates having lamination angle 0 = 45”. It is evident that the rigidity and hence the critical temperature decrease rapidly as the plate thickness ratio increases. The effect of the number of stacking layers N on T, is insignificant when a/t is large. Figure 4 shows the influence of the modulus ratio E,/E, on critical

641

buckling temperature. It is observed that in a simply supported plate of N = 4 and N = 8, T, increases with increase of modulus ratio and plotted curves are rather flat when E,/Ez 2 10. However, T, decreases as the modulus ratio increases when the plate is clamped. The effect of aspect ratio a/b on the critical temperature is illustrated in Fig. 5. It can be seen that T, goes up as the plate aspect ratio increases. It is concluded that there is no change of the buckling mode shape with the variation of aspect ratio, since the curves go up smoothly without any cusp. The effect of thermal expansion coefficient ratio aJa, on the critical temperature is shown in Fig. 6. The higher the ratio of thermal expansion coefficients, the lower the value of T,. Figure 7 shows the effect of boundary condition on the variation of critical temperature. The boundary condition has a strong impact on the critical temperature T,, as shown in Fig. 7. Finally, the variation of T, for different aspect ratios is presented in Fig. 7 for both simply supported and clamped plates with N = 4 in order to compare the effect of the boundary condition. It can be seen that the critical temperatures of clamped cases are always higher than those of the simply supported.

4,

1

(a)

N=S

t-9 0 x ii 2 l-

10

20

30

40

60

N=2

0

I

I

I

10

20

30

a/t

E1

40

IE2

10

(b)

I

16

N=4

10

20

30

40

50

a/t

Fig. 3. Effect of plate thickness on the critical buckling temperature of laminates (a/b = 1, 0 = 45”); (a) simply supported, (b) clamped.

00

10

20

30

40

El/E2

Fig. 4. Effect of modulus ratio on the critical buckling temperature of laminates (u/b = 1, a/t = 20, 0 =45”); (a) simply supported, (b) clamped.

J. CXEHet al.

6.6

22

“,

4.4

5

N=4

2.2

N=2

‘I---

0

0.6

1.2

1.8

2.4

_I

3

alb

5

g-

X c #$

6

30' 0

? I

I

0.6

1.2

I

I

7.3

2.4

I-

03

3

0

a/b

4

6

12

16

20

24

a2lat

Fig. 5. Effect of aspect

ratio on the critical buckling temperature of laminates (a/r = 20, B = 45”); (a) simply supported, (b) clamped.

Fig. 6. Effect of thermal expansion ratio on the critical buckling temperature of laminates (a/b = 1, a/r = 20, 6 = 45”); (a) simply supported, (b) clamped.

Part (b): Nonuniform temperature rise

To denotes the part of uniform temperature rise, and T, denotes temperature gradient, as shown in Fig. 8. Note that since the stresses are not uniformly distributed and prebuckling deformations are no longer zero, it is necessary to solve eqn (IO) in advance in order to obtain the critical buckling temperature from eqn (11). To validate the finite element analysis fo~ulation, the critical buckling temperatures of simply supported isotropic plates are compared with results of Gossard et al. [I] in Table 3. Both results are in exceilent agreement. The critical temperature gradient T,,, of antisymmetric angle-ply laminates for different lamin-

The finite element mesh tent-like

temperature

where

is a 4 x 4 mesh.

The

rise is

w, Y) = 14

12

10

g

8

j

6

X v-

4

2

0

0.8

1.2

1.6

2.4

:

e/b Fig. 7. InIIuence of boundary condition and aspect ratio on the ctitical buckling temperature of laminates (a/r = 20, N = 4, B = 45’).

Fig. 8. Nonuniform tent-like temperature distribution.

Buckling behavior of laminated plates

Table 3. Comparison between the present solution and that of ref. _ [l]_for an isotropic plate subjected to tent-like tempera-&

distrib
Simply supported laminated plate subjected to a tent-like temperature field o = 35.25, b = 22.5, t = 0.25, Y =0.33, a = 12.7 x 10-6, T, = 95.31, T, = 150 Nondimensional critical bucklinn- temoerature _ 4b2EaT,,f

Ref. _ 111 _

Present

5.39

5.45

ation angle 0 is presented in Fig. 9 for both simply supported and clamped plates. It can be seen that T,, reaches its maximum value at 0 = 45” and 0 = 90” for simply supported and clamped plates, respectively. Note at 0 = 45”, the critical temperature of two-layer simply supported laminates is only 58% of that of four-layer laminates. The reason is, as mentioned before, the two-layer laminated plates have a significant coupling effect. Figure 10 shows the influences of modulus ratio E,/E2 upon the critical temperature gradient T,,. It is observed that the T,, of a clamped plate decreases rapidly as the modulus ratio increases. The effects of temperature distribution on the critical temperature for different lamination angles is

0

l

I

I

I

I

15

30

45

60

75

90

643

represented in Fig. 11. It can be noted that the larger T,/T, is, the lower T,, will be. Furthermore+ the maximum value of the critical temperature always occurs at e = 45” in all cases. The critical temperature gradient T,n with T, = 0 and critical temperature T&, with T, = 0 are studied in Fig. 12 for simply supported laminates with N = 4. Both cases reach their maximum values at 8 = 45”. The critical temperature of laminates subjected to gradient temperature distribution is much higher than that of those subjected to uniform temperature rise because in-plane stresses of the latter are much higher. The critical temperature gradient TLC, of simply supported laminates versus the aspect ratio a/b is shown in Fig. 13. It must be noted that T,, increases with the increase of stacking layers and aspect ratio. Finally, Fig. 14 shows the effects of boundary condition on the critical temperature gradient of laminates with N = 4. It is evident that clamped plates have a much higher T,,, because of stiffer constraints. CONCLUSION

Thermal buckling, induced by uniform or nonuniform temperature rise in composite laminated plates, was studied by the finite element method using

0

1

I

I

I

10

20

30

40

30

40

8

hlE2

12,

I

(b)

N=8

9.6 -

2.4

t

0

15

30

45

90

75

90

0

Fig. 9. Effect of ply orientation on the critical temperature gradient of laminates (u/b = 1, a/t = 20, To= 0); (a) simply supported, (b) clamped.

0

10

20 El/E2

Fig. 10. Effect of modulus ratio on the critical temperature gradient of laminates (u/b = 1, a/t = 20, 0 = 45”, To = 0); (a) simply supported, (b) clamped.

644

W. J.

CliBN et al.

‘?2 X

9-

a'

0.6 -

,_!S42-

6

(b)

1 0

0.6

1.2

1.8

2.4

3.0

a/b Fig. 13. Effect of aspect ratio on the critical temperature gradient of laminates (a/r = 20, 0 = 45”, T, -O), simply supported. T,/To=2

1.2 -

o0

15

30

2. Clamped plates are stiffer than simply supported plates and their critical buckling temperatures are higher than those of simply supported.

45

60

75

90

U

Fig. 11. Effect of temperature distribution and ply orientation on the critical buckling temperature of the laminates (a/b = 1, n/t = 20, N = 4); (a) simply supported, (b) clamped. the eight-node Serendipity element. The study shows that: 1. Thermal buckling load is significantly influenced by lamination angle, modulus ratio, plate aspect ratio and the type of temperature distribution.

I 0

1

1

1

I

I

I

l

10 20

30

40

50

60

70

60

I

9

Fig. 12. Effect of ply orientation on critical buckling temperature under uniform and nonuniform temperature increase (u/b = 1, a/t = 20, N = 4, simply supported).

3. The critical buckling temperature increases with the number of stacking layers N. 4. The critical temperature graident T,, (with T,, = 0) is higher than the critical buckling temperature T,,(with T,= 0). 5. The bending-stretching coupling stiffnesses, which greatly affect the rigidity of the plates, reach their maximum values when N = 2 and decreases rapidly with increase of N.

1 0

I

I

I

I

I

15

30

45

60

76

90

e Fig. 14. Influence of boundary condition and ply orientation on the critical temperature gradient of laminates (a/b = 1, a/t =20, N =4, T,=O).

645

Buckling behavior of laminated plates REFERENCES

1. M. L. Gossard, P. Seide and W. M. Roberts, Thermal buckling of plates. NACA TN 2771 (1952). 2. N. J. Hoff, Thermal buckling of supersonic wing panels. I. Aeronaut. Sci. 23, 1019-1028 (1956). 3. J. M. Klosner and M. J. Forray, Buckling of simply supported plates under arbitrary symmetrical temperature distributions. J. Aeronaut. Sci. 25, 181-184 (1958). 4. P. Biswas, Thermal buckling of orthotropic plates. J. Appl. Me& 43, 361-363 (1976). 5. J. M. Whitney and J. E. Ashton, Effect of environment on the elastic response of layered composite plate. AIAA J. 7, 1708-1713-(1971). 6. M. S. Prabhu and S. Durvasula. Thermal buckling of restrained skew plates. J. En& Mech. ASCE ifJO, 1292-1295 (1974). 7. H. W. Bargmann, Thermal buckling of elastic plates. J. Thermal Stresses 8, 71-98 (1985). 8. H. Bednarczyk and M. Richter, Buckling of plated due to self-equilibrated thermal stresses. J. Thermal Stresses 8, 139-152 (1985). 9. T. R. Tauchert, Thermal buckling of thick antisymmetric angle-ply laminates. J. Thermal Srresses 10, 113-124 (1987). 10. K. R. Thangaratnam and J. Ramachandran, Thermal buckling of composite laminated plates. Compur. Struct. 32, 1117-1124 (1989). 11. J. S. Chang and S. Y. Leu, On the effect of thickness direction deformation to thermal stability of laminated plates. The Twelfth National Conference on the Theoretical and Applied Mechanics, Taipei, Taiwan, R.O.C., pp. 407-421 (1988). 12. L. W. Chen, E. J. Brunelle and L. Y. Chen, Thermal buckling of initially stressed thick plates. J. Mech. Des. 104, 557-564 (1982). 13. L. W. Chen and L. Y. Chen, Thermal buckling of laminated cylindrical plates. Composite Struct. 8, 189-205 (1987).

14. L. W. Chen and L. Y. Chen, Thermal buckling analysis of composite laminated plates by the Bnite element method. J. Thermal Stresses 12,41-56 (1989). 15. B. A. Boley and J. J. Weiner, Theory of Thermal Stress. John Wiley, New York (1960). APPENDIX

ah: ax [&I =

0

0

0

0

0

0~000, aNi _-ay 0

PA =

0

aNi

ax

0

0

0

alv,

ax

0 alv,

O O O Oay 000

alv,

F

an: -

ax