[American Institute of Aeronautics and Astronautics 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 16th AIAA/ASME/AHS Adaptive Structures Conference

#Professor and PhD, Department of Aerospace Engineering, *Graduate Student, Department of Aerospace Engineering, +Assoc. Professor and PhD, Department of Aerospace Engineering, Copyright © 2008 by the Authors

1 American Institute of Aeronautics and Astronautics

“Free Vibrations Analysis of Integrally-Stiffened and/or

Stepped-Thickness Plates or Panels with Two Side

Stiffeners”

Umur YUCEOGLU# Jaber JAVANSHIR* Sinan EYI+

Middle East Technical University

Ankara 06531, TURKEY

In this study, the theoretical analysis and the semi-analytical and numerical method of

solution for the “Free Vibration Analysis of Integrally-Stiffened and/or Stepped-Thickness

Plates or Panels with Two Side Stiffeners” are investigated in some detail. In general, the plate

elements of the system are considered as dissimilar, orthotropic “Mindlin Plates” with unequal

thicknesses. Thus, the transverse shear deformations and the transverse and rotary moments of

inertia of dissimilar plate elements are taken into account in the formulation. The dynamic

equations and the stress resultant-displacement expressions of the “Mindlin Plate Theory” for

individual plate elements, after some manipulations and combinations, are finally reduced to a

set of “Governing System of the First Order Ordinary Differential Equations” in “state vector”

forms. Aforementioned “Governing System of Equations” is suitable for making use of the

present method of solution which is the “Modified Transfer Matrix Method (MTMM) (With

Interpolation Polynomial)”. The mode shapes and the corresponding natural frequencies of the

stiffened plate or panel system under investigation are presented for various boundary

conditions. The effect of the fully “isotropic” and/or composite “orthotropic” material

characteristics of the individual plate elements on the mode shapes and the natural frequencies

are presented. It was found that the “isotropic” and “orthotropic” material properties

significantly influence the mode shapes as well as their natural frequencies. Additionally, some

parametric studies (such as the “Thickness ratio h(2)(=h(3))/h(1)”, the “Stiffener Length (or

width) Ratio ℓI/L”, etc.) on the natural frequencies are considered and the results are

graphically presented.

I. Introductory Remarks

The extensive practical applications of the “Integrally-Stiffened and/or Stepped-Thickness Plates or

Panels” can be found in the analysis and design of air and space vehicle (and also in high-speed hydrodynamic

vehicle) structures and related structural systems.

In general, the aforementioned plates or panel systems can provide significant advantages such as

economy in material usage, reduced dead-weight, optimized dynamic response, high resistance to fatigue and

fracture, relatively lower stress concentrations, and no mechanical type of connections between the plate

elements [1, 2, 3].

In these plate or panel systems, the main characteristic is that each system is manufactured or machined

out of one solid piece of “Advanced Metal Alloy” raw stock. In some applications, they may also be constructed

as one piece multi-step plates of “Advanced Composites” [2].

49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference <br> 16t7 - 10 April 2008, Schaumburg, IL

AIAA 2008-2011

Copyright © 2008 by © 2008 by the Authors . Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

American Institute of Aeronautics and Astronautics

2

The “Integrally-Stiffened and/or Stepped-Thickness Plates” may be categorized in several ways. For

instance, taking the step variation in one direction only in “thicknesses” of their plate elements, one may classify

or group aforementioned systems in four main “Groups” or “Types” as shown in Figure 1.

In spite of their practical importance and their certain advantages in aero- and hydrodynamic vehicle

structures, the research studies on their dynamic response in the open world-wide scientific and engineering

literature are relatively few and far between. In this connection, one may mention some investigations on “Type

1, 2 and 4” of the “Stepped-Thickness Plates” given in [4-11] and also more recently in Yuceoglu et al [12-14].

In the case of the “Stepped-Thickness or Stiffened Plates with Bonded Joints”, there are some recent

studies by Yuceoglu et al [15-22]. In the most of these studies, excluding Yuceoglu et al [12-22], the “Classical

Thin Plate Theory (CLPT)” is employed. It is, of course, a well-known fact that the results obtained by using

“CLPT” and by the “First Order Shear Deformation Plate Theories (FSDPT)” [23,24] are going to be

significantly different, especially in the case of the “Stepped-Thickness Plates” (with abrupt changes in

“thicknesses”).

Therefore, the main objectives of the present study are first to give a general theoretical analysis and,

subsequently, a general method of solution of the “Free Vibrations Analysis of Integrally-Stiffened and/or

Stepped-Thickness Plates or Panels with Two Side Stiffeners (or with Two Steps in Thicknesses)”. In other

words, the main concern here is the “Type 2.b Plates or Panel System” of Figure 1 and the configuration and the

geometry in Figure 2.The present theoretical formulation and proposed method of solution can easily handle

almost all the “Stepped-Thickness Plate or Panel Systems of Types 2.a, b, c” (see also Figure 1).

II. General Theoretical Formulation and Governing Equations

In the theoretical analysis, the individual plate elements of the Stepped-Thickness Plates or Panels” is to

be taken into account as the dissimilar, orthotropic plate or panels according to the “Mindlin Plate Theory”

which is one of the “FSDPT” [23, 24].

The general configuration, the geometric characteristics and the orthotropic materials directions of the

“Integrally- Stiffened and or stepped-Thickness Plates or Panels with Two Side Stiffeners” are given in detail in

Figure 2.a, and the longitudinal cross-section in the y-direction of the entire plate system is shown in Figure 2.b.

The coordinate systems are also included in Figures 2.a and 2.b. In the present problem the plate elements of the

aforementioned plate or panel system are assumed to be dissimilar orthotropic “Mindlin Plates” [23] with

unequal thicknesses. The transverse shear deformations and the transverse and the rotary moments of inertia of

the individual plate elements are included in the analysis. In the present study, the geometrically and materially

“Symmetric Two Side Stiffeners Case” is considered (i.e. they are “symmetric” with respect to the transverse

plane going through the mid-center of the entire plate system). In addition, the entire plate system is assumed to

be simply supported at (x=0, a) as shown in Figures 2.a and 2.b.

It is of interest to note here that the geometrically and material-wise “Non-Symmetric Two Side

Stiffeners Case” can also be handled in a similar manner without any difficulties in the formulation and in the

solution procedure employed here. In fact, the present authors also investigated this problem in detail and the

results will be reported later.

Referring to previous work by Yuceoglu et al [12-14] and in order to facilitate the direct application of

the present solution method, the “Domain Decomposition Technique” is utilized first. Thus, the entire stepped-

thickness plate system is divided into three “Parts” or “Regions” as shown in Figures 2.a and 2.b. These are

namely, Part І, Part ІІ and Part ІІІ regions in the y-direction. The next step is the “Non-Dimensionlization

Procedure” to be applied to the entire set of dynamic equations of orthotropic “Mindlin Plate Theory” [23] as

given in Appendix A. For this purpose “

,, , a” are chosen as main (or reference) parameters.


3

And also Il , IIl , IIIl , are selected as the “Length Reference Parameters” in the y-direction in each “Part” or

“Region” respectively. Then, all other quantities are non-dimensionalized with respect to these parameters. Then,

The dimensionless coordinates for part І, part ІІ and part ІІІ regions, respectively

,a/x=η IIIIII y l////=ξ (Part I)

,a/x=η IIIIII y l////=ξ (Part II)

,a/x=η IIIIIIIII y l////=ξ (Part III)

The dimensionless parameters related to orthotropic elastic constant of plate elements,

The dimensionless parameters related to the densities and the geometry of the plates.

where “a” is the width of the entire plate system

The dimensionless frequency parameter mnω of the entire “Stiffened Plate or Panel System”,

Taking into account the simple support conditions at (x=0, a), the generalized displacements and

generalized stress resultants can be expressed in Fourier Series in the x-directions for each element of the

“Stepped-Thickness Plate or Panel” system as shown in Appendix B. After this, and following some lengthy

algebraic combinations and manipulations, the sets of dynamic “Mindlin Plate” equations (as given in Appendix

A) can be reduced to a set of equations for each “Part” and “Region”. These equations can be recast as the

“Governing System of the First Order Ordinary Differential Equations” in the “compact matrix” or rather “state

vector” forms as follows:

(1)

(2) )2,1k,iand3,2,1j(,B

BB

)2,1k,iand3,2,1j(,B

BB

11)1(

)j()j(

11)1(

)j(ik)j(

ik

===

===

ll

ll

1hhh,h

hh,hhh

aL,aL,aL

1,,

1

11

1

33

1

22

1

11

1

13

1

22

====

===

=ρ

ρ=ρ

ρρ

=ρρ

ρ=ρ

ΙΙΙΙΙΙ

ΙΙΙΙ

ΙΙ

lll

(3)

)1,2,3...nm,(

Bh/a

mn

(1)11

21

2mn

41mn

=ω=Ω

ωρ=ω

(4)

(5)

region)PlateCentral(or IPartIn

[ ] ( )

....

,,,,))))(((())))((((

10,Iξat"ConditionsContinuity"thewith

1ξ0d

dI

11

=

≤≤=ξ

mnmnI

YCY


4

In the above equations (5-7), the dimensionless “fundamental dependent variables” or the unknown

“state vectors” of the problem under consideration are,

where (j=1, 2, 3) correspond to Part I, Part II and Part III regions, respectively in Figure 2.b. The “Coefficient

Matrices” [C ], [D ] and [E ] are of dimensions (6x6) and they include the dimensionless geometric and

material constants and also the dimensionless frequency parameter mnω of the entire plate or panel system.

It is important to note here that the above “Governing system of Equations” together with the “far left

end and the “far right end” (see also Figure 2.b) boundary conditions, establish the so-called “Two-Point

Boundary Value Problem of Mechanics”. This way, the original “Initial Value and Boundary Value Problem”

(i.e. the “Free Vibrations Problem” of the “Stepped-Thickness Plate or Panel System” under investigation) is

now, finally, reduced to a “Governing System” of ( 5-8 ) which is very suitable for the method of solution

proposed here.

III. Solution Method and Procedure

The present method of solution is a semi-analytical and numerical procedure called the “Modified

Transfer Matrix Method (MTMM) (with interpolation polynomials)” [12, 14]. This method has been essentially

developed by Yuceoglu et al [15-22 and 23, 26] to solve the vibration problems of “Integral-Stiffeners” and

“Bonded Joints” in plates and shallow shells. An earlier version of the method is given in Yuceoglu et al [25].

Recently, for the higher modes and natural frequencies (higher than six and up to fifteen or more), a very

accurate version of the present solution technique which is called the “Modified Transfer Matrix Method

(MTMM) (with Chebyshev Polynomials)” is also developed by Yuceoglu and Özerciyes [26].

Here in this study, the “Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials)”

[12-14] is employed for the “Stepped-Thickness Plate or Panel System” under consideration. This technique is a

combination of the “Classical Levy’s Method”, the “Transfer Matrix Method” and the “Integrating Matrix

Method (with Integration Polynomials)”. Without going into detail and referring to Yuceoglu et al [12-14] and

also to Yuceoglu et al [15-22], the “Governing System” given in equations (5-7), after some algebraic operations

and manipulations plus making use of the “Continuity Conditions” between plate elements can be finally

combined together in a final compact matrix form as,

(8)

III)II,I,k;1,2,3(j

ξη, k(j)

==

ψψ=Tj

mnyj

mnyj

mnyxj

mnj

mnyj

mnxmn QMMW )()()()()()( ,,;,,)(Y

(6)

region)Stiffener PlateLeft(or IIPartIn

[ ] ( )

1.IIξat "Conditions Continuity"

and0IIξatConditionsBoundaryArbitrary"with

1ξ0dξ

dII

22

II

=

=

≤≤= ,YDY )()(mnmn

regionStiffener PlateRight(or IIIPartIn ))))

(7)

[ ] ( )

1.IIIξat "Conditions Continuity"

and0IIIξatConditionsBoundaryArbitrary"with

1ξ0dξ

dIII

22

II

=

=

≤≤= ,,,,))))(((())))((((mnmn YEY


5

[ ] [ ] [ ]( )

[ ]

=

=

=ξ=ξ

=ξ=ξ

ΙΙΙΙΙ

ΙΙΙΙΙ

(2)(3)

(2)(3)

YQY

YUVWY

0011

0011

system, panelor plate entite For the

~~~~

~~~~~~~~~~~~

where [ ]01Q~ is now the “final form” of the discretized “Overall Global Modified Transfer Matrix” which

transfers the above “state variables” from the “initial end point (or far left end support ξII=0)” to the “final end point (or far right end support ξIII=1)” of the entire plate or panel system

The above “Overall Global Modified Transfer Matrix [ ] 01

~Q ” can further be reduced to (3x3) matrix

by inserting the “Boundary Conditions” at ξII=0 and ξIII=1. These operations simply yield,

The above equation (10) results in a complicated polynomial whose roots are the natural frequencies of the plate

or panel system, where the dimensionless natural frequencies are computed by searching the roots numerically

on the basis of the given “m” and the assigned “n” values. Then, they are sequenced according to their

magnitudes as shown above in (4) and (10).

IV. Some Numerical Results and Brief Conclusions

The present solution method is applied to a typical “Free Vibration Problem of the Integrally-Stiffened

Plate or Panel System” shown in Figure 2. The geometric dimensions and the material characteristics

(“Isotropic” Case and “Orthotropic” Case) are presented in Table 1. The only boundary conditions given on all

Figures are the support conditions in the y-direction. They are read from the left support to the right support. The

letter symbols indicate C=Clamped, S=Simple Support and F=Free support. In Figure 3, in the “Isotropic Al-Alloy Case”, the mode shapes and the corresponding natural

frequencies are plotted for the clamped support (CC) boundary conditions. The two side stiffeners are symmetric

with respect to a transverse plane going through mid-center of the plate system. In this particular case the

geometry, the material and the support conditions are “symmetric”. Therefore, in Figure 3, the “symmetric” and

the “skew-symmetric” modes follow each other as expected. In Figure 4, again in the “Isotropic Al-Alloy Case”, mode shapes and the corresponding natural

frequencies are presented for the clamped-simple support (CS) boundary conditions. It is obvious from Figure 4,

that the trend in mode shapes is not similar to those of Figure 3. This is because the boundary conditions are “not

symmetric”, although the geometry and material properties are “symmetric”. In Figure 5, for the “Isotropic Al-Alloy Case”, mode shapes and the corresponding natural frequencies

are shown for the simple-free (SF) boundary conditions. It can be seen that, since the support conditions are “not

symmetric” in this case, the mode shapes do not exhibit the “symmetric” and “skew-symmetric” properties as

observed in Figure 3.

In order to give an idea about the effect of the “orthotropic” material properties on the mode shapes and

the natural frequencies, Figure 6, 7, 8 are presented, in the “orthotropic” case, for the same (CC),(CS),(SF)

(9)

(10)

[ ]

[ ] 0Matrix Ceff. oftDeterminan

0

mn

mn

0

00

=ω=

⇒=ω

))))((((

))))((((

C

YC


6

boundary conditions, respectively. Comparing Figures 3 and 6, Figures 4 and 7 and Figures 5 and 8, one

clearly can observe that the mode shapes and the natural frequencies, after the second mode, are quite different

in the “isotropic” and the corresponding “orthotropic” cases. Also, due to the influence of the “orthotropic”

material properties, the natural frequencies in Figures 6, 7 and 8 are relatively higher than those of Figures

3, 4 and 5, respectively. In terms of some important parametric studies, in the “Isotropic Al-Alloy Case”, Figures 9 and 10 are

presented. In Figure 9, the “Dimensionless Natural Frequencies Ω” versus the “Thickness Ratio h2(=h3)/h1” are

plotted. As shown in Figure 9, the natural frequencies increase linearly as the h2(=h3)/h1 increases. In Figure 10, the “Dimensionless Natural Frequencies Ω” versus the “Length (or Width) Ratio ⁄ ”

are graphically presented. As ⁄ ratio increases, while =0.5 m is kept constant, the natural frequencies

decrease in a wavy pattern (i.e. non-linear fashion) as expected. As a further general conclusion, it can be stated that, the present theoretical formulation and the method

of solution are somewhat general and can easily be applied to other “Integrally-Stiffened and/or Stepped-

Thickness Plate or Panel Systems” of “Type 1,2,3” (see Figure 1) without any serious numerical and other

difficulties.

References

[1] Hoskins, B.C. and Baker, A. A., 1986 “Composite Materials for Aircraft Structures”, AIAA Educational Series, New

York.

[2] Marshall, I.H., and Demuts, E., (Editors), 1988, “Supportability of Composite Airframes and Aero-Structures”, Elsevier

Applied Science Publishers, New York.

[3] Niu C. Y. Michael, “Airframe Structural Design, Conmilit Pres Ltd.

[4] Chopra, 1974 “Vibration of Stepped-Thickness Plates” International Journal of Mechanical Sciences Vol.16, pp.337-344.

[5] Warburton G.B., 1975 Comment on “Vibration of Stepped-Thickness Plates” by I.Chopra, International Journal of

Mechanical Sciences Vol.16, pp.239.

[6] Yuan, J., and Dickson, S.M., 1992, The Flexural Vibration of Rectangular Plate Systems Approached by Artificial

Springs in The Rayleigh-Ritz Method”, Journal of Sound and Vibration, 159(1), pp 39-55.

[7] Bambil, D.V. et.al. 1991, “Fundamental Frequency of Transverse Vibration of Symmetrically Stepped Simply Supported

Rectangular Plates”, Jour. Of Sound and Vibration, Vol. 150, pp. 167-169.

[8] Takabatake, H., Imaizumi, T., Okatomi, K., 1995, “Simplified Analysis of Rectangular Plates with Stepped Thicknesses”,

ASCE Jour. of Struct. Engineering, Vol.117, pp. 1759-1779

[9] Shen, H.S., Chen,Y.,Young, J., 2003, “Bending and Vibrations Characteristics of a Strengthened Plate under Various

Boundary Conditions", Engineering Structures, Vol.25, pp.1157-1168.

[10] Li, Q.S., 2000, “Exact Solutions for Free Vibrations of Multi-Step Orthotropic Shear Plates.” Jour. of Struct.

Engineering and Mechanics, Vol.9, pp.269-288.

[11] Cheung, Y.K., Au, F.T.K., Zheng, D.V., 2000, “Finite Strip Method for the Free Vibrations and Buckling Analysis of

Plates with Abrupt Changes in Thickness”, Thin Walled Structures, Vol. 36, pp. 89-110.

[12] Yuceoglu, U., Güvendik, Ö. and Özerciyes, V., 2006, “On General Formulation of Free Vibrations of Stepped-

Thickness and Integrally-Stiffened Plates or Panels”, 2006 VI.th Kayseri Symposium in Aeronautics and Astronautics, May

12-14, 2006, Nevşehir, Turkey.

[13] Yuceoglu, U., Gemalmayan, N., Sunar, O., 2007, “Free Flexural Vibrations of Integrally-Stiffened and/or Stepped-

Thickness Rectangular Plates or Panels with a Central Plate Stiffener” “48th AIAA/ASME/ASCE/ASC/AHS, Structures,

Struc. Dynamics and Materials (SDM) Conference and Exhibit”, April 21-23, 2007, Waikiki, Hawaii, (AIAA Paper No:

AIAA-2007-2113).

[14] Yuceoglu, U., Gemalmayan, N., Sunar, O., 2007, “Free Bending Vibrations of Integrally-Stiffened and/or Stepped-

Thickness Rectangular Plates or Panels with a Non-Central Plate Stiffener”, “2007 Inter. Mech. Engineering Congress and

Exposition (IMECE-2007)”, November 10-16, 2007, Seattle, Washington, (ASME Paper No: IMECE 2007-41066)

[15] Yuceoglu, U. and Özerciyes, V., 1997, “Natural Frequencies and Mode Shapes of Composite Plates or Panels with a

Bonded Central Stiffening Plate Strip”, in the ASME-Noise Control and Acoustics Div. “Symposium on Vibroacoustic


7

Methods in Processing and Characterization of Advanced Materials and Structures, NCAD-Vol. 24”, pp.185-196,(an ASME

publication).

[16] Yuceoglu, U. and Özerciyes, V., 2000, “Natural Frequencies and Modes in Free Transverse Vibrations of Bonded

Stepped-Thickness and/or Stiffened Plates and Panels”, Proceed. Of the 41st AIAA/ ASME/ ASCE/ AHS/ ASC/ ”SDM

Conference”,AIAA. Paper No. AIAA-2000-1348, pp.

[17] Yuceoglu, U. and Özerciyes, V., 1996, “Free Bending Vibrations of Partially-Stiffened, Stepped-Thickness Bonded

Composite Plates”, in the ASME- Noise Control and Acoustics Div. “Advanced Materials for Vibro-Acoustic Applications

NCAD- Vol. 23”, pp.191-202 ,(an ASME publication) .

[18] Yuceoglu, U. and Özerciyes, V., 1998, “Free Bending Vibrations of Composite Base Plates or Panels Reinforced with a

Bonded Non-Central Stiffening Plate Strip”, in the ASME- Noise Control and Acoustics Div. “Vibroacoustic

Characterization of Advanced Materials and Structures” NCAD-Vol. 25”, pp. 233-243, (an ASME publication) .

[19] Yuceoglu, U. and Özerciyes, V., 1999,” Sudden Drop Phenomena in Natural Frequencies of Partially Stiffened,

Stepped-Thickness. Composite Plates or Panels”, Proceed. of the 40th AIAA/ ASME/ ASCE/ AHS/ ASC/ ”SDM

Conference”, AIAA. Paper No. AIAA-1999-1483, pp.2336-2347.

[20] Yuceoglu, U. and Özerciyes, V., 2000 “Sudden Drop Phenomena in Natural Frequencies of Composite Plates or Panels

bonded with a Central Stiffening Plate Strip”, Inter. Journal of Computers and Structures, Vol. 76 (1-3) (Special Issue),

pp.247-262.

[21] Yuceoglu, U. and Özerciyes, V., 2003,”Orthotropic Composite Base Plates or Panels with a Bonded Non-Central (or

eccentric) Stiffening Plate Strip”, ASME Journal of Vibration and Acoustics, Vol.125, pp. 228-243.

[22] Yuceoglu, U. ,Özerciyes, V.and Çil, K., 2004, “ Free Flexural Vibrations of Bonded Centrally Doubly Stiffened

Composite Base Plates or Panels” 2004 ASME Inter. Mech. Engineering Congress and Exposition (IMECE-2004).

[23] Mindlin, R.D., 1951, “Infuence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates”, ASME

Journal of Applied Mechanics, Vol. 18, pp.31-38

[24] Reissner, E., 1945, “The Effect of Transverse Shear Deformation on the Bending of Elastic Plates” ASME Journal of

Applied Mechanics, Vol. 12(2), pp A.69-A.77.

[25] Yuceoglu, U., Toghi, F., and Tekinalp, O., 1996, “Free Bending Vibrations of Adhesively-Bonded, Orthotropic Plates

with a Single Lap Joints”, ASME Journal of Vibration and Acoustics, vol.118, pp. 122-134.

[26]Yuceoglu, U., and Özerciyes, V., 2005 “Free Vibrations of Bonded Single Lap Joints in Composite Shallow Cylindrical

Shell Panels,” AIAA Journal, Vol.43 No.12, pp.2537-2548.


8

Table1. MATERIAL CONSTANTS AND DIMENSIONS OF “INTEGRALLY -STIFFENED AND/OR

STEPPED-THICKNESS PLATE OR PANEL SYSTEM”

“Isotropic” “Al-Alloy” System “Orthotropic” “Composite” System

Al-Alloy (Plate 1)

(j=1)

Al-Alloy (Plates 2 and 3)

(j=2,3)

Graphite-Epoxy (Plate 1)

(j=1)

Kevlar-Epoxy (Plates 2 and 3)

(j=2,3)

)j(xE =72.69 GPa

)j(yE =72.69 GPa

)j(xyG =25.78 GPa

)j(xzG =25.78 GPa

)j(yzG =25.78 GPa

)j(xyυ =0.313

)j(yxυ =0.313

)j(ρ =2.796 gr/cm3

1h =0,02 m

a =0.50 m ℓI 0.4

)j(xE =72.69 GPa

)j(yE =72.69 GPa

)j(xyG =25.78 GPa

)j(xzG =25.78 GPa

)j(yzG =25.78 GPa

)j(xyυ =0.313

)j(yxυ =0.313

)j(ρ =2.796 gr/cm3

2h = 3h =0,04 m

a =0.50 m ℓII ℓIII 0.3

)j(xE =11.71 GPa

)j(yE =137.8 GPa

)j(xyG =5.51 GPa

)j(xzG =2.5 GPa

)j(yzG =3.0 GPa

)j(xyυ =0.0213

)j(yxυ =0.25

)j(ρ =1.6 gr/cm3

1h =0,02 m

a =0.50 m ℓI 0.4

)j(xE =5.5 GPa

)j(yE =76.0 GPa

)j(xyG =2.10 GPa

)j(xzG =1.5 GPa

)j(yzG =2.0 GPa

)j(xyυ =0.024

)j(yxυ =0.34

)j(ρ =1.3 gr/cm3

2h = 3h =0,04 m

a =0.50 m ℓII ℓIII 0.3

L=Total Length of entire “Plate System”=1.00 m, =Mid-Center of Plate System=L/2

a=Width of entire “Plate System”=0.50 m


9

APPENDIX A

“Mindlin Plate Theory” as Applied to “Orthotropic Plates”

• Equations of Motion of “Mindlin Plates”

( )

( )

( )2

2

zz

yx

2

y23

zyzyy

yyx

2x

23

zxzxx

yxx

t

whqq

y

Q

x

Q

t12

hqq

2

hQ

y

M

x

M

t12

hqq

2

hQ

y

M

x

M

∂

∂ρ=−+

∂

∂+

∂

∂

∂

ψ∂ρ=++−

∂

∂+

∂

∂

∂

ψ∂ρ=++−

∂

∂+

∂

∂

−+

−+

−+

(A.1)

Where q’s are upper and lower surface Loads or surface stresses, respectively.

• Stress Resultant-Displacement Relations (in terms of Elastic Constants):

∂

ψ∂+

∂

ψ∂=

∂

∂+ψκ=

∂

ψ∂+

∂

ψ∂=

∂

∂+ψκ=

∂

ψ∂+

∂

ψ∂=

xyB

12

hM

y

wBhQ,

yB

xB

12

hM

x

wBhQ,

yB

xB

12

hM

yx66

3

yx

y442yy

y

22x

21

3

y

x552xx

y

12x

11

3

x

(A.2)

Where 2κ terms are the “Shear Correction Factors” of the “Mindlin Plate Theory”, and B’s are material

coefficients in the orthotropic stress-strain relations (Hooke’s Law) such that,

xy6622xy11yx2112

xz55

xyyx

y

22

yz44

yxxy

x11

GB,BBBB

GB,1

EB

GB,1

EB

=ν=ν==

=νν−

=

=νν−

=

(A.3)


10

• Stress Resultant-Displacement Relations (in terms of Stiffnesses):

1221

yx66yx

y44y

y

22x

21y

x55x

y

12x

11x

DD,xy

DM

y

wAQ,

yD

xDM

x

wAQ,

yD

xDM

=

∂

ψ∂+

∂

ψ∂=

∂

∂+ψ=

∂

ψ∂+

∂

ψ∂=

∂

∂+ψ=

∂

ψ∂+

∂

ψ∂=

(A.4)

Where the “Bending Stiffness D’s” and the “Shear Stiffness A’s” are,

552x5544

2y44

663

66ik

3

ik

BhA,BhA

)2,1k,i(12

BhD,

12

BhD

κ=κ=

=

==

(A.5)

• Mindlin Boundary Conditions

0w:)Clamped(C

0Mw:)portedsupsimply(S

0QMM:)free(F

tn

nt

nnnt

=ψ=ψ=

==ψ=

===

(A.6)

Where n and t are normal and tangential directions


11

APPENDIX B

“Classical Levy’s Solutions”

(Following the “Domain Decompositions Approach”, the “Classical Levy’s Solutions” in the x-direction,

corresponding, to each “Part” or “Region”):

• Displacement and Angles of Rotation,

)10(),10(),10(

e)msin()()t,,(

e)mcos()()t,,(

e)msin()(Wh)t,,(w

1m 1n

tik

)j(mnyk

)j(y

1m 1n

tik

)j(mnxk

)j(x

1m 1n

tik

)j(mn1k

)j(

mn

mn

mn

<ξ<<ξ<<ξ<

πηξψ=ξηψ

πηξψ=ξηψ

πηξ=ξη

ΙΙΙΙΙΙ

∞

=

∞

=

ω

∞

=

∞

=

ω

∞

=

∞

=

ω

∑ ∑

∑ ∑

∑ ∑

(B.1)

Where,

IIIPart for IIIk II,Part for IIk I,Part for Ik

IIIPart for 3j II,Part for 2j I,Part for 1j

===

===

• Stress Resultants,

)10(),10(),10(

e)mcos()(Qa

Bh)t,,(Q

e)msin()(Qa

Bh)t,,(Q

e)msin()(Ma

Bh)t,,(M

e)mcos()(Ma

Bh)t,,(M

e)msin()(Ma

Bh)t,,(M

1m 1n

tik

)j(mnx3

)1(11

41

k)j(

x

1m 1n

tik

)j(mny3

)1(11

41

k)j(

y

1m 1n

tik

)j(mny3

)1(11

51

k)j(

y

1m 1n

tik

)j(mnyx3

)1(11

51

k)j(

yx

1m 1n

tik

)j(mnx3

)1(11

51

k)j(

x

mn

mn

mn

mn

mn

<ξ<<ξ<<ξ<

πηξ=ξη

πηξ=ξη

πηξ=ξη

πηξ=ξη

πηξ=ξη

ΙΙΙΙΙΙ

∞

=

∞

=

ω

∞

=

∞

=

ω

∞

=

∞

=

ω

∞

=

∞

=

ω

∞

=

∞

=

ω

∑ ∑

∑ ∑

∑ ∑

∑ ∑

∑ ∑

(B.2)

Where,

IIIPart for IIIk II,Part for IIk I,Part for Ik

IIIPart for 3j II,Part for 2j I,Part for 1j

===

===


12

a) a) a) a)

b) b) b) etc

c) c)

d)

Figure 1. Various “Types” or “Classes” of “Integrally-Stiffened and/or Stepped-Thickness Plates or Panels”

INTEGRALLY-STIFFENED and/or

STEPPED-THICKNESS PLATE or PANEL SYSTEMS

(Rectangular Plates or Panels)

(Step(s) in one direction only)

(Type. 1)

One Step (or Single

Step)

(Type. 2)

Two Steps (or Double

Step)

(Type. 4)

More than Three

Steps (or Multi-Step)

(Type. 3)

Three Steps (or Triple

Step)


13

(a) General Configuration, Geometry, Coordinate Systems and Material Directions

(b) Longitudinal Cross-Section with Parts I, II, III and Coordinate Systems

Figure 2. “Integrally-Stiffened and/or Stepped-Thickness Plate or Panel System

with Two Side Stiffeners”


14

First Mode with 1= 11=42.119 Third Mode with 3= 21=249.660

Second Mode with 2= 12=109.545 Fifth Mode with 5= 22=593.318

Figure 3. Mode shapes and Dimensionless Natural Frequencies of “Integrally-Stiffened and/or Stepped-Thickness Plate or Panel System with Two Side Stiffeners”

(“Isotropic” Case) (Plate 1=Al-Alloy, Plate 2=Al-Alloy, Plate 3 =Al-Alloy)

(ℓI=0.40m, ℓII=0.3m, ℓIII=0.3m, b~

=0.50m, h1=0.02m, h2=h3=0.04, a=0.50m, L=1.00m)

(Boundary Conditions in y-direction CC)


15





(ℓI=0.40m, ℓII=0.3m, ℓIII=0.3m, b~

=0.50m, h1=0.02m, h2=h3=0.04, a=0.50m, L=1.00m)

(Boundary Conditions in y-direction CS)


16





(ℓI=0.40m, ℓII=0.3m, ℓIII=0.3m, b~

=0.50m, h1=0.02m, h2=h3=0.04, a=0.50m, L=1.00m)

(Boundary Conditions in y-direction SF)


17




(“Orthotropic” Case) (Plate 1=Graphite-Epoxy, Plate 2=Kevlar-Epoxy, Plate 3 = Kevlar-Epoxy)

(ℓI=0.40m, ℓII=0.3m, ℓIII=0.3m, b~

=0.50m, h1=0.02, h2=h3=0.04, a=0.50m, L=1.00m)

(Boundary Conditions in y-direction CC)


18





(ℓI=0.40m, ℓII=0.3m, ℓIII=0.3m, b~

=0.50m, h1=0.02, h2=h3=0.04, a=0.50m, L=1.00m)

(Boundary Conditions in y-direction CS)


19





(ℓI=0.40m, ℓII=0.3m, ℓIII=0.3m, b~

=0.50m, h1=0.02, h2=h3=0.04, a=0.50m, L=1.00m)

(Boundary Conditions in y-direction SF)


20

Figure 9. “Dimensionless Nat. Freq.’s Ω” versus “Thickness Ratio h2(=h3)/(h1)” in “Integrally-Stiffened and/or Stepped-Thickness Plate or Panel System with Two Side Stiffeners”

(“Isotropic” Case) (Plate1=Al-Alloy, Plate 2= Al-Alloy, Plate 3= Al-Alloy)

((ℓI+ℓII+ℓIII) =1.00m, h1=0.02m, (h2, h3) =varies, b~

=0.50m, a=0.50m, ℓI=0.4, ℓII =ℓIII=0.30m, L=1.00m)

(Boundary Conditions in y-direction (CC))

Figure 10. “Dimensionless Nat. Freq.’s Ω ” versus “Length (or Width) ⁄ ” in “Integrally-Stiffened

and/or Stepped-Thickness Plate or Panel System with Two Side Stiffeners”

(“Isotropic” Case) (Plate1=Al-Alloy, Plate 2= Al-Alloy, Plate 3= Al-Alloy)

((ℓI+ℓII+ℓIII) =1.00m, (ℓI, ℓII, ℓIII) =varies, b~

=0.50m, a=0.50m, h1=0.02m, h2=h3=0.04m, L=1.00m) (Boundary Conditions in y-direction (CC))

Documents

[American Institute of Aeronautics and Astronautics 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 16th AIAA/ASME/AHS Adaptive Structures Conference