Vietnam Journal of Mechanics, VAST, Vol.30, No.3 (2008), pp. 133–141

A NEW APPROACH FOR INVESTIGATING

CORRUGATED LAMINATED COMPOSITE

PLATES OF WAVE FORM

Dao Huy Bich

Vietnam National University, Hanoi

Khuc Van Phu

Military Logistic Academy

Abstract. Corrugated plates of wave form made of isotropic elastic material were con-

sidered as flat orthotropic plates with corresponding orthotropic constants determined

empirically by the Seydel’s technique. In some recent researches the extension of this

technique was given for corrugated laminated composite plates.

In the present paper a new approach for investigating corrugated composite plate of

wave form is proposed, regarding this plates as a combination of parts of shallow cylindri-

cal shells with alternative curvatures. It reduces to no use of Seydel’s empirical formulas

and sufficiently apply to composite plates. Based on this approach governing equations of

corrugated laminated composite plate of wave form are developed and application to the

non-linear stability problem of this plate is considered. Obtained results are compared

with those of Seydel’s technique.

1.

INTRODUCTION

Corrugated plates of wave form made of isotropic elastic material were considered as

flat orthotropic plates with corresponding orthotropic constants determined empirically

by Seydel’s technique. This approach was acceptable to solve many bending and stability

problems of corrugated isotropic elastic plates in practice [4, 5, 7]. However, the analysis

of corrugated laminated composite plates has received comparitively little attention.

In [1] the authors developed the Seydel’s technique to the bending problems of cor-

rugated laminated composite plates and cylindrical shells. In the stability problem of

corrugated laminated composite plates [2] besides bending stiffnesses to be extended it is

necessary to formulate extensional stiffnesses and to define more exactly the strain expres-

sion by including the curvature of middle surface of corrugated plate. But the extension of

Seydel’s technique to corrugated composite plates meets with difficulties in determination

of corresponding constants and experimental verificationof these constants. Consequently,

obtained calculation may not describe the real state of corrugated laminated plates.

In other hand, nowaday corrugated laminated composite plates and cylindrical shells

arewidelyused andtheirstaticanddynamicproblemswithgeometricalnon-linearityareof

significantpractical interest, particularlystabilityand post-buckling behavior ofcomposite

plates and shells is more important. Therefore it needs more accuracy in investigating

corrugated composite plates and shells.

In order to eliminate this restriction, a new approach for investigating corrugated

composite plates of wave form is proposed naturally in the present paper, regarding this

πx

3

2

l l

/

∂u

1 ∂w

∂w

∂ ∂ w ∂u ∂k

ε =

+ ,

2

∂ w

∂v ∂w ∂w

∂u

2

∂ w ∂u

134

Dao Huy Bich, Khuc Van Phu

plate as a combination of shallow cylindrical shell parts with alternative curvatures. It

reduces to no use of Seydel’s empirical formulas and can sufficiently apply not only to an

isotropic elastic corrugated plate, but to a composite corrugated plate as well.

Based on this approach governing equations of a corrugated laminated composite plate

of wave form are developed and an application to the non-linear stability problem of this

plate is considered. Obtained results are compared with those of Seydel’s technique.

2.

GOVERNING EQUATIONS

Consider a rectangular symmetrically laminated composite corrugated plate in the

form of a sine wave(see Fig. 1), each layer of which is an unidirectional composite material.

Suppose the portion of cross-section line of a corrugated plate in the plane (x,z) has the

form of a sine wave

z = H sin l with H << l,

so that the alternative curvature of cross-section line is

k = z00 ≈ z00 = −H.π2.sin πx.

(1 +z02) 2

(1)

Based on the new approach the non-linear strain-displacement relationships in the

middle surface and the changes of curvature and twist of a such corrugated plate now can

be written in the form

2

εx = ∂x + 2 ∂x −kw,

2

χx = −∂x k.u+ ∂x = − ∂x2 + k∂x +u∂x ,

∂v 1 ∂w2

y ∂y 2 ∂y

χy = −∂y2 ,

γxy = ∂y + ∂x + ∂x. ∂y ,

χxy = − 2∂x∂y +k∂y ,

where u, v denote displacements of the middle surface point along x, y directions and

y

z

b

s

l

O

x

a

H

Fig. 1. Model of a corrugated plate

+ − kw +A + ,

N = A

∂u

1 1

∂w ∂v ∂w

,

+ + .

N = A

∂ w ∂u ∂k ∂ w

2

2

∂ w

∂u ∂k

∂ w

∂ w ∂u

N M

∂N

M

x x

x

∂N

N

x

2

2 2

∂ M M

M

∂ ∂w ∂w

xy

∂

∂w ∂ w

∂w ∂ w

A new approach for investigating corrugated laminated composite plates of wave form

135

w - deflection of the plate respectively; εx,εy,γxy are strains in the middle surface and

χx,χy,χxy are changes of curvatures and twist of the plate.

The constitutive stress-strain relations for the plate material are omitted here for

brevity. Integrating the stress-strain equations through the thickness of the plate and

taking into account that in a multilayered symmetrically laminated material the coupling

stiffnesses are equal to zero, while the extensional stiffnesses A16, A26 and the bending

stiffnesses D16, D26 are negligible compared to the others, we obtained the expressions for

stress resultants and internal moment resultants of a corrugated composite plate of wave

form

"∂u 1 ∂w2 # "∂v 1 ∂w2#

x 11 ∂x 2 ∂x 12 ∂y 2 ∂y

" 2 # " 2#

Ny = A12 ∂x + 2 ∂x −kw + A22 ∂y + 2 ∂y ,

∂u ∂v ∂w ∂w

xy 66 ∂y ∂x ∂x ∂y

(2)

and

2 2

Mx = −D11.∂x2 +k∂x + u∂x −D12. ∂y2 ,

My = −D12. ∂x2 +k∂x +u∂x −D22. ∂y2 ,

2

Mxy = −D66. 2∂x∂y +k∂y ,

(3)

where Aij, Dij (i, j = 1, 2, 6) are extensional and bending stiffnesses of any laminated

plate, i.e. for a flat composite plate such as a corrugated one. The geometry of a plate

includes in the expressions of strains and curvature changes. Indeed, it is an advantage of

the new approach.

3.

FORMULATION OF EQUILIBRIUM EQUATIONS

The equilibrium equations of a corrugated plate of wave form subjected to uniformly

distributed biaxial compressive loads of intensities p and q respectively according to [6,8]

when considering the non-linear geometry are of the form

∂x + ∂∂yy − k∂∂xx + ∂∂y y = 0,

∂∂xy + ∂yy = 0,

∂∂x2x + 2 ∂x∂y + ∂∂y2y + ∂x Nx ∂x + Nxy ∂y

2 2

+ ∂y Nxy ∂x + Ny ∂y +p∂x2 +q ∂y2 = 0.

(4)