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#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 hydro- dynamic 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.
American Institute of Aeronautics and Astronautics
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
American Institute of Aeronautics and Astronautics
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
American Institute of Aeronautics and Astronautics
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
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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
American Institute of Aeronautics and Astronautics
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.
American Institute of Aeronautics and Astronautics
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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
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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)
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• 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
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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
===
===
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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)
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(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”
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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)
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First Mode with 1= 11=39.637 Third Mode with 3= 13=222.954
Second Mode with 2= 12=96.358 Fifth Mode with 5= 14=532.529
Figure 4. 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 CS)
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First Mode with 1= 11=35.692 Third Mode with 3= 13=116.352
Second Mode with 2= 12=64.212 Fifth Mode with 5= 21=249.347
Figure 5. 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 SF)
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First Mode with 1= 11=94.057 Third Mode with 3= 21=357.744
Second Mode with 2= 12=345.539 Fifth Mode with 5= 13=1013.243
Figure 6. Mode shapes and Dimensionless Natural Frequencies of “Integrally-Stiffened and/or Stepped-Thickness Plate or Panel System with Two Side Stiffeners”
(“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)
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First Mode with 1= 11=64.248 Third Mode with 3= 21=350.789
Second Mode with 2= 12=276.043 Fifth Mode with 5= 13=899.121
Figure 7. Mode shapes and Dimensionless Natural Frequencies of “Integrally-Stiffened and/or Stepped-Thickness Plate or Panel System with Two Side Stiffeners”
(“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 CS)
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First Mode with 1= 11=44.851 Third Mode with 3= 13=350.789
Second Mode with 2= 12=86.256 Fifth Mode with 5= 22=812.543
Figure 8. Mode shapes and Dimensionless Natural Frequencies of “Integrally-Stiffened and/or Stepped-Thickness Plate or Panel System with Two Side Stiffeners”
(“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 SF)
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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))
Recommended