Analytical investigation of thin rib-stiffened plates

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Analytical investigation of thin rib-stiffened plates

Article in International Review of Mechanical Engineering · January 2011

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(continued on inside back cover) ISSN 1970-8734

Vol. 5 N. 4 May 2011

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International Review of Mechanical Engineering (I.RE.M.E.), Vol. 5, N. 4 May 2011

Manuscript received and revised April 2011, accepted May 2011 Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved

Analytical Investigation of Thin Rib-Stiffened Plates

A. Karimin, M. Belhaq

Abstract – A buckled thin stiffened plate under uni-axial compression is analyzed analytically in this paper. In a first part, we present an exact solution and we examine the influence of the misplacement and mechanical characteristics of the stiffener on the critical buckling load, on the modes and on the buckling modes of the plate with a single rib-stiffener. In a second part, similar investigation is performed in the case of two rib-stiffeners. Specifically, attention is given to optimize expressions of the buckling mode, to examine the effect of the misplacement on the contribution coefficients of the buckling mode and the effect of stiffener parameters on the mode shapes. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Analytical Method, Buckling Mode, Stiffened Plate

Nomenclature

a Length of the plate

b Width of the plate

h Thickness of the plate

ν

Poisson ration

D=Eh

³

/12(1- ν

²

) Flexural rigidity of the plate

P Uniform compression in X-

direction

2 2

Pb

λ

D

=

π Buckling load

w Transverse displacement,

downward positive

η

Eigenfrequency

Ω

(ρhb

⁴

/Dπ

²

) η

²

a

j

Length of the jth span r

j

=a

j

/b Position of the stiffeners

M

x(1)

, M

x(N)

Bending moments in the first and the last spans of the continuous plate

R

x(j)

, R

x(j+1)

Shearing in the jth and j + 1 spans of the plate

Rx

( )

3 3

3 2 2

w w

D^⎡∂x

ν

^∂x y ^⎤

− ⎢ + − ⎥

∂ ∂ ∂

⎣ ⎦

(GJ)

j

Torsional rigidity of the jth rib (EI)j Flexural rigidity of the jth rib

I. Introduction

Stiffened plates are used for several purposes. Among these objectives: the diminution of the mass structure, the increase of the buckling load and the modification of the mechanical characteristic of the structure (eigenfrequency, mode shape...). The problem of the stiffened plates has been investigated by several authors.

Minguez [1] introduced the torsional rigidity of the ribs

in the energy method for studing the buckling of reinforced plate. Kakol [2] presented the stability analysis of stiffened plates by means of the finite strip method. Elishakoff and Starnes [3] investigated the effect of a small structural irregularity, due to the misplacement of stiffeners or interior supports, on both the buckling load and the buckling mode of the rib- stiffened plate. Kolli and Chandrashekhara [4] analyzed the nonlinear behavior of eccentrically stiffened laminated plates. The buckling mode localization in rib- stiffened rectangular plate with randomly misplaced stiffeners was considered in Xie [5]. While the buckling mode localization in rib-stiffened plates with randomly misplaced stiffeners was examined in Xie and Ibrahim [6] using a finite strip method. Xie and Elishakoff [7]

discussed the buckling mode localization in rib-stiffened plates with misplaced stiffeners by using Kantorovich approach. In Qing et al. [8], a novel mathematical model for the free vibration analysis of stiffened laminated plates was presented. Chen and Xie [9] studied the vibration localization in plates rib-stiffened in two orthogonal directions. Recently, Karimin and Belhaq [10]

examined analytically nonlinear dynamic of simply supported stiffened plate in a single mode approach.

In the present paper, we study analytically the buckling of a simply supported rectangular plate with N ribs parallel to (oy) under uni-axial compression. We focus our attention on the case where no support under the rib, i.e. the torsion and bending of ribs are taken into account [3]. The main purpose is to study the buckling mode under the effect of stiffener and the geometry of the plate in the case of a single rib-stiffener and in the case of a two rib-stiffener.

In the first part of this paper, the differential equation

of the deflection surface of the rectangular plate with

different boundary conditions is used [11]. In the case of

a single rib-stiffener, the exact solution is obtained and

A. Karimin, M. Belhaq

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 4

612 the effect of the torsional rigidity and flexural rigidity of the stiffener on the modes, the critical buckling load and the buckling modes are examined. Then, the optimal position of the stiffener leading to a simplified expression of buckling modes and to the large critical buckling load is obtained. In other words, we show that only the dominant contribution coefficients in the buckling mode expression can be retained for the analysis of stiffened plates. In the second part, a similar investigation is performed for the case of two rib- stiffener. Finally, the effect of the parameters of the stiffener on the mode shapes is analyzed.

II. Formulation

We consider a rib-stiffened plate subjected in its mid- plane to uniform compression P in the x-direction; see Fig. 1. From von Karman-type equations for the thin plate, the equation of the deflection surface of the plate can be written as:

4 4 4 2

4 2 2 2 4 2 0

w w w w

D P

x x y y x

⎛∂ + ∂ +∂ ⎞+ ∂ =

⎜ ⎟

⎜∂ ∂ ∂ ∂ ⎟ ∂

⎝ ⎠

(1)

 

Fig. 1. Uni-axial compressed rectangular stiffened plate with N ribs

The simply supported boundary conditions is along y

= 0 and y= b.

The solutions of Eq. (1) can be represented in the following form:

( ) ( )

^y

w x, y x sin

φ

^⎛

π

b^⎞

= ⎜⎝ ⎟⎠

(2)

Substitution of Eq. (2) into Eq. (1) leads to:

( ) ( ) ( )

4 2 2 4

4 2 2 2 4 0

d x P d x x

dx D b dx b

π π

φ

+^⎛⎜⎜ − ^⎞⎟⎟

φ

+

φ

=

⎝ ⎠

(3)

The corresponding characteristic equation reads:

 

2 4

4 2

2 4

2 0

2

Z P Z

D b b

π π

⎛ ⎞

+ ⎜⎜ − ⎟⎟ + =

⎝ ⎠

  (4)

 

and the solutions are written as:

 

^{2 2} 1 2

2 2 2 2

Z b

π

^⎡ ⎛

λ

⎞

λ λ

⎛ ⎞^⎤

= ⎢−⎜ − ±⎟ ⎜ − ⎟⎥

⎝ ⎠ ⎝ ⎠

⎢ ⎥

⎣ ⎦

  (5)

Even for the unstiffened plate, the buckling load P

cr

is always equal to or larger than

^{4 2}

2 D b

π . The roots of Eq.

(5) are given by Z

1

=iβ

1

, Z

2

=-iβ

1

, Z

3

=iβ

2

and Z

4

=-iβ

2

in which:

1 1 2

2 2 2

b

π λ λ λ

β

= ^⎛⎜⎝ − +^⎞⎟⎠ ^⎛⎜⎝ − ^⎞⎟⎠

 

₂ 1 2

2 2 2

b

π λ λ λ

β

= ^⎛⎜⎝ − −^⎞⎟⎠ ^⎛⎜⎝ − ^⎞⎟⎠

   

A solution of Eq. (1) is given by:

( ) ( )

^y

w x, y x sin

φ

^⎛

π

b^⎞

= ⎜ ⎟

⎝ ⎠

(6)

where:

( ) ( ) ( )

(

2

)

¹

(

2

)

¹

x Acos x B sin x C sin x D cos x

φ β β

β β

= + +

+ +

(7)

For the arbitrary jth span, the solution can be written as:

( ) ( )

j j j j y

w x , y x sin

φ

^⎛

π

b^⎞

= ⎜⎝ ⎟⎠

(8)

where:

( ) ( ) ( )

( ) ( )

1 1

2 2 0

j j j j j j

j j j j j j

x A cos x B sin x

C sin x D cos x x a

φ β β

β β

= + +

+ + ≤ ≤

(9)

The quantities A

j

, B

j

, C

j

and D

j

are unknown constants to be determined by using the boundary and the continuity conditions. The boundary conditions are:

1_x1 0 0

w = =

(10)

( )

1

1

2 2

1 1 1

2 2

0 1 0

x 0

x x

w w

M D

x

ν

y

= =

⎛∂ ∂ ⎞

= − ⎜⎜⎝∂ + ∂ ⎟⎟⎠ =

(11)

( ) ^{2 2}

2 2 0

N N

N n

N N N

x x a N x a

w w

M D

x

ν

y

= =

⎛∂ ∂ ⎞

= − ⎜⎜⎝∂ + ∂ ⎟⎟⎠ =

(12)

N N 0

N x a

w ₌ =

(13)

A. Karimin, M. Belhaq

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 4

Here, we consider the case where no support under the rib. Then, in addition to the torsion of ribs, the bending should be taken into account. By considering the rib-stiffened plate as a physically continuous plate with as many spans as the number of ribs, the stiffeners are accounted for through the conditions of continuity. These conditions between two consecutive spans j and j + 1 read:

1 1 0

j j j

j x a j x

w w

+ +

= = =

(14)

1

1

1 0

j j j

j j

j x a j x

w w

x x

+

+

= + =

∂ ∂

∂ = ∂

(15)

( ) ( )

( )

1

1

3 1

1

0 1 0

j j j

j

j

j j

x x x x a j j x

M M GJ w

x y

+

+

+ +

= = + =

− = ∂

∂ ∂

(16)

( ) ( )

( )

1

1

4 1

1 0 4

j j j 0

j

j j j

x x x x a j

x

R R EI w

y

+

+

+ +

= =

=

− = ∂

∂

(17)

The conditions can, in turn, be expressed by the following equations in terms of constants of integration:

1 1 0

A +C =

(18)

2 2

1 1A 2 1C 0

β

+

β

=

(19)

( ) ( )

(

¹ 2

) (

¹ 2

)

0

N N N N

N N N N

cos a A sin a B cos a C sin a D

β β

β β

+ +

+ + =

(20)

( ) ( )

( ) ( )

2 2

1 1 1 1

2 2

2 2 2 2

0

N N N N

N N N N

cos a A sin a B

cos a C sin a D

β β β β

β β β β

+ +

+ + = (21)

( ) ( ) ( )

( )

1 1 2

2 1 1 0

j j j j j j

j j j j

cos a A sin a B cos a C

sin a D A C

β β β

β

_{+ +}

+ + +

+ − − =

(22)

( ) ( )

( ) ( )

1 1 1 1

2 2 2 2

1 1 2 1 0

j j j j

j j j j

j j

sin a A cos a B

sin a C cos a D

B D

β β β β

β β β β

β

₊

β

₊

− + +

− + +

− − =

(23)

( ) ( )

( ) ( )

( ) ( )

2 2

1 1 1 1

2 2

2 2 2 2

2 2 2

1 1 1 1 2 1

2

2 1 0

j j j j

j j j j

j j j j

j j

cos a A sin a B

cos a C sin a D

A GJ B C

bD GJ D bD

β β β β

β β β β

β π β β

π β

+ + +

+

− − +

− − +

+ + +

+ =

(24)

( ) ( )

( ) ( )

( ) ( )

3 3

1 1 1 1

3 3

2 2 2 2

4 3

1 1 1

3

4 3

1 2 1

3

0

j j j j

j j j j

j j

j

j j

j

sin a A cos a B

sin a C cos a D

EI A B

b D

EI C D

b D

β β β β

β β β β

π β

π β

+ +

+ +

− +

+ − +

− + +

− + =

(25)

Introducing the following non-dimensional quantities:

( ) ( )

1 2

j j j

j j j

j j

GJ EI

r a , , ,

b bD bD

, i , b

τ ω

β π β

= = =

= =

1

2

1 2

2 2 2

1 2

2 2 2

λ λ λ

,

β

λ λ λ

β

⎛ ⎞

= − + ⎜ − ⎟

⎝ ⎠

⎛ ⎞

= − − ⎜⎝ − ⎟⎠

(26)

the boundary conditions and conditions of continuity, Eqs. (18)-(25), read:

1 1 0

A +C =

(27)

2 2

1 1A 2 1C 0

β

+

β

=

(28)

( ) ( )

(

¹ 2

) (

¹ 2

)

0

N N N N

N N N N

cos r A sin r B

cos r C sin r D

πβ πβ

πβ πβ

+ +

+ + =

(29)

( ) ( )

( ) ( )

2 2

1 1 1 1

2 2

2 2 2 2 0

N N N N

N N N N

cos r A sin r B

cos r C sin r D

β πβ β πβ

β πβ β πβ

+ +

+ + =

(30)

( ) ( ) ( )

( )

1 1 2

2 1 1 0

j j j j j j

j j j j

cos r A sin r B cos r C

sin r D A C

πβ πβ πβ

πβ

_{+ +}

+ + +

+ − − =

(31)

( ) ( )

( ) ( )

1 1 1 1

2 2 2 2

1 1 2 1

0

j j j j

j j j j

j j

sin r A cos r B

sin r C cos r D

B D

β πβ β πβ

β πβ β πβ

β

₊

β

₊

− + +

− + +

− − =

(32)

( ) ( )

( ) ( )

2 2

1 1 1 1

2 2

2 2 2 2

2 2

1 1 1 1 2 1

2 1 0

j j j j

j j j j

j j j j

j j

cos r A sin r B

cos r C sin r D

A B C

D

β πβ β πβ

β πβ β πβ

β τ β π β

τ πβ

+ + +

+

− − +

− − +

+ + + +

+ =

(33)

A. Karimin, M. Belhaq

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 4

614

( ) ( )

( ) ( )

3 3

1 1 1 1

3 3

2 2 2 2

3 3

1 1 1 1 2 1 0

j j j j

j j j j

j j j j j j

sin r A cos r B

sin r C cos r D

A B C D

β πβ β πβ

β πβ β πβ

πω

₊

β

₊

πω

₊

β

₊

−

+ − +

− + − + =

(34)

III. Buckling Problem

III.1. A Plate with a Single Rib-Stiffener

In order to investigate the variation of the buckling mode of the stiffened plate due to a small structural irregularity, we study the simplest case where there is a single stiffener which is slightly misplaced (Fig. 2).

This is also the case where the discreteness of the stiffener has most pronounced effect on the buckling load.

Fig. 2. Uni-axial compressed rectangular stiffened plate with a single misplaced rib

We will use the following non-dimensional notations for specifying the positions of the stiffeners:

1 2

1 2

a a a

r , r ,

b b δ b

= = = (35)

A plate with a single rib-stiffener corresponds to j = 1 and N = 2.

Therefore, Eqs. (27)-(34) can be written in the form:

0

F ∆ = (36)

where ∆ = (B

1

, D

1

, A

2

, B

2

, C

2

, D

2

)

^T

and F is a 6x6 matrix with elements as follows:

( )

( ) ( )

( )

( ) ( )

( ) ( )

( ) ( )

11 12 13 1 2

14 1 2 15 2 2

16 2 2 21 22

2 2

23 1 1 2 24 1 1 2

2 2

25 2 2 2 26 2 2 2

31 1 1 32 2 1 33

34 35 36 41 1

0 0

0 0

1

0 1 0

F , F , F cos r ,

F sin r ,F cos r ,

F sin r , F , F

F cos r , F sin r

F cos r , F sin r ,

F sin r ,F sin r , F ,

F , F , F ,F cos

πβ

πβ πβ

πβ

β πβ β πβ

β πβ β πβ

πβ πβ

β π

= = =

= =

= = =

= =

= =

= = = −

= = − = = ( β

1 1

r )

(37a)

( )

( )

( )

( )

( )

42 2 2 1 43 44 1

45 46 2 51 12 1 1

2 2

52 2 2 1 53 1 54 1 1

2 3

55 2 56 2 1 61 1 1 1

3 3

62 2 2 1 63 1 64 1

65 1 66 23

0

0 _,

F cos r , F , F ,

F , F F sin r ,

F sin r ,F , F ,

F , F ,F cos r ,

F cos r ,F ,F ,

F , F

β πβ β

β β πβ

β πβ β πβ τ

β πβ τ β πβ

β πβ πω β

πω β

= = = −

= = − = −

= − = =

= = = −

= − = − =

= − =

(37b)

Thus, the elements of matrix F are composed of the parameters δ, τ

1

, ω

1

,

λ

and r where r

1

= δr and r

2

=δ (1- r). A non-trivial solution of Eq. (36) is obtained by setting the determinant of F equal to zero, which yields a transcendental equation whose the smallest root corresponds to the critical buckling load λ.

For the unstiffened plate, the critical buckling load is always equal to or larger than 4 [12].

To determine the smallest critical buckling load, we first plot the torsional rigidity τ

₁

versus misplacement r using the vanishing of the determinant of F, Eq. (36).

This provides the critical buckling load λ.

In other words, for all values of r in the vicinity of r = 0.5, one can find   values of τ

1≥0 corresponding to Det(F)

~ 0.

Figures 3 and 4 show the variation of torsional rigidity

τ1

as   function of displacement r for δ=1.5 and δ = 2.5 and for different values of flexural rigidity ω

1

.

Figs. 3. Torsional rigidity τ1 versus misplacement r with δ = 1.5 and for different values of ω1

A. Karimin, M. Belhaq

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 4 Figs. 4. Torsional rigidity τ1 versus misplacement r with δ = 2.5 and for

different values of ω

Table I summarizes the results plotted in Figs. 3 and 4.

Figures 3 show that as the flexural rigidity is increased, the torsional rigidity is increased, whereas the critical buckling load remains constant.

Figures 4 show that the flexural rigidity and torsional rigidity vary in the opposite direction, i.e. for increasing flexural rigidity, the torsional rigidity decreases.

Here, the critical buckling load decreases (see Table I). Note that the flexural rigidity has no effect for values greater than 3 (see Fig. 4(a)).

TABLEI

MODES AND CRITICAL BUCKLING LOAD FOR DIFFERENT VALUES OF ∆ AND Ω1

δ ω1 λcr δ

β

₁ δ

β

₂

1.5 1 to 100 4.38 2.031964 1.107302

2.5 1 4.42 3.438067 1.817881

2.5 4.21 3.137607 1.991963

10 4.38 3.386607 1.845504

100 4.38 4.929111 1.825887

3 1 5.07 5.002296 1.799173

2.5 5.14 5.002296 1.799173

10 5.14 5.002296 1.799173

100 5.14 5.002296 1.799173

In general, the stiffeners used to increase the critical buckling load in the plates have the geometry of

δ

≥3.

In this case, one has noticed that flexural rigidity has no effect for values greater than 2 (see Fig. 5(b)).

Figs. 5. Torsional rigidity τ1 versus misplacement r with δ = 3

Substituting

λcr

and the corresponding values of (

δ

,

ω1

) into Eq. (36), we obtain the buckling mode:

( )

¹

( )

¹

1 1 1 1 2

1 1

0

x x

x B sin f r sin

a a

x a

φ

= ^⎡⎢⎣ ^⎛⎜⎝

πδβ

^⎞⎟⎠+ ^⎛⎜⎝

πδβ

^⎞⎟⎠^⎤⎥⎦

≤ ≤

(38)

( )

( ) ( ) ( ) ( )

1 1 2

2 1 2

2 2 1

3 2 2

4 2 2

2 2

0

g r cos x

a g r sin x

x B a ,

g r cos x

a g r sin x

a x a

πδβ πδβ φ

πδβ πδβ

⎡ ⎛ ⎞ + ⎤

⎢ ⎜⎝ ⎟⎠ ⎥

⎢ ⎥

⎢+ ⎛ ⎞+⎥

⎢ ⎜⎝ ⎟⎠ ⎥

⎢ ⎥

= ⎢⎢⎢+ ⎛⎜⎝ ⎞⎟⎠+⎥⎥⎥

⎢ ⎛ ⎞ ⎥

⎢+ ⎜ ⎟ ⎥

⎢ ⎝ ⎠ ⎥

⎣ ⎦

≤ ≤

(39)

where the coefficients f and g

i

(i = 1...4) are the contributions to the buckling mode. Now, we are interested in the variation of the contributions to the buckling mode as function of a small misplacement of the stiffener.

Figures 6-8 illustrate the contribution coefficients of

the buckling mode as a function of the misplacement of

stiffeners for δ = 3 and for different values of ω

1

. Figures

6(a) and 7(a) show the increase in the contribution f of

the second mode β

2

.

A. Karimin, M. Belhaq

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 4

616

Figs. 6. Contribution coefficients of the buckling modes versus misplacement r with δ = 3, ω1 = 0 and ¸λ_cr = 4.35

Figs. 7. Contribution coefficients of the buckling modes versus misplacement r with δ = 3, ω1 = 0.5 and λcr = 4.93

Figure 8(a) shows that the second mode has a strong contribution in the interval (r= 0.4, r=0.46) so that the contribution of this mode can be neglected outside this range (i.e.

1

( )

1 1 1

x

¹

x B sin

φ ≈ ⎛ ⎜ πδβ a ⎞ ⎟

⎝ ⎠ ). Figure 6(b)

indicates that the contribution coefficients g

3

and g

4

can be neglected. In Fig. 7(b) we can see that the contribution coefficients g

2

can be ignored. Figure 8(b) illustrates that for values of r ≥ 0.46, the deflection of the second span can be neglected ( φ

₂ ≈0

).

Figs. 8. Contribution coefficients of the buckling modes versus misplacement r with δ = 3, ω1 = 1 and λcr = 5.07

III.2. A plate with a Two Rib-Stiffener

Here, we consider a three-span continuous plate with stiffeners attached to the same positions as the interior supports. The plate is subjected to the uni-axial uniform compression in the direction perpendicular to the stiffeners. For this specific problem, a set of twelve algebraic equations can be established in the form of Eq.

(36). In this case, ∆= (B

1

, D

1

, A

2

, B

2

, C

2

, D

2

, A

3

, B

3

, C

3

, D

3

)

^T

and F is a 10x10 matrix. For two rib-stiffener, we have:

3

1 2

1 2 3

a

a a a

r , r , r ,

b b b

δ

b

= = = =

(40)

In this case, we set:

( )

¹

1 2 3

1 1

2

r r, r r r , r a

δ δ a

= = = − = (41)

The elements of matrix F are composed of the

parameters

δ, τ1, ω1,τ2, ω2, r and λ. First, we plot τ2

A. Karimin, M. Belhaq

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 4

versus misplacement r for different values of δ, τ

1, ω1, τ2

and

ω2

. Then, the plots provide the smallest critical buckling load. Thus, the buckling mode is established as follows:

( )

¹

( )

¹

1 1 1 1 2

1 1

0

x x

x B sin f r sin

a a

x a

φ

= ^⎡⎢⎣ ^⎛⎜⎝

πδβ

^⎞⎟⎠+ ^⎛⎜⎝

πδβ

^⎞⎟⎠^⎤⎥⎦

≤ ≤

(42)

( )

( ) ( ) ( ) ( )

1 1 2

2 1 2

2 2 1

3 2 2

4 2 2

2 2

0

g r cos x

a g r sin x

x B a ,

g r cos x

a g r sin x

a x a

πδβ πδβ φ

πδβ πδβ

⎡ ⎛ ⎞ + ⎤

⎢ ⎜⎝ ⎟⎠ ⎥

⎢ ⎥

⎢+ ⎛ ⎞+⎥

⎢ ⎜⎝ ⎟⎠ ⎥

⎢ ⎥

= ⎢⎢⎢+ ⎛⎜⎝ ⎞⎟⎠+⎥⎥⎥

⎢ ⎛ ⎞ ⎥

⎢+ ⎜ ⎟ ⎥

⎢ ⎝ ⎠ ⎥

⎣ ⎦

≤ ≤

(43)

( )

( ) ( ) ( ) ( )

1 1 3

2 1 3

3 3 1

3 2 3

4 2 3

3 3

0

h r cos x a h r sin x

x B a ,

h r cos x

a h r sin x

a x a

πδβ πδβ φ

πδβ πδβ

⎡ ⎛ ⎞ + ⎤

⎢ ⎜ ⎟ ⎥

⎝ ⎠

⎢ ⎥

⎢+ ⎛ ⎞+⎥

⎢ ⎜⎝ ⎟⎠ ⎥

⎢ ⎥

= ⎢⎢⎢+ ⎛⎜⎝ ⎞⎟⎠+⎥⎥⎥

⎢ ⎛ ⎞ ⎥

⎢+ ⎜ ⎟ ⎥

⎢ ⎝ ⎠ ⎥

⎣ ⎦

≤ ≤

(44)

We are interested in the variation of the buckling mode with a small misplacement of the two rib-stiffener.

We plot f, g

i

and h

i

as a function of r to show the contribution coefficients of the bucking mode for values picked from Fig. 10. Figures 9-11 illustrate the torsional rigidity τ

2

versus misplacement r with

δ

= 3, in the cases where the misplacement r is in the vicinity of 1/2 and 1/3. These figures show that the critical buckling load for displacement r near 1/ 2 is larger than that for r in the vicinity of 1/3. It can be noticed that an increase in the flexural rigidity leads to the increase of the critical buckling load. Figures 12-14 illustrate the contribution coefficients f, g

i

and h

i

(i=1...4) of the buckling mode versus misplacement r in the vicinity of 1/2 and 1/3.

These figures show that the use of two stiffeners can reduce the amplitude of the buckling mode. In term of the critical buckling load, the position r in the middle of the plate (r near 1/2) is better than the position where r is near 1/3. Figure 13(a) shows that all coefficients have to be taken into consideration. In contrast, Fig. 13(b) indicates that g

2

, g

3

and g

4

can be neglected in the interval (r = 0.36; r = 0.46). In this case, we can simplify

the expression of φ

₂

, (i.e. φ

2

( )

x2 ≈ B g r1 1

( )

cos ₁^x2 πδβ a

⎛ ⎞

⎜ ⎟

⎝ ⎠

).

Finally, Fig. 14(a) shows that only h

4

can be neglected, while Fig. 14(b) indicates that only h

2

can be ignored.

Figs. 9. Torsional rigidity τ2 versus misplacement r with δ = 3. (a):

misplacement r in the vicinity of 1/2, (b): misplacement r in the vicinity of 1/3

Figs. 10. Torsional rigidity τ2 versus misplacement r with δ = 3. (a):

misplacement r in vicinity of 1/2, (b): misplacement r in the vicinity of 1/3

A. Karimin, M. Belhaq

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 4

618

Figs. 11. Torsional rigidity τ2 versus misplacement r with δ = 3. (a):

misplacement r in vicinity of 1/2, (b): misplacement r in the vicinity of 1/3

Figs. 12. Contributions coefficient f of the buckling mode versus misplacement r with δ = 3, ω1 = 1, τ1 = 0.5 and ω2 = 0.8, (a):

misplacement r in the vicinity of 1/2, (b): misplacement r in the vicinity of 1/3

Figs. 13. Contributions coefficient gi of the buckling mode versus misplacement r with δ = 3, ω1 = 1, τ1 = 0.5 and ω2 = 0.8, (a):

misplacement r in the vicinity of 1/2, (b): misplacement r in the vicinity of 1/3

Figs. 14. Contributions coefficient hi of the buckling mode versus misplacement r with δ = 3, ω1 = 1, τ1 = 0.5 and ω2 = 0.8, (a):

misplacement r in the vicinity of 1/2, (b): misplacement r in the vicinity of 1/3

A. Karimin, M. Belhaq

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 4

IV. Linear Vibration Problem

In this section, the natural frequencies and the mode shapes of the thin rib-stiffener with a single stiffener can be obtained from the following deflection equation of the thin rectangular plate:

2 4 4 4

2 4 2 2 4

2 2

2

cr

0

w w w w

h D

t x x y y

P w x

ρ ∂ ∂ + ⎛ ⎜ ⎜ ⎝ ∂ ∂ + ∂ ∂ ∂ + ∂ ∂ ⎞ ⎟ ⎟ ⎠ +

+ ∂ =

∂ (45)

The solution of Eq. (45) can be written as:

( ) ( ) ^y

w x, y,t x,t sin φ ⎛ π b ⎞

= ⎜ ⎟

⎝ ⎠ (46)

Substitution of Eq. (46) into Eq. (45) leads to:

( ) ( )

( )

4 2 2

4 2 2

4 4

2

0 Pcr

d x,t d x,t

dx D b dx

b x,t

φ π φ

π φ

⎛ ⎞

+⎜⎜ − ⎟⎟ +

⎝ ⎠

+ =

(47)

We follow the same procedure of calculation as above, β

1

and β

2

take the form:

1 1

2 2

1 2

2 2 2

1 2

2 2 2

cr cr cr

cr cr cr

b b

b b

λ λ λ

π π

β β

λ λ λ

π π

β β

⎛ ⎞ ⎛ ⎞

= ⎜ ⎝ − + ⎟ ⎠ ⎜ ⎝ − ⎟ ⎠ + Ω =

⎛ ⎞ ⎛ ⎞

= ⎜ ⎝ − − ⎟ ⎠ ⎜ ⎝ − ⎟ ⎠ + Ω =

(48)

where:

4 2 2

hb D

ρ η

Ω = π and:

( ) ^x,t ( ) ^{x e}

^{i tη}

φ = ψ

In this case, the Det (F) is plotted versus Ω for different values of r, λ

cr

,ω

1

, and τ

1

(r).

Figure 15(a) shows that for low values of ω

1

, the structure can have three frequencies (Ω

1

,

Ω2, Ω3

) corresponding to the three modes shape.

It illustrates also the effect of misplacement r on the first mode.

In Fig. 15(b) is shown that the structure can have only two modes (Ω

1

, Ω

2

).

These figures show that in stiffened plates we can obtain two or three mode shapes and the

eigenfrequencies can be modified by changing the parameters of stiffener.

Figs. 15. Determinant of the matrix F versus natural frequencies - with δ = 3 and different values of misplacement r, (a): ω1 = 0.01 and λcr =

4.35 (b): ω1= 0.1 and λcr = 4.75

V. Conclusion

In this paper, we have investigated analytically the buckling of rib-stiffened plates with misplaced stiffeners in one direction. In the case of a single rib-stiffener, the exact solution is obtained and the effect of the torsional rigidity and flexural rigidity of the stiffener on the modes, the critical buckling load and on the buckling modes is examined. We show that for a plate with a geometric parameter δ = 3, the critical buckling load increases with increasing the flexural rigidity for values up to 2.

It is also shown that the values of contribution

coefficients to the buckling modes can reach maximum

values for certain values of misplacement r. In other

words, only the dominant coefficients in the contribution

buckling mode expressions can be retained so that the

analysis for stiffened plates based on these optimized

expressions is simplified. Similar investigation is

performed in the case of two rib-stiffener. In this later

case, it was demonstrated that the misplacement r located

in the middle of the plate is better than the misplacement

r located near 1/3. We notice that the use of two (or

several) stiffeners reduces the amplitude of the response

and changes the system characteristics.Further, it is

A. Karimin, M. Belhaq

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 4

620 shown that in stiffened plates, two or three mode shapes can be obtained and the eigenfrequencies can be modified by changing the parameters of stiffener. This study gives an analytical expression of the buckling modes and their variations with respect to misplacement r allowing us the understanding of the local phenomena problem.

References

[1] J. M. Minguez, Torsional rigidity of ribs in the buckling of reinforced panels-Analytical study, Computers et Structures, Vol.

28, n. 1, pp. 47-51, 1988.

[2] W. Kakol, Stability analysis of stiffened plates by finite strips, Thin-Walled Structures, Vol. 10, pp. 277-297, 1990.

[3] I. Li, Y. W. Elishakoff, J. H Starnes J.R., Buckling mode localization in elastic plates due to misplacement in the stiffener location, Chaos, Solitons and Fractals, Vol. 5, n. 8, pp. 1517- 1531, 1995.

[4] M. Kolli, K. Chandrashekhara, Nonlinear static and dynamic analysis of stiffened laminated plates, International Journa Non- Linear Mechanics, Vol. 32, n. 1, pp. 89-101, 1997.

[5] W. C Xie, Buckling mode localization in rib-stiffened plates with randomly misplaced stiffeners, Computers et Structures, Vol. 67, pp. 175-189, 1998.

[6] W. C. Xie, A. Ibrahim, Buckling mode localization in rib-stiffened plates with misplaced stiffeners a finite strip approach, Chaos, Solitons and Fractals, Vol. 11, pp. 1543-1558, 2000.

[7] W.C. Xie, I. Elishakoff, Buckling mode localization in rib- stiffened plates with misplaced stiffeners-Kantorovich approach, Chaos, Solitons and Fractals, Vol. 11, pp. 1559-1574, 2000.

[8] G. Qing, J. Qiu, Y. Liu, Free vibration analysis of stiffened laminated plates, International Journal of Solids and Structures, Vol. 43, pp. 1357-1371, 2006.

[9] Z. Chen, W. C. Xie, Vibration localization in plates rib-stiffened in two orthogonal directions, Journal of Sound and Vibration, Vol. 280, pp. 235-262, 2005.

[10] A. Karimin, M. Belhaq, Effect of stiffener on nonlinear characteristic behavior of a rectangular plate: A single mode approach, Mechanics Research Communications, Vol. 36, pp.

699-706, 2009.

[11] C. Y. Chia, Nonlinear analysis of plates, McGraw-Hill, New York, 1980.

[12] S. P. Timoshenko, J. M. Gere, Theory of elastic stability, McGraw-Hill, New York, 1961.

Authors’ information

Abdellah Karimin was born in El jadida (Morocco), in January 1, 1980. He got his Ph.D.

in Mechanics at the Hassan II University, Casablanca, Morocco in 2010. The author’s major field of study is mechanics of structure and nonlinear dynamics.

He is the author of two publications appeared in Chaos, Soliton & Fractal and Mechanics research communications. His current research includes perturbation methods, nonlinear dynamics with application in MEMS.

Mohamed Belhaq was born in Casablanca (Morocco), in September 9, 1953. He got his Ph.D. in Mechanics at the University of Paul Sabatier, Toulouse, France in 1982. The author’s major field of interest is nonlinear mechanics, perturbation method and stability analysis.

He is the author of more than 60 publications appeared in international journals related to nonlinear dynamics, physics and mechanics. His current research includes perturbation methods, nonlinear dynamics, fast and slow phenomena, heteroclinic bifurcation with application in MEMS.

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