A GENERALIZED APPROACH TO MODELLING DISPLACEMENTS IN PLATES MADE OF NON-HOMOGENEOUS MATERIALS

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MODELOWANIE INŻYNIERSKIE ISNN 1896-771X 32, s. 239-246, Gliwice 2006

A GENERALIZED APPROACH

TO MODELLING DISPLACEMENTS IN PLATES MADE OF NON-HOMOGENEOUS MATERIALS

JAKUB KASPRZAK

MARIAN OSTWALD

Institute of Applied Mechanics, Poznan University of Technology

Summary. In this paper a generalized way of modelling displacements in plates made of non-homogeneous materials is presented. It is assumed that the mechanical properties of a plate are symmetrical in respect to the mid-plane. The principle of stationarity of the total potential energy of a plate is used to formulate the system of equilibrium equations and boundary conditions for rectangular plates. These derived equations are used for solving bending and buckling problem of a plate in order to compare a few deformation hypotheses. Plates with the constant, continuous and discrete changes of mechanical properties are considered.

1. INTRODUCTION

Rectangular plates are one of the basic structural elements. Nowadays, rectangular plates are made of uniform iso(orto)thropic materials. They may be constructed as sandwich plates built of two stiff facings and a light core placed between the facings or as laminated plates built of several layers made of fibre-reinforced composite material. Recently, plates made of porous materials have also started to be applied. In this last case, the weight of plates is reduced with only slight decrease in their stiffness achieved by appropriate controlling the density of material.

The simplest way of modelling the mechanics of non-homogeneous multilayered plates is through application of Classical Thin Plate Theory with the modified form of stiffness coefficients, like in case of Classical Laminated Plate Theory [5]. This approach is based on Kirchhoff-Love hypothesis and does not include shear effect, therefore obtained results are valid only for thin plates with slight differences in mechanical properties. In order to overcome this limitation other theories including shear effect have been proposed. The overview of these proposals is presented in [11], where authors beside a historical review present their own solutions for beams and plates.

Significant amount of research was also dedicated to modelling composite laminated plates due to their wide use. The review of such problems and a rich bibliography may be found in the review works of Carrera [1-3] and in the monograph of Reddy [10]. Comparisons of several hypotheses are presented in [4, 6, 9]. The article of Magnucki, Malinowski and Kasprzak [8] is a work dedicated to porous plates. Mechanics of plates is also discussed in the monograph of Kączkowski [7] and encyclopaedic monograph edited by Woźniak [12].

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In this work a generalized approach to modelling the mechanics of non-uniform plates is presented. It is shown that the form of the system of equilibrium equations and boundary conditions does not depend on the kind of non-homogeneity and assumed deformation hypothesis.

2. GENERALIZED DISPLACEMENTS OF A PLATE

Classical Thin Plate Theory is based on Kirchhoff-Love hypothesis. Horizontal displace- ments u(x,y,z) and v(x,y,z) in a bent or buckled plate are the result of rotation of a cross- section of the plate, which stays flat and perpendicular to the mid-plane of the plate. Since the rotation angle is small these displacements are replaced by the following perpendicular translations

(

x^,y^,z

)

zw,x^, v

(

x^,y^,z

)

zw,y^,

u =− =−

where w = w(x,y) is the deflection function of the plate. In the presented generalized approach the assumption about the flatness of a cross-section is abandoned. Horizontal displacements are defined as the sum of translations being the result of rotation of a cross-section and some correction described by functions Ψ(x,y,z) and Φ(x,y,z) for displacements in X and Y direc- tions. It is assumed that these functions may be defined as the sum of products of two types of functions, unknown displacement functions ψi(x,y), ϕi(x,y) and a priori assumed functions fi(z) which describe the shape of deformation of a cross-section of a plate. The displacement field of a plate is presented in Fig. 1 and has the form

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

(

^, ^,

) (

^, ^,

) ( ) ( )

^,

( ) ( )

^, ^,

, ,

,

, , ,

,

0 1

, ,

0 1

, ,

∑

=

⋅

=

⋅ +

−

= +

−

=

⋅

=

⋅ +

−

= +

−

=

n

i

i i

n

i

i i

y y

n

i

i i

n

i

i i

x x

z f y x z

f y x zw

z y x zw

z y x v

z f y x z

f y x zw

z y x zw

z y x u

y x w z y x w

ϕ ϕ

Φ

ψ ψ

Ψ (1)

where

( )

^, , ^, 0

( )

^, , ^, 0

( )

^.

0 x y =−w_x ϕ x y =−w_y f z =z

ψ

Further in this work, n is called the order of a hypothesis. It should be noted that this notion is used differently than by other researchers, e.g. Reddy [10] uses the same name for the degree of a polynomial describing the deformation of a cross-section of a plate.

- wz ,x

u t x

A’

A

B

B’

w Ψ

Fig.1. Generalized displacements of a plate

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Using linear geometrical relationships and displacements (1) strains are defined as follows

( )

. ,

, ,

,

1 , ,

, 1

, ,

,

0

, , ,

, 0

, ,

0 , ,

∑

=

⋅

= +

=

⋅

= +

=

+

= +

=

⋅

=

⋅

=

n

i

z i i x

z xz n

i

z i i y

z yz

n

i

i x i y i x

y xy n

i

i y i y

y n

i

i x i x

x

f w

u f

w v

f v

u f

v f

u

ψ γ

ϕ γ

ϕ ψ γ

ϕ ε

ψ ε

(2)

Stresses are based on linear physical relationships and strains (2), i.e.

0 , , 0

0 0



 



⋅



 



=



 





















⋅

















=

⋅

=

















xz yz xz yz xz

yz

xy y x

xy y

y xy

y xy x

xy y x

G G G

E E

Q γ

γ τ

τ γ

ε ε

∆

∆ ν

∆ ε

τ σ σ

where ∆ = 1 – νxy2

Ey / Ex and Ex = Ex(z), Ey = Ey(z), νxy = νxy(z), Gxy = Gxy(z), Gyz = Gyz(z) Gxz = Gxz(z) are symmetrical functions of z variable that describe the change of mechanical properties on the thickness of a plate.

The i-th order generalized internal moments and forces are defined as follows

. ,

, ,

,

2

2 , 2

2 ,

2

2 2

2

∫

−

⋅

=

⋅

=

⋅

=

⋅

=

⋅

=

t

z i yz i

y t

t

z i xz i

x

t

t i xy i

xy t

t i y i

y t

t i x i

x

dz f Q

dz f M

τ τ

τ σ

σ

(3)

Their relationship with the displacement functions may be stated in the follow matrix form

, 2

0 0

66 22 12

12 11

0 66 0 22 0 12

0 12 0 11

0 66 0 22 0 12

0 12 0 11

00 66 00 22 00 12

00 12 00 11

0 0 0































 +

















−

⋅

















































































=

















































n,x n,y

n,y n,x ,xy ,yy ,xx

nn nn nn

nn nn

n n n

n n

n n n

n n

n xy

n y n x xy

y x

w w w

D D D

D D

D D D

D D

M M M M M M

ψ ψ

ϕ ψM L

M O

M

L

M (4a)

, 0

0 0

0

0 0 0

0

1 1

44 55 1

44 1 55

1 44 1

55 11

44 11 55 1

1















 







 





⋅















 



 



 







 



 



 





=





































n n nn

nn n

n

n n

n y n x y x

H H H

H

H H H

H

Q Q Q Q

ϕ ψ ϕ ψ M L

M O

M

L

M (4b)

where stiffness coefficients are defined as follows

. ,

,

2

, , 55

2

, , 44

2

∫ ∫

∫

− −

−

⋅

=

⋅

=

⋅

=

t

z j z i xz ij

t

z j z i yz ij

t

j i kl ij

kl Q f f dz H G f f dz H G f f dz

D (5)

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It is noteworthy that except for internal moments Mx0

, My0

, Mxy0

and forces Qx0

, Qy0

the other of internal moments and forces do not have any physical meaning and are only some generali- zation.

2. EQUILIBRIUM EQUATIONS OF A PLATE

Equilibrium equations of a rectangular plate are derived with the help of the principle of stationarity of the total potential energy which states that

(

U−T

)

=⁰

δ , (6)

where

−

∫ ∫∫

+ +

+

=

2

2 0 0

2 1 ^t

t b a

xz xz yz yz xy xy z z y y x

x dxdydz

U σ ε σ ε σ ε τ γ τ γ τ γ

is the total elastic strain energy of a plate and

∫∫

^⋅ ⁺ ^⋅ ⁺ ^⋅ ⁺ ^⋅ ^⋅

=

b a

y x xy y y x

x w N w N w w dxdy

N w q T

0 0

, , 2

, 2

1 2

1

is the work of transverse load q and in-plane load Nx, Ny and Nxy.

Substituting (2) into the equation (6) and using the relationships (4) and basic operations of variational calculus, the follow system of equilibrium equations is obtained after grouping ex- pressions by the variations of the unknown functions w, ψi, ϕi







 =





= +

−

= +

+

n Q i

M M

Q M

M

w N w N w

N q M

M M

i y i

x xy i

y y

i x i

y xy i

x x

yy y xy xy xx

x yy

y xy xy xx

x

, , 1 for 2

2

, ,

, 0

,

K ^. (7)

These equations may be expressed directly by the unknown displacement functions after taking into account relationships (4).

The use of the principle of stationarity of the total potential energy makes it possible to formulate boundary conditions on the edges of a rectangular plate. Boundary conditions for three common cases of support of edges parallel to the Y axis are presented below:

§ free edge

n i i xy i

x x

y xy y xy x

x M M M M M

M 1

0 0

, 0

, 0, 0, 0, 0

= =

=

= +

+

§ simply supported edge

n i i i

x

x M

M

w 1

0 0, 0, 0

,

0 = = = =

= ϕ

§ clamped edge

n i i i

wx

w=0, _, =0, ψ =0,ϕ =0 ₌₁

Boundary conditions for edges parallel to the X axis may be obtained by replacing indexes x with y and functions ψi with ϕi in above equations.

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Conditions which should be satisfied by functions fi describing the shape of the deformation of a cross-section of a plate are formulated as well. Firstly, they should be continuous (C⁰ class) because of continuity of displacements. Secondly, taking into account the fact that

(

x^,y^,t ²

)

= _xz

(

x^,y^,−t ²

)

= _yz

(

x^,y^,t ²

)

= _yz

(

x^,y^,−t ²

)

=⁰

xz τ τ τ

τ then

( )

² ,

(

²

)

⁰^.

, t = f −t =

f_i_z _i_z

Moreover, fi should be asymmetrical functions of z due to the symmetry of changes of me- chanical properties. Some researchers [9] suggest that they should be selected in a way which ensure that stresses τyz, τxz are continuous. However, this condition is not fulfilled if a plate is multilayered with the discontinuous functions of mechanical properties E(z), G(z).

3. CASE STUDY

The above generalized approach and comparison of a few deformation hypotheses is presented for a square plate simply supported on all edges. In the first case the plate is loaded transversely with load

(

^x ^a

) (

^y ^a

)

q

q= ₀sin π sin π .

In the second case the plate is loaded with uniform compressive load Nx in the direction of the X axis.

Both problems are solved with the help of Navier’s method. It is assumed that unknown displacement functions have the follow form

( ) ( ) ( )

( )

^,^, ^sin^cos

( ) (

^cos^sin

)

^for^for ¹¹^,^, ^,^, ^.^,

, sin

sin

, ₀

n i

a y a

x y

x

n i

a y a

x y

x

a y a

x w

y x w

i i

K K

=

π π

Φ ϕ

π π

Ψ ψ

π π

These functions satisfy all boundary conditions for simply supported edges and are a closed- form solution of the system (7).

The follow first order deformation hypotheses (n = 1) are compared:

§ sinusoidal (sine 1) [6] ^f1

( )

^z =^t ^{π sin}

( )

^π^z ^t ,

§ polynomial (poly 1) [6] ^f¹

( )

^z ⁼^z

[

¹⁻⁴ ³

( )

^z ^t ²

]

^,

the second order hypotheses (n = 2):

§ sinusoidal (sine 2) f1

( )

z =t π^sin

( )

πz t ^, f2

( ) ( ) (

z =t ³π ^sin ³πz t

)

^,

§ polynomial (poly 2) ^f¹

( )

^z ⁼^z

[

¹⁻⁴ ³

( )

^z ^t ²

]

^, ^f²

^{( )}

^z ⁼^z

[

¹⁻⁸

^{( )}

^z ^t ² ⁺¹⁶

^{( )}

^z ^t ⁴

]

^,

§ trigonometric (trig 2) [7] f1

( )

z =t π^sin

( )

πz t ^, f2

( ) ( ) (

z =t ²π ^sin ²πz t

)

^cos²

( )

πz t ^, and the sixth order hypothesis (n = 6):

§ sinusoidal (sine 6) ^fi

( )

^z =^t

[ (

²ⁱ−¹

)

π

]

^sin

[ (

²ⁱ−¹

)

π^z ^t

]

^cos²

( )

π^z ^t ^, dla i = 1,...,6.

In the Figure 2 and Tables 1 and 2, the ratio of the maximum deflection wmax of a plate ob- tained with the help of a particular deformation hypothesis to the maximum deflection wmax,K-L

obtained with the aid of Classical Thin Plate Theory is presented as a function of the ratio of the length a of an edge to the thickness t of a plate. The presented graph and values are also the ratios of the critical load Nx

cr

K-L of a plate obtained with the help of Classical Thin Plate

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Theory to the critical load Nxcr

obtained with the aid of a particular theory. Results are presented for three types of plates:

§ an isothropic plate with constant mechanical properties, i.e.

( )

^z ⁼^E

( )

^z ⁼^E⁼^const^,^ν

( )

^z ⁼⁰^.³^,^G

( )

^z ⁼^G

( )

^z ⁼^G

( )

^z ⁼^E

(

²⁺²^ν

)

^,

E_x _y _xy _xy _yz _xz

§ a porous plate with mechanical properties which vary on the thickness of the plate in a continuous way, i.e.

( )

^z ⁼^E

( )

^z ⁼^E

[

¹⁻⁰^.⁹⁹^cos

( )

^π^z ^t

]

^,^ν

( )

^z ⁼⁰^.³^, ^G

( )

^z ⁼^G

( )

^z ⁼^G

( )

^z ⁼^E

(

²⁺²^ν

)

^,

E_x _y _xy _xy _yz _xz

§ a laminated plate with mechanical properties which vary on the thickness of the plate in a discrete way, i.e. a 5-layered cross-ply plate (0°, 90°, 0°, 90°, 0°) made of orthotropic material with the follow mechanical properties

( )

^, 2

( )

1 ²⁰^, 12

( )

⁰^.³^, 12

( )

13

( )

³ ⁴ 2^, 23

( )

³ ²⁰ 2^.

1 z E E z E z G z G z E G z E

E = = ν = = = =

0 20 40 60 80 100

a/t

1 1.05 1.1 1.15 1.2 1.25

wmax/wmax,K-L & Nxc r K -L/Nxcr

Fig. 2. The ratios wmax / wmax,K-L and Nx cr

K-L/ Nx

cr as the function of a / t for the isothropic plates Solutions obtained with the help of the higher-order hypotheses are virtually the same for the isothropic plates.

Table 1. The ratios wmax / wmax,K-L and Nxcr

K-L/ Nxcr

for the porous plates

a / t 5 10 25 50 75 100

sine 1 1,814 1,205 1,033 1,008 1,004 1,002

poly 1 1,762 1,191 1,031 1,008 1,003 1,002

sine 2 2,330 1,347 1,056 1,014 1,006 1,004

poly 2 2,141 1,276 1,047 1,012 1,001 1,000

trig 2 2,575 1,417 1,068 1,017 1,008 1,004

sine 6 3,115 1,577 1,094 1,024 1,010 1,006

In the case of laminated plates the second order zig-zag hypothesis is studied as well. For this hypothesis functions fi(z) have the follow form

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( )

 ∈ −

= ( ) 10 else

10 , 10

1 sign z t

t t z z z

f ,

( )















∈

−

∈

−

∈ +

−

∈

−

=

2 , 10 3 5

10 3 , 10 10

10 , 10 0

10 , 10 3 10

10 3 , 2 5

2

t t z t

z

t t z

t t z t z

t t z t

z

f .

Table 2. The ratios wmax / wmax,K-L and Nxcr

K-L/ Nxcr

for laminated plates

a / t 5 10 25 50 75 100

sine 1 2,2054 1,3117 1,0506 1,0127 1,0056 1,0032 poly 1 2,2133 1,3142 1,0510 1,0128 1,0057 1,0032 sine 2 2,3574 1,3488 1,0564 1,0141 1,0063 1,0035 poly 2 2,2844 1,3301 1,0534 1,0134 1,0059 1,0033 trig 2 2,4870 1,3835 1,0621 1,0156 1,0069 1,0039 zig-zag 2 3,0204 1,5437 1,0895 1,0225 1,0100 1,0056 sine 6 2,9921 1,5286 1,0867 1,0218 1,0097 1,0054

4. CONCLUSIONS

The generalized approach presented in this work makes it possible to formulate equilibrium equations and boundary conditions for rectangular plates independently of an assumed deformation hypothesis because only stiffness coefficients (5) depend on it. The number of equilibrium equations and boundary conditions are not dependent on this hypothesis but on the number of functions used for their description. This approach enables modelling displacements in plates made of various non-homogenous materials with the help of nonlinear and zig-zag (bro- ken line) deformation hypotheses.

Obtained results show that for thick and medium thickness (a / t ≤ 20) plates Kirchhoff- Love hypothesis is not valid even for uniform isothropic plates as differences between results obtained with the help of Classical Thin Plate Theory (n = 0) and the higher order hypotheses (n ≥ 1) for the thick isothropic plate are bigger than 20%. However, it does not make any difference which of higher-order hypothesis is used for modelling displacements in isothropic plates.

In the case of porous and laminated plates differences between results obtained with the help of Kirchhoff-Love hypothesis and higher-order hypotheses are higher than 200% for thick plates (a / t = 5). They become smaller with the increase of the ratio a / t and for thin plates (a / t ≥ 50) they do not exceed 2.5%. Hypotheses which are based on trigonometric functions usually give bigger deflection and smaller critical load than those based on polynomial functions. The biggest deflection and the smallest critical load of laminated plates are obtained with the use of the second order zig-zag hypothesis.

In presented cases it may be assumed that the difference between results obtained with the help of Kirchhoff-Love hypothesis and higher-order hypotheses is inversely proportional to the square of the ratio a / t.

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ACKNOWLEGDEMENTS

The work has been supported by the Ministry of Education and Science – Grant of No.

4 T07A 051 29.

REFERENCES

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2. Carrera E.: Developments, ideas, and evaluations based upon Reissner’s mixed variational theorem in the modelling of multilayered plates and shells, Applied Mechanics Reviews, 54(4), p. 301-329, 2001.

3. Carrera E.: Historical review of Zig-Zag theories for multilayered plates and shells, Ap- plied Mechanics Reviews, 56, p. 287-308, 2003.

4. Chattopadhyay A., Gu H.: Exact elasticity solution for buckling of composite laminates, Composite Structures, 34(3), p. 291-299, 1996.

5. Jones R.M.: Mechanics of Composite Materials, McGraw-Hill, Washington, 1975.

6. Idlbi A., Karama M., Touratier M.: Comparison of various laminated plate theories”, Composite Structures, 37(2), p. 173-184, 1997.

7. Kączkowski Z., Płyty: Obliczenia statyczne, wyd. 3, Arkady, Warszawa 2000.

8. Magnucki K., Malinowski M., Kasprzak J.: Bending and buckling of a rectangular porous plate, Steel and Composite Structures, 6(4), p. 319-333, 2006.

9. Noor A.K., Malik M.: An assessment of five modelling approaches for thermo-mechanical stress analysis of laminated composite panels, Computational Mechanics, 25, p. 43-58, 2000.

10. Reddy J.N.: Mechanics of laminated composite plates and shells: theory and analysis, Ed. 2, CRC Press, 2004.

11. Wang C.M., Reddy J.N., Lee K.H.: Shear deformable beams and plates, Ed. 1, Elsevier, Oxford, 2000.

12. Woźniak C., (red).: Mechanika sprężystych płyt i powłok, tom VIII Mechanika techniczna, PWN, Warszawa, 2001.

UOGÓLNIONE PODEJŚCIE DO MODELOWANIA PRZEMIESZCZEŃ W PŁYTACH WYKONANYCH

Z MATERIAŁÓW NIEJEDNORODNYCH

Streszczenie. W pracy przedstawiono uogólniony sposób modelowania przemieszczeń w płytach wykonanych z materiałów niejednorodnych o właściwościach symetrycznych względem płaszczyzny środkowej. Za pomocą zasady stacjonarności energii potencjalnej sformułowano układ równań równowagi oraz warunki brzegowe dla płyt prostokątnych. Wyprowadzone równania wykorzystano do rozwiązania dwóch zagadnień ugięcia i stateczności płyty w celu porównania różnego rodzaju hipotez deformacyjnych dla płyt o stałych, ciągłych oraz skokowych zmianach właściwości mechanicznych na grubości płyty.