Effect of stiffener on nonlinear characteristic behavior of a rectangular plate: A single mode approach

Effect of stiffener on nonlinear characteristic behavior of a rectangular plate: A single mode approach

Abdellah Karimin, Mohamed Belhaq^*

Laboratory of Mechanics, University Hassan II-Casablanca, Morocco

a r t i c l e i n f o

Article history:

Received 13 January 2009 Available online 1 April 2009

Keywords:

Stiffened plate Natural frequency Nonlinear behavior

a b s t r a c t

In this paper, nonlinear dynamic of an excited square plate with a single stiffener is inves- tigated using a single mode approach. In a first part, we present an exact solution of the buckling mode of the stiffened plate under uni-axial compression. The differential equation of the deflection surface of the rectangular plate with different boundary conditions is used and the effect of the stiffener upon natural frequency of the plate is examined. In a second part, we analyze the dynamic of the excited mode near the principal resonance. Specifi- cally, we study the influence of stiffener on nonlinear characteristic behavior of the excited plate.

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1. Introduction

Stiffened plates are widely used in structures in order to increase their strength and enhance their performance. Appli- cations include aircraft, helicopters and spacecraft for quoting a few. The problem of stiffened plates has been studied fol- lowing three different trends. One approach consists of replacing the stiffened plate by an ‘equivalent’ orthotropic plate after the stiffeners are smeared out, in an energetic sense, over the entire surface of the plate (Brush and Almroth, 1975; Mcfarland et al., 1972). The second approach is based on energy consideration and treats the contribution of the plate and the stiffener separately (Bulson, 1969; Liew and Wang, 1990). The third approach is the analytical method for equally spaced stiffeners by the analytical finite difference calculus (Wah and Calcote, 1970; Bleich, 1952).Elishakoff et al. (1995)investigated the effect of 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.Tao et al. (2004)analyzed the nonlinear dynamic buckling of stiffened plates under the in-plane impact load, whereasChen et al. (2006)studied the effect of nonlinear contact upon natural frequency of del- aminated stiffened composite plate.

In the present work, we study analytically nonlinear dynamic of a simply supported stiffened plate in a single mode ap- proach. A similar approach was used to analyze nonlinear flexural vibration of a thin circular ring (Rougui et al., 2008). Spe- cifically, we examine the influence of a single-rib stiffener on the natural frequency of the buckling mode and on the nonlinear behavior of the stiffened plate under uni-axial compression.

In the first part of the paper, the differential equation of the deflection surface of the rectangular plate (Chia, 1980) with different boundary conditions is used. Here, we focus our attention on the case of a simple support under the rib (Mcfarland et al., 1972). The exact solution to the problem is obtained and an ordinary parametric nonlinear differential equation of the buckling mode shape is established using a Galerkin’s method. Therefore, the effect of the stiffener upon natural frequency of the stiffened plate is examined. In the second part, we study the nonlinear behavior of the buckling mode shape considering the Mathieu–Duffing oscillator model obtained via the Galerkin’s method. Using the multiple scale technique (Nayfeh and

0093-6413/$ - see front matterÓ2009 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mechrescom.2009.03.003

*Corresponding author.

E-mail address:mbelhaq@yahoo.fr(M. Belhaq).

Contents lists available atScienceDirect

Mechanics Research Communications

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / m e c h r e s c o m

Mook, 1979), the slow flow system describing the modulation of the amplitude and the phase of the periodic response of the considered mode is derived. The influence of the location of the stiffener on the hardening and softening behavior are exam- ined in the parameter plane corresponding to nonlinear component versus torsional rigidity.

2. Formulation 2.1. Linear case

We consider a rib-stiffened plate subjected in its mid-plane to uniform compressionPin thex-direction; seeFig. 1. From von Karman-type equations for the thin plate, we obtain the following equation of the deflection surface of the plate

D @⁴w

@x⁴ þ2 @⁴w

@x²@y²þ@⁴w

@y⁴

! þP@²w

@x²¼0 ð1Þ

wherewis the transverse displacement, downward positive andDis the flexural rigidity of the plate. The in-plane excitation of the thin plate may be expressed in the form P¼P0PDcosðXtÞ. We introduce the transformations x¼^x_a;y¼^y_b;w¼^w_h;P¼^a_D²Pandk¼_b^a, and for simplicity, we drop overbars. Hence, Eq.(1)can be written in the nondimen- sional form as

@⁴w

@x⁴þ2k² @⁴w

@x²@y²þk⁴@⁴w

@y⁴þP@²w

@x² ¼0 ð2Þ

The solution of Eq.(2)can be represented by

wðx;yÞ ¼XðxÞsinðpyÞ ð3Þ

Substituting Eq.(3)into Eq.(2)leads to d⁴

dx⁴XðxÞ þP2k²p^2d

2

dx²XðxÞ þk⁴p⁴XðxÞ ¼0 ð4Þ

The corresponding characteristic equation reads

s⁴þP2k²p²s²þk⁴p⁴¼0 ð5Þ

and the solutions are written as

s²¼ P 2k²p²

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P

2 P 22k²p²

s

ð6Þ

where the roots are given by s1¼ib₁;s2¼ ib₁;s3¼ib₂ and s4¼ib₂ with b₁¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð^P₂k²p²Þ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P 2 P 22k²p²

r q

and

b₂¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P 2k²p²

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P 2 P

22k²p²

r q

. A solution of Eq.(2)is given by

wðx;yÞ ¼ðAcosðb₁xÞ þBsinðb₁xÞ þCcosðb₂xÞ þDsinðb₂xÞÞsinðpyÞ ð7Þ

a/2 a/2

b P

P

X

Y

d

x₁ x2

a₁ a

2

Fig. 1.Uni-axial compressed rectangular stiffened plate with a single misplaced rib.

whereA;B;CandDare unknown constants to be determined using the continuity and boundary conditions. For the arbitrary jth span, the solution is written as

wjðxj;yÞ ¼ Ajcosb1xj

þBjsinb1xj

þCjcosb2xj

þDjsinb2xj

sinðpyÞ; 06xj6nj ð8Þ wheren_jis the length of thejth span andjranges from 1 toNfor anN-span plate. For the plate simply supported along its periphery, the boundary conditions are

w1jx1¼0¼0 M^ð1Þ_{x x}

1¼0¼ Dh a²

@²w1

@x²₁ þmk²@²w1

@y²

!_x

1¼0

¼0

M^ðNÞ_x

xN¼nN

¼ Dh a²

@²wN

@x²_N þmk²@²wN

@y²

!_x

N¼nN

¼0 wNjxN¼nN¼0

ð9Þ

whereM^ð1Þ_x andM^ðNÞ_x are the bending moments in the first and last spans of the continuous plate;mis the Poisson ratio,his the thickness andn_j¼^a_a^j. In view of Eq.(8), the above boundary conditions become

A1þC1¼0 ð10Þ

b²₁þmk²p²

A1þb²₂þmk²p²C1¼0 ð11Þ

b²₁þmk²p²

cosðb1nNÞANþb²₁þmk²p²sinðb1nNÞBNþ ðb²₂þmk²p²Þcosðb2nNÞCNþb²₂þmk²p²sinðb2nNÞDN¼0 ð12Þ cosðb1nNÞANþsinðb1nNÞBNþcosðb2nNÞCNþsinðb2nNÞDN¼0 ð13Þ As to the continuity conditions between two successive spans, we consider a simple support under the rib. In this case, the flexural rigidity of the stiffener is not large enough, and a vertical support is installed under the stiffener to suppress the transverse displacement. Thus, the continuity conditions between the two typical neighboring spansjandjþ1 are

wjþ1jxjþ1¼0¼0 ð14Þ

wjj_xj¼nj¼0 ð15Þ

@wj

@xj

xj¼nj

¼@wjþ1

@xjþ1

xjþ1¼0

ð16Þ

M^ðjþ1Þ_x

xjþ1¼0M^ðjÞ_x

xj¼nj

¼kðGJÞ_j bD

@³wjþ1

@xjþ1@y² _x

jþ1¼0

ð17Þ

where ðGJÞ_j denotes the torsional rigidity of thejth rib. Substituting Eq.(8) into Eqs. (14)–(17)leads to the following equations:

Ajþ1þCjþ1¼0 ð18Þ

cos b1nj

Ajþsin b1nj

Bjþcos b2nj

Cjþsin b2nj

Dj¼0 ð19Þ

b1sin b1nj

Ajþb1cos b1nj

Bjb2sin b2nj

Cjþb2cos b2nj

Djb1Bjþ1b2Djþ1¼0 ð20Þ b²₁þmk²p²cos b1nj

Ajb²₁þmk²p²sin b1nj

Bjb²₂þmk²p²cos b2nj

Cjb²₂þmk²p²sin b2nj

Dj

þb²₁þmk²p²Ajþ1þb²₂þmk²p²Cjþ1þab₁Bjþ1þab₂Djþ1¼0 ð21Þ wherea¼^ðGJÞ_bD^jkp2. A plate with a single rib-stiffener corresponds toj¼1. Therefore, Eqs.(10)–(13)and Eqs.(18)–(21)can be written in the form

FD¼0 ð22Þ

whereFis a matrix of 88 elements andD¼ðA1;B1;C1;D1;A2;B2;C2;D2Þ^T.

Let us consider a square plate. Intuitively, the single stiffener should be placed as close as possible to the mid cross-sec- tion of the plate to produce the highest reinforcement on the plate. We will use the following nondimensional notations for specifying the positions of the stiffeners

n1¼1

2þd; n2¼1

2d; d¼d

a ð23Þ

Thus, the elements of matrixFare composed of the parametersd;Panda. A non-trivial solution of Eq.(22)is obtained by setting the determinant ofFequal to zero, which yields a transcendental equation whose the smallest root corresponds to the critical buckling loadP.

However, for the unstiffened plate, the buckling loadPis always equal to or larger than 4k²p²(Timoshenko and Gere, 1961). To determine the critical buckling load, we first plot the torsional rigidityaversus misplacementdusing the vanishing of the determinant ofF(seeFig. 2). This figure provides the critical buckling loadP0corresponding to the smallest critical buckling load given byd¼0;aP0 and DetðFÞ 0. In this case, theb₁andb₂modes areb₁¼2pandb₂¼^p₂. The buckling mode for the mid-span (seeFig. 1) are expressed as

w1ðx1;yÞ ¼B1 sin 2ð px1Þ þf1sin px1

2

h i

sinðpyÞ; 06x161

2þd ð24Þ

w2ðx2;yÞ ¼B1 f3 cos px2

2

cos 2ð px2Þ

þf2sin 2ð px2Þ þf4sin px2

2

h i

sinðpyÞ; 06x261

2d ð25Þ

Note that the plate deflection shape is a combination of a simply supported mode and a clamped mode in the transverse direc- tion. The sine-terms represent the simply supported deflection mode, while the cosine-terms represent the clamped mode.

2.2. Nonlinear case

From von Karman-type equations for the thin plate (Chia, 1980), the equations of motion for a rectangular thin plate are given by

Dr⁴wþqh@²w

@t² @²w

@x²

@²/

@y²@²w

@y²

@²/

@x²þ2 @²w

@x@y

@²/

@x@yþl^@w

@t ¼0 ð26Þ

r⁴/¼Eh @²w

@x@y

!2

@²w

@x²

@²w

@y² 2

4

3

5 ð27Þ

whereqis the density,D¼Eh³=ð12ð1m²ÞÞis the bending rigidity,Eis the Young’s modulus,mis the Poisson’s ratio,/is the stress function, andlis the damping coefficient. We introduce the following variables and parameter transformations:

s¼1 a²

ffiffiffiffiffiffi D

qh s

t; X¼a² ffiffiffiffiffiffi

qh D r

X; /¼ /

k²Eh³; l¼ a²

p^{ffiffiffiffiffiffiffiffiffiffi}hqDl; ¼12ð1m²Þh²

ab ; C¼k⁴ab

h² ð28Þ

Eqs.(26) and (27)can be rewritten in nondimensional form as

@²w

@s^2þl^@w

@s^þ

@⁴w

@x⁴þ2k² @⁴w

@x²@y²þk⁴@⁴w

@y⁴þC @²w

@x²

@²/

@y²þ2@²w

@x@y

@²/

@x@y@²w

@y²

@²/

@x²

" #

¼0 ð29Þ

@⁴/

@x⁴þ2k² @⁴/

@x²@y²þk⁴@⁴/

@y⁴¼ @²w

@x@y

!2

@²w

@x²

@²w

@y² ð30Þ

The boundary conditions satisfied by the stress function/may be expressed as Z 1

0

@²/1

@y² dy¼ 1

12ð1m²Þk⁴P; atx1¼0 Z 1

0

@²/2

@y² dy¼ 1

12ð1m²Þk⁴P; atx2¼1 2d

ð31Þ

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−20 0 20 40 60 80 100 120 140

δ

α

P=2.2π² P=2.25π² P=2.5π²

Fig. 2.The nondimensional torsional rigidityaversus misplacementdwith different values of buckling loads.

and Z ¹₂þd

0

@²/₁

@x² dx¼0; aty¼0; y¼1 Z ¹₂d

0

@²/₂

@x² dx¼0; aty¼0; y¼1

ð32Þ

Substituting Eqs.(24) and (25)into Eq.(30), considering the boundary conditions, Eqs.(31) and (32), and integrating, we obtain the stress function as follows:

/1¼/10cos 4ð px1Þ þ/11cosðpx1Þ þ/12cos 5px1

2

þ/13cos 3px1

2

þ/14cosð2pyÞ þ/15cos 5px1

2

cosð2pyÞ þ/16cos 3px1

2

cosð2pyÞ 1

24ð1m²Þk⁴Py² ð33Þ

/2¼/20sinð4px2Þ þ/21cosð4px2Þ þ/22cosð2pyÞ þ/23cosðpx2Þ þ/24sin 5px2

2

þ/25cos 5px2

2

þ/26sin 3px2

2

þ/27cos 3px2

2

þ/28sin 5px2

2

cosð2pyÞ þ/29cos 5px2

2

cosð2pyÞ þ/210sin 3px2

2

cosð2pyÞ þ/211cos 3px2

2

cosð2pyÞ 1

24ð1m²Þk⁴Py² ð34Þ

where

/₁₀¼ 1

128B²₁; /₁₁¼1

8f²₁B²₁; /₁₂¼ 1

25f1B²₁; /₁₃¼ 1

9f1B²₁; /₁₄¼ 1 16k⁴ 2þ1

8f²₁

B²₁; /₁₅¼ 9

800k²f1B²₁; /₁₆¼ 25

288k²f1B²₁ ð35Þ

/20¼f2f3

64 B²₁; /21¼f2²f²₃

128 B²₁; /22¼ 1 16k⁴

17

8 f²₃þ2f2²þ1 8f²₄

B²₁; /23¼1

8f²₄f²₃ B²₁; /₂₄¼ 1

25ðf3f4f3f2ÞB²₁; /₂₅¼ 1

25 f²₃þf2f4

B²₁; /₂₆¼ 1

9ðf2f3þf3f4ÞB²₁ ð36Þ /27¼1

9 f²₃f2f4

B²₁; /28¼ 9

800k²ðf3f4f2f3ÞB²₁; /29¼ 9

800k² f²₃þf2f4

B²₁; /₂₁₀¼ 25

288ðf2f3þf3f4ÞB²₁; /₂₁₁¼ 25

288f2f4þf²₃ B²₁

For simplicity, we drop the overbars in the following analysis. Using Galerkin’s method, substituting Eqs.(24) and (25)and Eqs.(33) and (34)into Eq.(29)and integrating, we obtain the dimensionless damped Mathieu–Duffing equation

B1ðtÞ þ€ lD1B1ðtÞ þ_ x²þPDD2cosðXtÞ

B1ðtÞ þD3B1ðtÞ³¼0 ð37Þ

whereD₁;D₂andD₃depend on system parametersaandd.

60 62 64 66 68 70 72 74

0 5 10 15 20 25

P₀

ω

δ=−0.01 ,α=0 δ=−0.09 ,α=0 δ=−0.09 ,α=40

Fig. 3.The nondimensional frequencyxversus static loadP0for different values of misplacementdand torsional rigiditya.

Fig. 3illustrates the nondimensional frequency as function of static loadP0for different values ofdanda. This figure shows that the location of misplacementdwith respect to the medium of the plate influences the natural frequency. Also, asais increased froma¼0 toa¼40, and ford¼0:09, the natural frequencyxdecreases.

3. Dynamic analysis

In this section, we examine periodic responses of the buckling mode shape modelled by Eq.(37). Applying the multiple scales method, approximate periodic solutions of Eq.(37)can be sought in the form

B1ðsÞ ¼B01ðT0;T1Þ þB11ðT0;T1Þ þoð²Þ ð38Þ whereT0¼sis a fast time scale andT1¼sis a slow time scale describing the envelope of the response. In terms of the variableTn, the time derivative becomes_d^d_s¼D0þD1þoð²Þ, whereDn¼_@T^@

n. We perform our analysis in the vicinity of the principal resonance 2:1 which is expressed as

X¼2xþr ð39Þ

Substituting Eq.(38)into Eq.(37), using Eq.(39), and equating coefficients of like powers of, we obtain at different order of

Order⁰:

D²₀B01þx²B01¼0 ð40Þ

Order¹:

D²₀B11þx²B11¼ lD1D0B01D3B³₀₁PDD2cosðXT0ÞB012D0D1B01 ð41Þ The solution of Eq.(40)can be expressed as

B01ðT0;T1Þ ¼AðT1Þe^ixT0þcc ð42Þ

whereccdenotes the complex conjugate of the preceding terms. The quantityAðT1Þis determined by eliminating the secular terms at the next level of approximations. Substituting Eq.(42)into Eq.(41), eliminating secular terms, lettingA¼^aðT₂^1Þe^ihðT1Þ, whereaandhare real functions, and separating real and imaginary parts, we obtain the following modulation equations of amplitude and phase:

da dT1

¼ lD1

2 aPDD2

4x ^asinðcÞ ð43Þ

adc

dT1

¼r

2a3D3

8x^a

3PDD2

4x ^acosðcÞ ð44Þ

wherec¼rT1h. Equilibria of the slow flow system(43) and (44)correspond to periodic solutions of Eq.(37). These are obtained by setting, in Eqs.(43) and (44),_dT^da

1¼_dT^dc

1¼0. We obtain

lD1

2 a¼ PDD2

4x ^asinðcÞ ð45Þ

r

2a¼3D3

8x^a

3þPDD2

4x ^acosðcÞ ð46Þ

This system has a trivial solutiona¼0 and a non-trivial one given by

lD1

2 2

þ r

23D3

8x^a

2

2

¼ PDD2

4x

2

ð47Þ It follows from Eq.(47)that:

r¼3D3

4x^a

22 PDD2

4x

2

lD1

2 2!¹₂

ð48Þ In order that Eq.(48)has real solutions,P_Dmust exceed the critical value

PDcr ¼2lx^D1 D2

ð49Þ Figs. 4 and 5given by Eq.(48)illustrate the frequency response fora¼10 anda¼15, respectively, and for different values of d.Fig. 4indicates that for a torsional rigiditya¼10, the behavior of the mode is softening for bothd¼ 0:01 andd¼ 0:09, whereasFig. 5shows that fora¼15, the behavior is softening (ford¼ 0:01) and hardening (ford¼ 0:09).Fig. 6summa- rizes these results. In this figure, we plot the variation of the cubic nonlinear component¼³₈^D_x³versus the torsional rigidity

a. The sign ofdetermines effectively the behavior of the system. The hardening behavior corresponds to >0 and the soft-

ening one corresponds to <0. It can be concluded fromFig. 6that when misplacementdis located near the middle of the plate (solid line), the response of the system changes its characteristic from softening to hardening for a smaller value ofa

comparing to misplacement located far from the middle of the plate (dashed line). Also, it can be seen that for the values ofa

lying between a1 anda2, corresponding to the change of sign of, the behavior of the mode is hardening ( >0) for d¼ 0:01 and it is softening ( <0) ford¼ 0:09 (seeFig. 6).

−8 −6 −4 −2 0 2 4 6 8

0 0.01 0.02 0.03 0.04 0.05 0.06

σ

a

Fig. 4.Frequency response curves fora¼10 with different values of misplacementd: (‘-’ ford¼ 0:01) and (‘- -’ ford¼ 0:09).

−8 −6 −4 −2 0 2 4 6 8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

σ

a

Fig. 5.Frequency response curves whena¼15 with different values of misplacementd: (‘-’ ford¼ 0:01) and (‘- -’ ford¼ 0:09).

10 15 20 25

−4000

−3000

−2000

−1000 0 1000 2000 3000 4000

α

ϒ

α₁ α₂

δ=−0.01 δ=−0.09

Fig. 6.Regions of hardening and softening behavior.

4. Conclusion

In this work, we have investigated the nonlinear flexural dynamic of a stiffened plate under uni-axial excitation in a single mode approach. First, the buckling mode is obtained and the ordinary differential equation of the mode shape is established.

In the second part, we have examined the nonlinear behavior of the buckling mode shape near the principal resonance 2:1 using a perturbation analysis. We have shown that hardening and softening behaviors can occur in the response of the stiff- ened plate depending on the location of misplacement. Specifically, for certain range of torsional rigiditya, the nonlinear characteristic behavior of the plate may change from softening to hardening or vise versa by changing only the location of misplacementd.

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