Research Article Design of Corrugated Plates for Optimal ...downloads.hindawi.com/journals/aav/2016/4290247.pdf · Design of Corrugated Plates for Optimal Fundamental Frequency NabeelAlshabatat

Research ArticleDesign of Corrugated Plates for OptimalFundamental Frequency

Nabeel Alshabatat

Department of Mechanical Engineering, Tafila Technical University, Tafila 66110, Jordan

Correspondence should be addressed to Nabeel Alshabatat; [email protected]

Received 28 March 2016; Revised 16 June 2016; Accepted 27 June 2016

Academic Editor: Marc Thomas

Copyright © 2016 Nabeel Alshabatat. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper investigates shifting the fundamental frequency of plate structures by corrugation. Creating corrugations significantlyimproves the flexural rigidities of plate and hence increases its natural frequencies. Two types of corrugations are investigated:sinusoidal and trapezoidal corrugations. The finite element method (FEM) is used to model the corrugated plates and extract thenatural frequencies and mode shapes. The effects of corrugation geometrical parameters on simply supported plate fundamentalfrequency are studied. To reduce the computation time, the corrugated plates are modeled as orthotropic flat plates withequivalent rigidities. To demonstrate the validity of modeling the corrugated plates as orthotropic flat plates in studying the freevibration characteristics, a comparison between the results of finite element model and equivalent orthotropic models is made. Acorrespondence between the results of orthotropic models and the FE models is observed. The optimal designs of sinusoidal andtrapezoidal corrugated plates are obtained based on a genetic algorithm.The optimization results show that plate corrugations canefficiently maximize plate fundamental frequency. It is found that the trapezoidal corrugation can more efficiently enhance thefundamental frequency of simply supported plate than the sinusoidal corrugation.

1. Introduction

Plate structures have been widely used in structural, naval,automobile, and aerospace engineering. However, platesexhibit poor vibration performance due to their lateralflexibility (i.e., plates can be deformed easily in the out-of-plane direction under static or dynamic loads).The vibrationproperties of plates can be modified by different methods.One objective of these methods is to shift the plate naturalfrequencies away from the frequency of the excitation forceto avoid resonance. One cost effective method to improve thevibration characteristics of plates is to modify their shapes[1]. In this study, shape modification is used to optimize thevibration characteristics of plates. In particular, corrugation isused in plate construction to maximize the first plate naturalfrequency.

Corrugation of plate-like structures is commonly used toincrease their strength for static loading conditions such asroofing in engineering structures, container walls, and bridgebulkheads. In the late 1960s, NASA developed corrugated

panel structures to use them in many aerospace applicationssuch as the space shuttle and hypersonic aircraft. The cor-rugated nature of these panels makes them suitable for hightemperature applications because the curved sections permitpanel thermal expansion without inducing a significantthermal stress [2]. In 1970, Plank et al. [3] studied the bestprimary structure of aMach 8 hypersonic transportwing; andthey examined the ultimate load, wing flutter, panel flutter,fatigue, and creep for different wing structures. Recently,corrugated plates are used in constructing morphing wings[4].

A precise analysis of corrugated plates can be achievedby using the finite element method. However, the finiteelement method does not efficiently model the corrugatedplates because it requires significant computing times.Amorepractical method to design and analyze corrugated platesis to treat them as orthotropic flat plates with equivalentrigidities (i.e., the corrugated plate is flexible in the corru-gation direction and rigid in the crosswise direction). Afterthat, an approximate solution can be obtained by solving

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2016, Article ID 4290247, 9 pageshttp://dx.doi.org/10.1155/2016/4290247

2 Advances in Acoustics and Vibration

zy

x

Lx

Ly

𝜆F

C

Figure 1: Sinusoidal corrugated plate.

the equivalent orthotropic plate problem. This approach ofapproximation is computationally more efficient with a littleloss of precision.

The literature is rich in studies related to the behaviorof corrugated plates under static loading. However, there isa limited number of papers which study the vibration ofsuch plates. Samanta and Mukhopadhyay [5] presented for-mulas for the equivalent rigidities of trapezoidal corrugatedplates based on energy principles; and they carried out freevibration analysis using finite element method. In addition,they carried out a nonlinear geometric analysis. Based on thenonlinear bending theory of thin shallow shells, Wang et al.[6] investigated the large amplitude vibration of corrugatedcircular plates with shallow sinusoidal corrugations undertemperature changes. Hamilton’s principle was used to derivethe governing equations. Liew et al. [7] studied the freevibration of stiffened and unstiffened corrugated plates byusing a mesh-free Galerkin method based on the first ordershear deformation theory. Yucel andArpaci [8] used the finiteelement method to study the free vibration of different typesof trapezoidal and sinusoidal corrugated plates.

The objective of this study is to design corrugated platesand to optimize their fundamental frequencies. A free vibra-tion analysis of sinusoidal and trapezoidal corrugated platesbased on both the finite element method and equivalentorthotropic models is made in this paper. To demonstrate thevalidity of modeling the corrugated plates as orthotropic flatplates in studying the free vibration characteristics, a com-parison between the results of the finite element model andequivalent orthotropic models is made. Parametric analysisand optimization examples are studied to provide a guidefor design engineers. The optimal designs of the corrugatedplates are found by using a genetic algorithm (GA).

2. Theoretical Background

Corrugated plates are assumed to have sinusoidal or trape-zoidal corrugation in one direction as shown in Figures 1 and2. The corrugated plate can be described by the number ofcorrugation 𝑁

𝑐, the pitch 𝐶, the height 𝐹, and the trough

zy

x

Lx

Ly

𝜆F

C

𝜃

Figure 2: Trapezoidal corrugated plate.

angle 𝜃 (for trapezoidal corrugation). In this work, FEM isused for the modal analysis of corrugated plates. In additionto FEM, the corrugated plates are studied analytically basedon modeling the corrugated plates as orthotropic flat plateswith the same thickness and equivalent rigidities.

2.1. Free Vibration Analysis of Orthotropic Plate. Using Love-Kirchhoff ’s hypotheses, the equation of the free vibration oforthotropic rectangular thin plate of length 𝐿

𝑥and width 𝐿

𝑦

can be written as

𝐷11

𝜕4

𝑤

𝜕𝑥4+ 2𝐻

𝜕4

𝑤

𝜕𝑥2𝜕𝑦2+ 𝐷22

𝜕4

𝑤

𝜕𝑦4+ 𝜌ℎ

𝜕2

𝑤

𝜕𝑡2= 0, (1)

where 𝑤 is the lateral displacement of the plate at point(𝑥, 𝑦); 𝐷

11and 𝐷

22are the flexural rigidities in the 𝑥 and 𝑦

directions, respectively; 𝐻 is the equivalent rigidity; ℎ is thethickness; and 𝜌 is the mass density of the plate. The solutionof the harmonic motion of the plate has the form

𝑤 (𝑥, 𝑦, 𝑡) = 𝑊 (𝑥, 𝑦) cos (𝜔𝑡) , (2)

where 𝑊(𝑥, 𝑦) is the natural mode. Substituting (2) into (1)results in the following:

𝐷11

𝜕4

𝑊

𝜕𝑥4+ 2𝐻

𝜕4

𝑊

𝜕𝑥2𝜕𝑦2+ 𝐷22

𝜕4

𝑊

𝜕𝑦4− 𝜌ℎ𝜔

2

𝑊 = 0. (3)

The solution of (3) depends on the boundary conditions ofthe plate. For a rectangular plate simply supported on all thefour sides, the solution of (3), which satisfies the boundaryconditions, can be written as

𝑊(𝑥, 𝑦) =

∞

∑

𝑚=1

∞

∑

𝑛=1

𝐴𝑚𝑛

sin(𝑚𝜋𝑥𝐿𝑥

) sin(𝑛𝜋𝑦

𝐿𝑦

) , (4)

Advances in Acoustics and Vibration 3

where𝐴𝑚𝑛

is a scale factor of (𝑚, 𝑛) mode shape. Substitutionof (4) into (3) gives the circular frequency 𝜔

𝑚𝑛which is

associated to the𝑊𝑚𝑛

mode shape as

𝜔𝑚𝑛

=𝜋2

√𝜌ℎ

⋅ √𝐷11(𝑚

𝐿𝑥

)

4

+ 2𝐻(𝑚

𝐿𝑥

)

2

(𝑛

𝐿𝑦

)

2

+ 𝐷22(𝑛

𝐿𝑦

)

4

.

(5)

After calculating the equivalent rigidities of the corru-gated plate, the free vibration characteristics can be estimatedby solving (1) either theoretically for some boundary condi-tions or numerically for others.

2.2. Equivalent Orthotropic Properties of Corrugated Plates.As mentioned previously, corrugated plates can be approxi-mated as orthotropic flat plates. The suitable selection of theequivalent rigidities plays a significant role in the accuracy ofthe model. There are different equivalent rigidities availablein the literature. The most complete and the most recentequivalent rigidities are adopted in this study to calculate thenatural frequencies. The equivalent models are summarizedhere. The derivation of these models is not considered to bewithin the scope of this study.

The earliest complete estimation of the equivalent rigidi-ties for sinusoidal corrugation known to the author is foundin the works of Huber [9] and Seydel [10]. For manyyears, researchers followed the classical formulas of Huberand Seydel. Briassoulis [11] reviewed the classical formulasfor the equivalent rigidities and developed more accurateexpressions for the rigidities of corrugated plates:

𝐷11=𝐶

𝜆

𝐸ℎ3

12 (1 − ]2),

𝐷22=𝐸ℎ𝐹2

2+

𝐸ℎ3

12 (1 − ]2),

𝐷66=

𝐸ℎ3

24 (1 + ]),

𝐷12= ]𝐷11,

(6)

where 𝐶 is the corrugation pitch; 𝐹 is the height of cor-rugation (see Figure 1); 𝜆 is the developed length of unitcorrugation; 𝐼 is the moment of inertia of the plate alongthe corrugation direction; 𝐸 and ] are the modulus ofelasticity andPoisson’s ratio of platematerial, respectively. Fortrapezoidal corrugated plates, Samanta and Mukhopadhyay[5] presented the following bending rigidities:

𝐷11=𝐶

𝜆

𝐸ℎ3

12,

𝐷22=𝐸𝐼

𝐶,

𝐷66=𝜆

𝐶

𝐸ℎ3

24 (1 + ]),

𝐷12= 0.

(7)

Liew et al. [7] used the following formulas for flexuralrigidities of trapezoidal corrugated plates:

𝐷11=𝐶

𝜆

𝐸ℎ3

12 (1 − ]2),

𝐷22=𝐸ℎ

𝐶𝛼 +

𝐸ℎ3

12 (1 − ]2),

𝐷66=

𝐸ℎ3

24 (1 + ]),

𝐷12= ]𝐷11,

(8)

where 𝛼 is a parameter that depends on the geometry ofcorrugation as given in [12].

Xia et al. [13] investigated the equivalent flexural rigiditiesfor the geometry of arbitrary corrugations. The suggestedbending rigidities are

𝐷11=𝐶

𝜆

𝐸ℎ3

12 (1 − ]2),

𝐷22=

𝐸𝐼

1 − ]2+ 𝐴

𝐸ℎ3

12 (1 − ]2),

𝐷66=𝜆

𝐶

𝐸ℎ3

24 (1 + ]),

𝐷12= ]𝐷11,

(9)

where 𝐴 depends on the geometry of corrugation. Forsinusoidal corrugation, 𝐴 is given by

𝐴 = ∫

0.5

−0.5

1

√1 + ((2𝜋𝑇/𝐶) cos (2𝜋𝑋))2𝑑𝑋. (10)

For trapezoidal corrugation 𝐴 is given by

𝐴 = 𝐶 −8

3

𝐹

tan (𝜃). (11)

Very recently, Ye et al. [14] derived an equivalent plate modelfor general corrugation. For shallow sinusoidal corrugation,the bending rigidities are

𝐷11=𝐶

𝜆

𝐸ℎ3

12 (1 − ]2),

𝐷22= 𝐸𝐼 + 𝐴

𝐸ℎ3

12+ ]2𝐷

11,

𝐷66=𝜆

𝐶

𝐸ℎ3

12,

𝐷12= ]𝐷11.

(12)


(a) (b)

Figure 3: Typical finite element models for (a) sinusoidal and (b) trapezoidal corrugated plates.

Approximate values for the moment of inertia 𝐼 and thedeveloped length of a unit corrugation 𝜆 are available invarious references [15, 16]. For more accuracy, the valuesof 𝐼 and 𝜆, in this study, are evaluated numerically. Themass density used in the equivalent orthotropic models isestimated by (𝜆/𝐶) multiplied by the actual mass density ofthe corrugated plate.

3. Parametric Analysis

To design a corrugated plate, three independent geometricalparameters need to be determined such as the thickness of theplate ℎ, the number of corrugation 𝑁, and the corrugationheight 𝐹, or the corrugation pitch 𝐶 instead of numberof corrugation 𝑁, in addition to the trough angle 𝜃 fortrapezoidal corrugation. The plate under consideration is asimply-supported square plate with length 𝐿

𝑥= 𝐿𝑦= 1m

and thickness ℎ = 0.001m that is made of steel with modulusof elasticity 𝐸 = 210GPa, Poisson’s ratio ] = 0.3, and massdensity 𝜌 = 7800 kg/m3.

The finite element method (FEM) is used for the modalanalysis of corrugated plates. In particular, the plates aremodelled by using ANSYS Parametric Design Language(APDL) in which the model can be built in terms of thegeometrical parameters of corrugation. The shell element“shell63” with four nodes and six degrees of freedom pernode is used to mesh the solid models. Convergence studiesare performed and the finite element of 10mm size is usedthroughout the parametric analysis. Typical finite elementmeshes of sinusoidal and trapezoidal corrugated plates areshown in Figure 3.

To gain some knowledge about the fundamental fre-quency of corrugated plates, it is important to study theeffect of changing the corrugation geometrical parameterson the plate fundamental frequency. To study the effect ofthe corrugation height of sinusoidally corrugated plate on

FE model

4

6

8

10

12

14

16

18

20

22

24

Fund

amen

tal f

requ

ency

ratio

6 8 10 12 14 16 18 204Corrugation height (mm)

(N)FE = 6

(N)FE = 10

(N)FE = 20

Figure 4: Effect of corrugation height on the fundamental fre-quency ratio for different number of corrugations (sinusoidalcorrugation).

the fundamental frequency ratio, the number of corrugationis held constant (e.g., 𝑁 = 6 or 10 or 20 corrugations).The corrugation height varies from 5mm to 20mm. Theresults are shown in Figure 4. The fundamental frequencyratio represents the ratio between the fundamental frequencyof the corrugated plate and the fundamental frequency ofthe flat plate with the same lengths and thickness. Increas-ing the corrugation height would significantly increase thefundamental frequency ratio of the sinusoidally corrugatedplate. This increase in the fundamental frequency resulted


FE model

15

20

25

35

30

Fund

amen

tal f

requ

ency

ratio

8 10 12 14 16 18 206Number of corrugations

(F)FE = 15mm(F)FE = 20mm(F)FE = 30mm

Figure 5: Effect of number of corrugations on the fundamentalfrequency ratio for different corrugation heights (sinusoidal corru-gation).

from the obvious increase in themoment of inertia due to thecorrugation. Figure 5 shows the change in the fundamentalfrequency ratio of sinusoidally corrugated plates with varyingthe number of corrugations from 6 to 20 and the height ofcorrugation is held constant (e.g., 𝐹 = 15 or 20 or 30mm).It is shown that, for small corrugation height, the frequencyratio decreases as𝑁 increases though the frequencies are stillhigher than those of the uncorrugated plate. For a highercorrugation number, increasing the number of corrugations(6 ≤ 𝑁 ≤ 14) decreases the fundamental frequencyratio. In all cases, the significant increase in the numberof corrugations has a minor effect on the fundamentalfrequency ratio.

The effects of changing the corrugation height, number,and the trough angle on the fundamental frequency ratioof trapezoidally corrugated plate are shown in Figures 6and 7. Figure 6 shows that the fundamental frequency ratioof trapezoidally corrugated plate increases significantly byincreasing the corrugation height and the trough angle. Forsmall corrugation height, the trough angles have almost thesame effect on the fundamental frequency ratio. As shownin Figure 7, the fundamental frequency ratio of trapezoidallycorrugated plate is sensitive to the increase in the number ofcorrugations, especially for plates with small trough angles.By increasing the number of corrugations, the fundamentalfrequency ratio decreases significantly when 𝜃 = 45

∘.In addition to changing the natural frequencies, creating

corrugations changes the platemode shapes. For example, thefirst four mode shapes of a simply supported plate before andafter creating 10 sinusoidal corrugations with 20mm heightsare shown in Figure 8. It is expected that this change in modeshapes results in a change in the sound radiation when theplate is subjected to an excitation force.


5

10

15

20

25

30

Fund

amen

tal f

requ

ency

ratio

FE model (N = 10)

𝜃 = 45∘

𝜃 = 60∘

𝜃 = 90∘

Figure 6: Effect of corrugation height on the fundamental fre-quency ratio for different trough angles (trapezoidal corrugation).

15

20

25

30

Fund

amen

tal f

requ

ency

ratio

10 15 205Number of corrugations

FE model (F = 20mm)

𝜃 = 45∘

𝜃 = 60∘

𝜃 = 90∘

Figure 7: Effect of number of corrugations on the fundamental fre-quency ratio for different trough angles (trapezoidal corrugation).

Modeling and vibration analysis of corrugated platesusing FEM can be very computational expensive and timeconsuming especially for the cases of high number of cor-rugations (i.e., high number of corrugations per unit lengthrequires finermesh).Thus, using FEM in design optimizationor in any rigorous analysis of corrugated plates is inefficient.Treating the corrugated plate as a flat orthotropic plate isan economic alternative for using FEM. In the optimization


Flat plate Sinusoidal corrugated plate

First mode First mode

Second mode Second mode

Third mode Third mode

Fourth mode Fourth mode

Figure 8: First four mode shapes of flat and sinusoidal corrugatedsimply supported plate.

section, the corrugated plates will be treated as orthotropicplates with equivalent rigidities. To select the most appro-priate equivalent rigidities for the current study, the effect ofgeometric parameters of corrugation on plate fundamentalfrequency is investigated using equivalent orthotropic flatplate models. The rigidities in these models are based on themost complete and most recent equivalent rigidities ((6)–(12)).

4 6 8 10 12 14 16 18 20

FE versus orthotropic models (10 corrugations)

Corrugation height (mm)

4

6

8

10

12

14

16

18

20

22

24

Fund

amen

tal f

requ

ency

ratio

FEMBriassoulis [11]

Xia et al. [13]Ye et al. [14]

Figure 9: Effect of corrugation height on the fundamental fre-quency ratio based on finite element and equivalent orthotropicmodels (sinusoidal corrugation).

15

16

17

18

19

20

21

22

23

24

25

Fund

amen

tal f

requ

ency

ratio


FEMBriassoulis [11]

Xia et al. [13]Ye et al. [14]

FE versus orthotropic models (F = 20mm)

Figure 10: Effect of number of corrugations on the fundamentalfrequency ratio based on finite element and equivalent orthotropicmodels (sinusoidal corrugation).

The accuracy of the equivalent orthotropic models forcorrugated plates is examined by comparing their results withthose of the FE models. Comparison of the fundamentalfrequency ratio of the sinusoidal corrugated plates, computedby FEM and different orthotropic models based on theequivalent rigidities of Briassoulis [11], Xia et al. [13], andYe et al. [14], is shown in Figures 9 and 10. It is clearthat the orthotropic model that is based on the equivalentrigidities of Ye et al. [14] yields the most best agreement withthe fundamental frequency values obtained using FEM. In


5

10

15

20

25

30

Fund

amen

tal f

requ

ency

ratio


FE versus orthotropic models (N = 10 and 𝜃 = 60∘)

FEMSamanta and Mukhopadhyay [5]

Xia et al. [13]Liew et al. [7]

Figure 11: Effect of corrugation height on the fundamental fre-quency ratio based on finite element and equivalent orthotropicmodels (trapezoidal corrugation).

general, the differences between the results of FEM and thoseof the equivalent orthotropic models get larger by increasingthe corrugation height and number of corrugations.

For trapezoidal corrugated plates, orthotropic equivalentplates that are based on the equivalent rigidities of Samantaand Mukhopadhyay [5], Xia et al. [13], and Liew et al. [7] arecompared with FE models of the actual corrugated plates inFigures 11 and 12. Like in the sinusoidal corrugated plates, abetter correlation between the orthotropic models and theFE models can be obtained when the corrugation heightdecreases because the derivations of the equivalent rigidi-ties are based on the assumption of shallow corrugations.Figure 12 shows that the correlation in the results of theorthotropic model that is based on equivalent rigidities ofSamanta and Mukhopadhyay [5] improves with the increasein the number of corrugations.

In the optimization examples presented in the nextsection, the equivalent rigidities of Ye et al. [14] and Samantaand Mukhopadhyay [5] are used for sinusoidal corrugatedplates and trapezoidal corrugated plates, respectively.

4. Optimization Examples

In the previous section, an insight is gained on the effect ofcorrugation geometric parameters on a plate fundamentalfrequency. In this section, the focus will be on maximiz-ing the plate fundamental frequency by using sinusoidalor trapezoidal corrugations. We consider two optimizationproblems.The first is tomaximize the fundamental frequencyof a square corrugated plate. The second is to maximizethe fundamental frequency-to-mass ratio of the same plate.Three dimensions of sinusoidal corrugated plates, includingplate thickness ℎ, height of corrugation 𝐹, and number of

15

20

25

30

Fund

amen

tal f

requ

ency

ratio


FEMSamanta and Mukhopadhyay [5]

Xia et al. [13]Liew et al. [7]

FE versus orthotropic models (F = 20mm and 𝜃 = 60∘)

Figure 12: Effect of number of corrugations on the fundamentalfrequency ratio based on finite element and equivalent orthotropicmodels (trapezoidal corrugation).

corrugations 𝑁, are selected as design variables. In additionto these design variables, the trough angle 𝜃 is considered asa design variable for designing trapezoidal corrugated plates.During optimization, the number of corrugation is restrictedto integer numbers. The equivalent orthotropic plate modelis used to model the corrugated plates in the optimizationproblems.

Here, a genetic algorithm (GA) is used for optimizationexamples. The GAs are stochastic optimization methods thatare based on the concept of biological evolution. It wasintroduced by Holland in 1975 [17]. In comparison withtraditional optimization methods, the GAs work over a setof candidate points, which cover the entire search spaceinstead of a single candidate point at each iteration. It,therefore, increases the probability to reach to the globaloptimum. GAs are not gradient-based methods. Thus, theycan work with discrete design variables and do not requirethe objective functions to be differentiable. Also, the GAscan work with nonconvex problems. In general, GA usesa set of candidate points (called population) which coverthe entire search space. Then, the objective function (fitnessfunction) is evaluated for each candidate point. The pointswith relatively good objective functions are used to generatea new set of candidate points through three mechanismsknown as reproduction, crossover, and mutation. In thereproduction method, the candidate point in the currentgeneration with the best objective function is automaticallypassed to the next generation. The crossover method createsa new point by combining the vectors of two candidate pointsin the current generation.Themutationmethod creates a newpoint by randomly changing some components of a candidatepoint in the current generation. The algorithm continuesiteration (generation) until it reaches the stopping criteria.The population size for each generation is assumed to be 70


with a crossover probability of 80% and mutation probabilityof 20%. The stopping criterion is chosen to be either thetolerance of less than 10−4 for the objective function or themaximum number of generations not to exceed 100.

4.1. Maximizing the Fundamental Frequencies of CorrugatedPlates. In this section, examples are presented to show theefficacy of designing the corrugated plates to maximize theirfundamental frequencies. The maximization is applied ontwo designs: using sinusoidal corrugations and using trape-zoidal corrugations. The optimization problem is defined asfollows:Maximize: 𝜔

1

Subject to: 𝐹lb ≤ 𝐹 ≤ 𝐹ub

𝑁lb ≤ 𝑁 ≤ 𝑁ub

ℎlb ≤ ℎ ≤ ℎub

(𝐹

𝐶)lb≤ (

𝐹

𝐶) ≤ (

𝐹

𝐶)ub

𝜃lb ≤ 𝜃 ≤ 𝜃ub

(In trapezoidal corrugation)

𝐹lb ≤ 𝐹 ≤ (𝐶

4) tan (𝜃)

(In trapezoidal corrugation) ,

(13)

where lb and ub are the lower and upper bounds of designvariables, respectively. Throughout this study, the bounds ofdesign variables are selected to be 𝐹lb = 0, 𝐹ub = 30mm,𝑁lb = 6, 𝑁ub = 30, ℎlb = 0.5mm, ℎub = 5mm, (𝐹/𝐶)lb =

0, (𝐹/𝐶)ub = 0.5, 𝜃lb = 0∘, and 𝜃ub = 90

∘. The optimaldesign variables that maximize the fundamental frequencyof sinusoidal corrugated plate are 𝐹 = 30mm, 𝑁 = 6,and ℎ = 5mm. The fundamental frequency of the optimalsinusoidal corrugated plate is 165.4Hz.

The optimal design variables that maximize the funda-mental frequency of trapezoidal corrugated plate are 𝐹 =

30mm, 𝑁 = 6, ℎ = 5mm, and 𝜃 = 90∘. The fundamental

frequency of the optimal trapezoidal corrugated plate is208.5Hz. It can be concluded that themaximum fundamentalfrequency occurs at the upper bounds of the corrugationheight 𝐹 and trough angle 𝜃 and at the lower bound of thenumber of corrugations 𝑁. These results can be anticipatedfrom the parametric study.

4.2. Maximizing the Fundamental Frequency-to-Mass Ratio ofCorrugated Plates. As shown in the previous optimizationexample, the fundamental frequency increases when theheight of corrugation, the trough angle, and the thicknessincrease. However, increasing the aforementioned designvariables increases the mass of the plate. For example,the masses of the optimal plates in the previous exampleare 49.4 kg and 66.8 kg for the sinusoidal and trapezoidalcorrugated plates, respectively. The fundamental frequency-to-mass ratios in the previous example are 3.35Hz/kg and

A

B

−220

−210

−200

−190

−180

−170

−160

−150

−140

−130

−120

10 20 30 40 50 60 700Mass (kg)

−𝜔

1(H

z)

Figure 13: Pareto front for the fundamental frequency and the massof the trapezoidal corrugated plates.

3.12Hz/kg for the optimal sinusoidal and trapezoidal corru-gated plates, respectively. In this example, the fundamentalfrequency-to-mass ratio of simply supported plate is max-imized by creating sinusoidal or trapezoidal corrugations.The optimization problem is the same as the problem thatis illustrated in the previous example but with maximizing𝜔1/mass instead of maximizing 𝜔

1.

The GA is used to solve the optimization problem. Thedesign parameters of the optimal sinusoidal corrugated plateare 𝐹 = 30mm, 𝑁 = 6, and ℎ = 0.5mm. The maximum𝜔1/mass is 33.2Hz/kg (i.e., 𝜔

1is 164Hz and the mass

is 4.94 kg). The optimal design variables, which maximizethe fundamental frequency-to-mass ratio of the trapezoidalcorrugated plate, are 𝐹 = 30mm, 𝑁 = 6, ℎ = 0.5mm,and 𝜃 = 53

∘. The maximum 𝜔1/mass is 34.5Hz/kg (i.e., 𝜔

1

is 183Hz and the mass is 5.30 kg). Pareto front of the massand the fundamental frequency of the trapezoidal corrugatedplate is shown in Figure 13. In this figure, point A stands forthe maximum fundamental frequency (i.e., 𝜔

1= 208.5Hz

and 𝑚 = 66.8 kg), and point B stands for the maximumfundamental frequency-to-mass ratio (i.e., 𝜔

1= 183Hz and

𝑚 = 5.30 kg).

5. Conclusions

Designing sinusoidal and trapezoidal corrugated plates foroptimal fundamental frequency was investigated. The fun-damental frequencies of simply supported corrugated plateswere determined numerically by using the finite elementmethod and equivalent orthotropic models. The corrugationof the plate increases the area moment of inertia and henceincreases its bending rigidities. In addition to altering thenatural frequencies of plates, corrugations change the platemode shapes.

A comparative study shows that using the equivalentorthotropic models to study the free vibration of corrugatedplates significantly reduces the computational time. Theparametric analysis shows the efficacy of the corrugation


technique on increasing the fundamental frequency of simplysupported plates. Two optimization problems were consid-ered: the maximization of corrugated plate fundamentalfrequency and maximization of the corrugated plate funda-mental frequency-to-mass ratio. A future extension of thisworkwill focus on the experimental verification of the results.Future work will consider the design of corrugated platessubjected to a dynamic load with respect to the minimalsound radiation.

Competing Interests

The author declares that they have no competing interests.

References

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[13] Y. Xia, M. I. Friswell, and E. I. Saavedra Flores, “Equivalentmodels of corrugated panels,” International Journal of Solids andStructures, vol. 49, no. 13, pp. 1453–1462, 2012.

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Research Article Design of Corrugated Plates for Optimal ...downloads.hindawi.com/journals/aav/2016/4290247.pdf · Design of Corrugated Plates for Optimal Fundamental Frequency NabeelAlshabatat