free flexural vibration response of integrally- stiffened and ......vibration response of...

Free Flexural Vibration Response of Integrally-Stiffened and/or Stepped-Thickness CompositePlates or PanelsJaber Javanshir, Touraj Farsadi and Umur YuceogluDepartment of Aerospace Engineering, Middle East Technical University, Ankara, 06531, Turkey

(Received 29 September 2012; accepted 21 May 2013)

This study is mainly concerned with a general approach to the theoretical analysis and the solution of the freevibration response of integrally-stiffened and/or stepped-thickness plates or panels with one or more integral platestiffeners. In general, the Stiffened System is considered to be composed of dissimilar Orthotropic Mindlin Plateswith unequal thicknesses. The dynamic governing equations of the individual plate elements of the system andthe stress resultant-displacement expressions are combined and algebraically manipulated. These operations leadto the new Governing System of the First Order Ordinary Differential Equations in state vector forms. The newgoverning system of equations facilitates the direct application of the present method of solution, namely, theModified Transfer Matrix Method (MTMM) (with Interpolation Polynomials). As shown in the present study, theMTMM is sufficiently general to handle the free vibration response of the stiffened system (with, at least, one or upto three or four Integral Plate Stiffeners). The present analysis and the method of solution are applied to the typicalstiffened plate or panel system with two integral plate stiffeners. The mode shapes with their natural frequenciesare presented for orthotropic composite cases and for several sets of support conditions. As an additional example,the case of the stiffened plate or panel system with three integral plate stiffeners is also considered and is shownin terms of the mode shapes and their natural frequencies for several sets of the boundary conditions. Also, someparametric studies of the natural frequencies versus the aspect ratio, stiffener thickness ratio, stiffener length (orwidth) ratio and the bending stiffness ratio are investigated and are graphically presented.

1. INTRODUCTORY REMARKS AND BRIEFREVIEW

The so-called integrally-stiffened and/or stepped-thicknessplate or panel systems of various configurations are primarilyused in air and space flight vehicle structures and substructures.They may also be utilized in high speed hydrodynamic vehi-cle structural systems.1–3 Their applications in engineering aredue to their advantageous properties of light-dead-weight, free-dom from mechanical (riveted, bolted, or welded) connections,and of their favourable stiffness characteristics in appropriateplaces. A typical and well-known example is their utilizationas aircraft wing cover panels with multi-stiffeners.3

These integrally-stiffened plate or panel systems are man-ufactured by means of the CAD-CAM process as one-pieceplate systems out of a solid (or raw stock) of advanced Metal-Alloy plates. In some applications, they may also be manu-factured as one-piece advanced composite plate systems withsome stepped thicknesses.2, 3

The integrally-stiffened plate or panel systems may gener-ally be categorized or grouped in terms of four main groupsas shown in Fig. 1. For the purposes of the present study, theGroup I systems may further be organized in terms of the num-ber of steps in their configurations, as shown in Fig. 2. In thepresent study, the integrally-stiffened plate or panel systems(one-step, two-step, three-step, four-step, etc., and all in onedirection) are to be analysed by means of a general approachto their free dynamic response.

Some significant research studies on the aforementionedintegrally-stiffened and/or stepped-thickness plates are avail-able in world-wide scientific and engineering literature.4–19 In

this brief literature survey, one may include the analytic so-lutions.4, 5, 10, 17, 19 The Raleigh-Ritz Method,7, 9, 11, 12 the FiniteElement Method (FEM),18 the Finite Strip Method (FSM),13, 15

the Kantrovich Method,8 and the Superposition Method,16

have all been studied. More recently, there appeared somestudies by Yuceoglu et al.,20–24 which employed the ModifiedTransfer Matrix Method (MTMM).20–33

The main concern of this study is to present a general ap-proach to the free vibration response of integrally-stiffenedand/or stepped-thickness, rectangular Mindlin plates or panels.Thus, the present work aims to achieve two objectives: (1) Ageneral theoretical analysis, and (2) A general method of solu-tion consistent with (1) for the various types of the integrally-stiffened plate or panel systems as defined and shown in Fig. 2.

In the present theoretical analysis, as a general approach,the integrally-stiffened and/or stepped-thickness plate or panelsystem of Group I, Type 4 (or Four-Step case) in Fig. 3 will beconsidered. Later on, it will be shown that the Lower-Step orHigher-Step cases may easily be obtained from the aforemen-tioned case. Thus, the Group I, Type 4 (or Four-Step case) ispresented in terms of its general configuration, material direc-tions, coordinate systems, and the longitudinal cross-section inFig. 3. Each stiffened plate system is assumed to be a com-bination of Mindlin plates34 with dissimilar orthotropic mate-rial properties and unequal thicknesses. In both figures, eachplate or panel system is considered to be simply supported atx = 0, a, while the boundary conditions in the y-directioncan be arbitrarily specified within the Mindlin plate theory,34

which is one of the First Order Shear Deformation Plate The-ories (FSDPTs).34, 35

114 (pp. 114–126) International Journal of Acoustics and Vibration, Vol. 19, No. 2, 2014

J. Javanshir, et al.: FREE FLEXURAL VIBRATION RESPONSE OF INTEGRALLY-STIFFENED AND/OR STEPPED-THICKNESS COMPOSITE. . .

Figure 1. Groups (or classes) of stiffened plate or panel systems.22

Figure 2. Various types or classes of integrally-stiffened and/or stepped-thickness plates or panels.23

The importance of the transverse shear deformations on thedynamic response of plates are pointed out by Whitney andPagano,36 and also by Librescu et al.37 A brief review of someof the various Higher Order Shear Deformation Plate Theories(HSDPTs) can be found in Subramanian.38

2. GENERAL ANALYSIS AND DERIVATIONOF GOVERNING EQUATIONS

The first step in the present theoretical analysis is to applythe Domain Decomposition technique for the entire plate orpanel system under consideration. Thus, depending on a par-ticular type, the plate elements of the system are consideredto form Part I, Part II, Part III, etc. regions (or domains) asshown in the longitudinal cross-section of the stiffened systemof Fig. 3.

In the second step, the sets of the dynamic equations ofthe orthotropic Mindlin plates and also the stress resultants-displacement expressions are taken into account as given in

Appendix A. The entire system of the governing partial differ-ential equations (PDEs), in a special form (which is very suit-able for the present method of solution—i.e., MTMM), is pre-sented in Appendix B. The aforementioned governing PDEsmay be written in a compact matrix form, in terms of the statevectors of each part in a type or problem under considerationhere.

Hence, in this study, the Group I and Type 4 (or Four-StepCase) is to be considered (see also Fig. 3). Then, referring toAppendix B:

• In Part I region (or Left Plate Stiffener)

1

lI

∂

∂ξI

Y(1)

=

[C′(∂

∂η,∂2

∂t2, . . .

)]Y(1)(ξI, η)

; (1)

• In Part II region (or Right Plate Stiffener)

1

lII

∂

∂ξII

Y(2)

=

[D′(∂

∂η,∂2

∂t2, . . .

)]Y(2)(ξII, η)

; (2)

International Journal of Acoustics and Vibration, Vol. 19, No. 2, 2014 115


Figure 3. Integrally-stiffened and/or stepped-thickness plate or panel systemwith two plate stiffeners: (a) general configuration, geometry, coordinate, andmaterial directions; (b) longitudinal cross-section with Parts I, II, III, IV, V andcoordinate systems.

• In Part III region (or Far Left Plate Element)

1

lIII

∂

∂ξIII

Y(3)

=

[E′(∂

∂η,∂2

∂t2, . . .

)]Y(3)(ξIII, η)

; (3)

• In Part IV region (or Far Right Plate Element)

1

lIV

∂

∂ξIV

Y(4)

=

[F′(∂

∂η,∂2

∂t2, . . .

)]Y(4)(ξIV, η)

; (4)

• In Part V region (or Middle Plate Element)

1

lV

∂

∂ξV

Y(5)

=

[G′(∂

∂η,∂2

∂t2, . . .

)]Y(5)(ξV, η)

. (5)

In the above, ξk = yklk

(k = I, II, III, IV,V) and η = xa , and the

fundamental dependent variables (or the state vectors) of thestiffened system are given as

Y(j)(ξk, η)

=ψ(j)x , ψ(j)

y ,W (j);M(j)yk ,M

(j)y , Q(j)

y

T;

(j = 1, 2, 3, 4, 5, k = I, II, III, IV,V) . (6)

It should be pointed out that the above Eqs. (1)–(5) are cou-pled through the continuity conditions at each intersection be-tween the plate elements of the entire stiffened plate system.It is also important to note here that, although the above equa-tions are written for Type 4 (or the Four-Step case), they caneasily be reduced to Type 2 (or the Two-Step case). They canalso easily be extended to Type 6 (or the Six-Step case) or withthe appropriate continuity conditions at the interfaces of theindividual plate elements. Therefore, the analysis as presentedhere is quite general and it can be further extended, if desired,to other higher order-step cases in a similar manner.

The next step in the analysis is the Non-Dimensionalizationprocedure of the governing PDEs of Eqs. (1)–(5) in each oftheir respective regions. For this purpose the plate elementof Part I is chosen as the reference plate. Therefore, B11(1),h1, ρ1, and a are chosen as the main (or reference) quanti-ties (or parameters). Additionally, l′I, l

′II, l′III, l

′IV, l′V are se-

lected as the length reference parameters in the y-direction ineach part (or region), respectively. All other quantities are non-dimensionalized with respect to these parameters.

The dimensionless coordinates in Part I, Part II, . . . , Part Vregions, respectively, are

η = x/a; ξI = yI/l′I; (in Part I)

η = x/a; ξII = yII/l′II; (in Part II)

η = x/a; ξIII = yIII/l′III; (in Part III)

η = x/a; ξIV = yIV/l′IV; (in Part IV)

η = x/a; ξV = yV/l′V. (in Part V) (7)

The dimensionless parameters related to the orthotropic elasticconstant of plate elements are

B(j)

ik = B(j)ik /B

(1)11 ; (j = 1, 2, 3, . . . and i, k = 1, 2, . . .);

B(j)

ll = B(j)ll /B

(1)11 ; (j = 1, 2, 3, . . . and l = 1, 2, . . .); (8)

and the dimensionless parameters related to the densities andthe geometry of the plates are

ρ2 =ρ2ρ1

; ρ3 =ρ3ρ1

; ρ4 =ρ4ρ1

; ρ5 =ρ5ρ1

; ρ1 =ρ1ρ1

= 1;

LII =l′IIa

; LIII =l′IIIa

; LIV =l′IVa

; LV =l′Va

; LI =l′Ia

;

h2 =h2h1

; h3 =h3h1

; h4 =h4h1

; h5 =h5h1

; h1 =h1h1

= 1;

(9)

where a is the width of the entire plate system.The dimensionless natural frequency parameter ωmn of the

entire stiffened plate or panel system, (i.e., Type 4 (or Four-Step case)) is

ωmn =ρ1a

4ω2mn

h21B(1)11

; Ω = ωmn; (m,n = 1, 2, 3, . . .) .

(10)Here, the non-dimensional natural frequencies ωmn are ar-ranged in the ascending order of Ω1 < Ω2 < Ω3 < . . . Theymay be obtained once m is assigned and n is calculated ac-cordingly.

Now, considering the simple support conditions at x = 0, a,in the x-direction, then the generalized displacements andthe stress resultants can be expressed in the Classical Levy’sMethod in Fourier series in each part (or region for each plateelement). These series are given in Appendix C. Thus, by sim-ply substituting Classical Levy’s Solutions into the system ofEqs. (1)–(5), and making use of some algebraic manipulationsand combinations, the governing system of the first order or-dinary differential equations in the state-vector forms are ob-tained. Thus, for the Type 4 (or Four-Step case):

• In Part I region (or Left Plate Stiffener)

d

dξI

Y

(1)

mn

=[C′]

Y(1)

mn

; (0 ≤ ξI ≤ 1) ; (11)

with the continuity conditions at ξI = 0, 1.

116 International Journal of Acoustics and Vibration, Vol. 19, No. 2, 2014


• In Part II region (or Right Plate Stiffener)

d

dξII

Y

(2)

mn

=[D′]

Y(2)

mn

; (0 ≤ ξII ≤ 1) ; (12)

with the continuity conditions at ξII = 0, 1.

• In Part III region (or Far Left Plate Element)

d

dξIII

Y

(3)

mn

=[E′]

Y(3)

mn

; (0 ≤ ξIII ≤ 1) ; (13)

with the arbitrary boundary conditions at ξIII = 0 and thecontinuity conditions at ξIII = 1.

• In Part IV region (or Far Right Plate Element)

d

dξIV

Y

(4)

mn

=[F′]

Y(4)

mn

; (0 ≤ ξIV ≤ 1) ; (14)

with the arbitrary boundary conditions at ξIV = 1 and thecontinuity conditions at ξIV = 0.

• In Part V region (or Middle Plate Element)

d

dξV

Y

(5)

mn

=[G′]

Y(5)

mn

; (0 ≤ ξV ≤ 1) ; (15)

with the continuity conditions at ξV = 0, 1.

In the above, the dash (—) over the matrices indicates thatthese matrices are non-dimensionalized. The dimensionlessfundamental dependent variables or the state vectors of theproblem are now given as

Y(j)

mn (ξk)

=ψ(j)

mnx, ψ(j)

mny,W(j)

mn,M(j)

mnyx,M(j)

mny, Q(j)

mny

T;

(j = 1, 2, 3, 4, 5; k = I, II, III, IV,V) . (16)

The above dimensionless coefficient matrices[C′],[D′], . . . ,[

G′]

are of dimensions (6 × 6). They include the dimension-less geometric and material characteristics of the individualplate elements of the entire stiffened system. They also in-clude the unknown dimensionless frequency parameter ωmnof the system.

It is important to recall that the reduced system of Eqs. (11)–(15) is coupled by means of the continuity conditions at the in-terfaces of the plate elements of the system. At this point, theInitial Value and Boundary Value Problem of the free dynamicresponse of the stiffened plate or panel system under study arenow reduced or converted to the so-called Two-Point BoundaryValue Problem of Mechanics and Physics in terms of the gov-erning system of Eqs. (11)–(15). This is a very important stepin the theoretical analysis of the problems considered here.

In order to show that the present analytical formulation isquite general, the Group I and Type 2 (or Two-Step case), asshown in Fig. 4, may be considered. For this purpose, one mayreplace the following quantities as

l′I → l′I and[C′]→ [

C′]

;

l′III → l′II and[E′]→ [

D′]

;

l′IV → l′III and[F′]→ [

E′]

;

l′V → 0 and[G′]→ 0; (17)

Figure 4. Integrally-stiffened and/or stepped-thickness plate or panel systemwith a non-central plate stiffener: (a) general configuration, geometry, coordi-nate, and material directions; (b) longitudinal cross-section with Parts I, II, IIIand coordinate systems.

and the dimensionless coordinates as

ξI → ξI; η = x/a;

ξIII → ξII; η = x/a;

ξIV → ξIII; η = x/a;

ξV → 0; η = x/a. (18)

Making use of Eqs. (17) and (18) and inserting these intothe governing system of the First Order Ordinary DifferentialEquations (ODEs) in Eqs. (11)–(15), they can be reduced tothe Group I and Type 2 (or Two-Step case) of Fig. 4. Hence:

• In Part I region (or Non-Central Plate Stiffener)

d

dξI

Y

(1)

mn

=[C′]

Y(1)

mn

; (0 ≤ ξI ≤ 1) ; (19)

with the continuity conditions at ξI = 0, 1.

• In Part II region (or Left Plate Element)

d

dξII

Y

(2)

mn

=[D′]

Y(2)

mn

; (0 ≤ ξII ≤ 1) ; (20)

with the arbitrary boundary conditions at ξII = 0 and thecontinuity conditions at ξII = 1.

• In Part III region (or Right Plate Element)

d

dξIII

Y

(3)

mn

=[E′]

Y(3)

mn

; (0 ≤ ξIII ≤ 1) ; (21)

with the arbitrary boundary conditions at ξIII = 1 and thecontinuity conditions at ξIII = 0.



In the above, the dimensionless fundamental dependent vari-ables or the state vectors are now expressed as

Y(j)

mn (ξk)

=ψ(j)

mnx, ψ(j)

mny,W(j)

mn,M(j)

mnyx,M(j)

mny, Q(j)

mny

T;

(j = 1, 2, 3; k = I, II, III) . (22)

Again, it should be noted that the Initial Value and Bound-ary Value Problem corresponding to the configuration in Fig. 4is finally reduced to a Two-Point Boundary Value Problem ofMechanics and Physics in terms of the new set of the govern-ing systems of the First Order ODEs in Eqs. (19)–(21) for theGroup I and Type 2 (or Two-Step case); see also Yuceoglu etal.25–33

In a similar manner, one can write the governing system ofthe First Order ODEs for the Group I Type 6 (Six-Step case)(or higher order cases).

3. GENERAL METHOD OF SOLUTION ANDNUMERICAL PROCEDURE

The solution technique to be employed is the ModifiedTransfer Matrix Method (MTMM) (with Interpolation Polyno-mials). The state vector forms of the governing system of equa-tions as given in Eqs. (11)–(15), facilitates the present methodof solution. This semi-analytical and numerical solution pro-cedure is a combination of the Classical Levy’s Method, theTransfer Matrix Method, and the Integrating Matrix Method(with Interpolation Polynomials).

The aforementioned method has been developed and beenefficiently utilized for some classes of plate and shallow shellvibration problems by Yuceoglu et al.25–33, 39, 40 It can be foundin more detail in Yuceoglu and Ozerciyes,25–33 Yuceoglu etal.,39 and in earlier versions in Yuceoglu et al.40

The two other more recently developed versions of thismethod are the MTMM (with Chebyshev Polynomials) andthe MTMM (with Eigenvalue Approach). These recent ver-sions by Yuceoglu and Ozerciyes39 are also highly accurateand more suitable for computing the higher natural frequen-cies and modes (higher than the sixth mode and up to fifteen orhigher modes).

In the present study, in connection with the stiffened platesor panels, the MTMM (with Interpolation Polynomials) is tobe used. Referring to the previous studies on the integrally-stiffened and/or stepped-thickness plates by Yuceoglu etal.,20–24 the main steps of the method are briefly explainednext. The very first step is to discretise the state vectors andthe Coefficient Matrices as defined in Eqs. (11)–(15). Thediscretization procedure is simply to divide Part I, Part II, . . . ,Part V regions, into sufficient number of points (or stations)along ξI, ξII, . . . , ξV directions, respectively. Following the dis-cretization, in the second step, the discretized versions of thegoverning system of Eqs. (11)–(15), are pre-multiplied by theappropriate Global Integrating Matrices [LI] , [LII] , . . . , [LV]in their respective parts (or regions). The global integratingmatrices include integrating sub-matrices [l] for each part, re-spectively. Then, in the Part I region (Left Plate Stiffener), thefollowing equation is obtained

Y(1)

mn

−Y

(1)

ξI=0

= [LI]

[C′]

Y(1)

mn

. (23)

In Eq. (23), the discretizations are shown by the · sign on ma-trices along the ξ-directions; the subscript ξI = 0 means thatthe matrix is evaluated at the initial end point at ξI = 0. Theabove expression can be further rearranged between the initialend point ξI = 0 and a general station ξI > 0 in the Part I re-gion. Then, dropping the mn subscripts for convenience, oneobtains:

• In Part I region (Left Plate Stiffener)Y

(1)

ξI

=[U]

Y(1)

ξI=0

;

[U]

=

([I]− [LI]

[C′])−1

; (24)

where[U]

is the discretized Global Modified Transfer Ma-trix for Part I region and [I] is the unit matrix. Similarly, onecan write the following expressions for the Part II, Part III, . . . ,Part V regions:

• In Part II region (Right Plate Stiffener)Y

(2)

ξII

=[V]

Y(2)

ξII=0

;

[V]

=

([I]− [LII]

[D′])−1

; (25)

• In Part III region (Far Left Plate)Y

(3)

ξIII

=[W]

Y(3)

ξIII=0

;

[W]

=

([I]− [LIII]

[E′])−1

; (26)

• In Part IV region (Right Plate)Y

(4)

ξIV

=[S]

Y(4)

ξIV=0

;

[S]

=

([I]− [LIV]

[F′])−1

; (27)

• In Part V region (Middle Plate)Y

(5)

ξV

=[T]

Y(5)

ξV=0

;

[T]

=

([I]− [LV]

[G′])−1

; (28)

where[U],[V],[W],[S], and

[T]

are the discretizedforms of the Global Modified Transfer Matrices for their re-spective parts (or regions), the matrices [LI], [LII], [LIII], [LIV],and [LV] are the Integrating Matrices with roman subscriptsindicating the particular part (or region) in which they oper-ate. As mentioned before, they include the Integrating Sub-Matrices [l].

Here, one can express the relations between the state vectorsat the initial end points ξI, ξII, ξIII, ξIV, ξV = 0, and the statevectors at the final end points ξI, ξII, ξIII, ξIV, ξV = 1 in eachPart I, Part II, . . . , Part V regions, respectively. Thus, droppingthe · sign for convenience, the following equation systems areobtained:



Table 1. Material constants and dimensions of the stiffened plate or panel system.

Orthotropic Composite System with Two Plate Stiffeners Orthotropic Composite System with Three Plate StiffenersGraphite-Epoxy Kevlar-Epoxy Kevlar-Epoxy Graphite-Epoxy Kevlar-Epoxy Kevlar-EpoxyPlate 1 = Plate 2 Plate 3 = Plate 4 Plate 5 Plate 1 = Plate 2 = Plate 3 Plate 4 = Plate 5 Plate 6 = Plate 7

(j = 1, 2) (j = 3, 4) (j = 5) (j = 1, 2, 3) (j = 4, 5) (j = 6, 7)E

(j)x = 11.71 GPa E

(j)x = 5.5 GPa E

(j)x = 5.5 GPa E

(j)x = 11.71 GPa E

(j)x = 5.5 GPa E

(j)x = 5.5 GPa

E(j)y = 137.8 GPa E

(j)y = 76.0 GPa E

(j)y = 76.0 GPa E

(j)y = 137.8 GPa E

(j)y = 76.0 GPa E

(j)y = 76.0 GPa

G(j)xy = 5.51 GPa G

(j)xy = 2.10 GPa G

(j)xy = 2.10 GPa G

(j)xy = 5.51 GPa G

(j)xy = 2.10 GPa G

(j)xy = 2.10 GPa

G(j)xz = 2.5 GPa G

(j)xz = 1.5 GPa G

(j)xz = 1.5 GPa G

(j)xz = 2.5 GPa G

(j)xz = 1.5 GPa G

(j)xz = 1.5 GPa

G(j)yz = 3.0 GPa G

(j)yz = 2.0 GPa G

(j)yz = 2.0 GPa G

(j)yz = 3.0 GPa G

(j)yz = 2.0 GPa G

(j)yz = 2.0 GPa

ν(j)xy = 0.0213 ν

(j)xy = 0.024 ν

(j)xy = 0.024 ν

(j)xy = 0.0213 ν

(j)xy = 0.024 ν

(j)xy = 0.024

ν(j)yx = 0.25 ν

(j)yx = 0.34 ν

(j)yx = 0.34 ν

(j)yx = 0.25 ν

(j)yx = 0.34 ν

(j)yx = 0.34

ρ(j) = 1.6 g/cm2 ρ(j) = 1.3 g/cm2 ρ(j) = 1.3 g/cm2 ρ(j) = 1.6 g/cm2 ρ(j) = 1.3 g/cm2 ρ(j) = 1.3 g/cm2

h1 = 0.04 m h2 = h3 = 0.02 m h2 = h3 = 0.02 m h1 = 0.04 m h2 = h3 = 0.02 m h2 = h3 = 0.02 ma = 0.50 m a = 0.50 m a = 0.50 m a = 0.50 m a = 0.50 m a = 0.50 m

• In Part I region (Left Plate Stiffener)Y

(1)

ξI=1

=[U1,1

]01

Y

(1)

ξI=0

; (0 ≤ ξI ≤ 1) ; (29)

• In Part II region (Right Plate Stiffener)Y

(2)

ξII=1

=[V1,1

]01

Y

(2)

ξII=0

; (0 ≤ ξII ≤ 1) ; (30)

• In Part III region (Far Left Plate Element)Y

(3)

ξIII=1

=[W1,1

]01

Y

(3)

ξIII=0

; (0 ≤ ξIII ≤ 1) ; (31)

• In Part IV region (Far Right Plate Element)Y

(4)

ξIV=1

=[S1,1

]01

Y

(4)

ξIV=0

; (0 ≤ ξIV ≤ 1) ; (32)

• In Part V region (Middle Plate Element)Y

(5)

ξV=1

=[T1,1

]01

Y

(5)

ξV=0

; (0 ≤ ξV ≤ 1) ; (33)

where the subscripts 01 mean that the final forms of the dis-cretized Global Modified Transfer Matrices

[U],[V],[W],[

S], and

[T]

are transferring the state vectors from the initialend point 0 to the final end point 1 in each part (or region),respectively.

At this point, the continuity conditions in terms of the statevectors between (Part III, Part I), (Part I, Part V), (Part I,Part II), etc. are inserted into the above Eqs. (29) through (33).Some further combinations and manipulations yield the finalequation for the entire panel system as

Y(4)

ξIV=1

=[

S] [

V] [

T] [

U] [

W]

01

Y

(3)

ξIII=0

;

Y(4)

ξIV=1

=[

Q]

01

Y

(3)

ξIII=0

. (34)

It is important to observe here that[Q]01

is the final form ofthe discretized Overall Global Modified Transfer Matrix. Thismatrix transfers the above state variables from the initial endpoint (far left end support) ξIII = 0 to the final end point (farright end support) ξIV = 1 of the entire plate or panel systemof Group I and Type 4 (or Four-Step case). The above OverallGlobal Modified Transfer Matrix

[Q]01

can further be reduced

to a (3×3) matrix by making use of or inserting of the Bound-ary Conditions at the initial end point (or far left end support)ξII = 0 and the final right end point (far right end support)ξIII = 1. This operation yields the following:

[C0 (ωmn)]Y0

= 0 ;

Determinant of Coeff. Matrix∣∣ [C0 (ωmn)]

∣∣ = 0; (35)

where the above Coefficient Matrix [C0] implicitly includesthe unknown natural frequency parameter (ωmn). Thus, Y0is not necessarily zero, then the determinant of the CoefficientMatrix must be equal to zero, yielding the polynomial whoseroots are the natural frequencies of the entire plate or panelsystem:

Ω = ωmn; (m,n = 1, 2, 3, . . .) ;

Ω1 < Ω2 < Ω3 < . . . ; (36)

where the dimensionless natural frequencies ωmn are com-puted by searching the roots numerically on the basis of thegiven m and the assigned n values. After then, they are se-quenced or reorganized (or renamed) according to their mag-nitudes as shown above with a single subscript in Eq. (36).

4. SOME NUMERICAL RESULTSAND CONCLUSIONS

The present general approach to the analytical formulationand the method of solution are applied to the problem of thefree dynamic response of integrally-stiffened and/or stepped-thickness rectangular plate or panel system with two plate stiff-eners given in Fig. 3; i.e., Group I and Type 4 (or Four-Stepcase). The present analytical formulation can easily be ex-tended to the Type 6 (or the Six-Step case).

In the set of the present numerical examples, the entire plateor panel system is assumed to be made of orthotropic compos-ite system (Orthotropic case). The two and three plate stiff-eners are considered as made of Graphite-Epoxy and the plateelements of the system are chosen as Kevlar-Epoxy. Thus, it ispossible to compare, in terms of the mode shapes and the cor-responding natural frequencies, and consequently reach someimportant conclusions.

The material and the dimensional characteristics of or-thotropic composite cases are given in Table 1. The bound-ary conditions on all figures are shown only as the far leftand the far right support conditions of the whole stiffened sys-tem. They read from left to right along the y-direction as C =Clamped, S = Simple, and F = Free support conditions.



Figure 5. Mode shapes and dimensionless natural frequencies of integrally-stiffened and/or stepped-thickness rectangular plate or panel system with two platestiffeners (orthotropic case). Plate 1 = Plate 2 = Graphite-Epoxy, Plate 3 = Plate 4 = Plate 5 = Kevlar-Epoxy; lI = 0.15 m, lII = 0.15 m, lIII = 0.233 m,lIV = 0.233 m, lV = 0.234 m; a = 0.50 m, h1 = h2 = 0.04 m, h3 = h4 = h5 = 0.02 m, L = 1.00 m; boundary conditions in y-direction CC.

In Figs. 5–8 for the Orthotropic Composite Plate or PanelSystem, the mode shapes and their natural frequencies areshown for the (CC) and (CS) boundary conditions, respec-tively. The symmetric and/or skew-symmetric mode shapesare observed when the Complete Symmetry conditions hold,as in Fig. 5. Otherwise, in Fig. 6, this is not the case.

The significant effects of several important parameters onthe dimensionless natural frequencies of the present stiffenedsystem are also investigated. These parameters are the aspectratio a/L, the stiffeners length (or width) ratio lI(= lII)/L, thestiffener thickness ratio h3(= h4 = h5)/h1(= h2) and alsothe bending stiffness ratio D(3)

22 /D(1)22 (= D

(2)22 ). Each individ-

ual parameter versus the dimensionless natural frequencies arecomputed and presented in the next set of Figs. 9–12.

The aspect ratio a/L is the parameter which significantlyaffects the natural frequencies of the entire system. Thus, inFig. 9, the natural frequency curves are highly non-linear andthey increase sharply as the aspect ratio a/L increases, regard-less of the support conditions of the system.

The stiffeners length (or width) ratio lI(= lII)/L versus thedimensionless natural frequencies for each set of the support

conditions are shown in Fig. 10. Regardless of the supportconditions, the main characteristic observed is that the dimen-sionless natural frequencies exhibit linear behavior and theyincrease gradually with the increasing parameter under study.The bending stiffness ratio D(3)

22 /D(1)22 (= D

(2)22 ) versus the di-

mensionless natural frequencies are shown in Fig. 12. Re-gardless of the boundary conditions the natural frequencies in-crease gradually and almost linearly.

On the basis of the present Governing Equations and thepresent Solution Method, it can further be concluded that theseare fairly general for the Integrally-Stiffened and/or Stepped-Thickness Plates Systems considered in this study.

REFERENCES1 Hoskins, B. C. and Baker, A. A. Composite Materials for

Aircraft Structures, AIAA Educational Series, New York,(1986).

2 Marshall, I. H. and Demuts, E. (Editors) Supportabilityof Composite Airframes and Aero-Structures, Elsevier Ap-plied Science Publihers, New York, (1988).



Figure 6. Mode shapes and dimensionless natural frequencies of integrally-stiffened and/or stepped-thickness rectangular plate or panel system with two platestiffeners (orthotropic case). Plate 1 = Plate 2 = Graphite-Epoxy, Plate 3 = Plate 4 = Plate 5 = Kevlar-Epoxy; lI = 0.15 m, lII = 0.15 m, lIII = 0.233 m,lIV = 0.233 m, lV = 0.234 m; a = 0.50 m, h1 = h2 = 0.04 m, h3 = h4 = h5 = 0.02 m, L = 1.00 m; boundary conditions in y-direction CS.

3 Niu, M. C.-Y. Airframe Structural Design, Commilit PresLtd., Hong Kong, (1988).

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8 Cortinez, V. H. and Laura, P. A. A. Analysis of VibratingRectangular Plates of Discontinuously varying thickness bythe Cantorovich Method, Journal of Sound and Vibrations,137 (3), 457–461, (1990).

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10 Harik, I. E., Liu, X., and Balakrishnan, N. Analytic Solutionfor Free Vibration of Rectangular Plates, Journal of Soundand Vibration, 153 (1), 51–62, (1992).

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13 Guo, S. J., Khane, A. I., and Moshreff-Torbati, M. VibrationAnalysis of Stepped-Thickness Plates, Journal of Soundand Vibration, 204 (4), 645–657, (1997).



Figure 7. Mode shapes and dimensionless natural frequencies of integrally-stiffened and/or stepped-thickness plate or panel system with three plate stiffeners(orthotropic case). Plate 1 = Plate 2 = Plate 3 = Graphite-Epoxy, Plate 4 = Plate 5 = Plate 6 = Plate 7 = Kevlar-Epoxy; lI = lII = lIII = 0.140 m,lIV = lV = lVI = lVII = 0.145 m; a = 0.50 m, h1 = h2 = h3 = 0.04 m, h4 = h5 = h6 = h7 = 0.02 m, L = 1.00 m; boundary conditions in y-directionCC.

14 Li, Q. S. Exact Solutions for Free Vibration of Multi-StepOrthotropic Shear Plates, Journal of Structural Engineeringand Mechanics, 9, 269–288, (2000).

15 Cheung, Y. K., Au, F. T. K., and Zheng, D. V. FiniteStrip Method for the Free Vibrations and Buckling Analysisof Plates with Abrupt Changes in Thickness, Thin WalledStructures, 36, 89–110, (2000).

16 Gorman, D. J. and Singhal, R. Free Vibration Analysisof Catilever Plates with Step Discontinuities in Propertiesby Superposition Method, Journal of Sound and Vibration,253 (3), 631–652, (2002).

17 Xiang, Y. and Wang, C. M. Exact Buckling and VibrationAnalysis for Stepped Rectangular Plates, Journal of Soundand Vibration, 250 (3), 503–517, (2002).

18 Hull, P. V. and Buchanan, G. R. Vibration of ModeratelyThick Square Orthotropic Stepped-Thickness Plates, Jour-nal of Applied Acoustics, 64, 753–763, (2003).

19 Xiang, Y. and Wei, G. W. Exact Solutions for Buckling andVibration of Stepped-Rectangular Mindlin Plates, Interna-tional Journal of Solids and Structures, 41 (3), 279–294,(2004).

20 Yuceoglu, U., et al. Further Generalization of Analysisand Solution for Free Flexural Vibrations of Integrally-Stiffened and/or Stepped-Thickness Plates or Panels, VIKayseri Havacilik Sempozyumu (VIth Kayseri Symposiumon Aeronautics and Astronautics), Erciyes University, Kay-seri, Turkey, (2006).

21 Yuceoglu, U., Gemalmayan, N., and Sunar, O. FreeFlexural Vibrations of Integrally-Stiffened and/or Stepped-Thickness Plates or Panels with a Central Plate Stiffener,48th AIAA/ASME/ASCE/AHS/ASC/ Structures, Struct. Dy-namics, Materials (SDM) Conference and Exhibit, Waikiki,Hawaii, (2007).

22 Yuceoglu, U., Gemalmayan, N., and Sunar, O. FreeFlexural Vibrations of Integrally-Stiffened and/or Stepped-Thickness Plates or Panels with a Non-Central Plate Stiff-



Figure 8. Mode shapes and dimensionless natural frequencies of integrally-stiffened and/or stepped-thickness plate or panel system with three plate stiffeners(orthotropic case). Plate 1 = Plate 2 = Plate 3 = Graphite-Epoxy, Plate 4 = Plate 5 = Plate 6 = Plate 7 = Kevlar-Epoxy; lI = lII = lIII = 0.140 m,lIV = lV = lVI = lVII = 0.145 m; a = 0.50 m, h1 = h2 = h3 = 0.04 m, h4 = h5 = h6 = h7 = 0.02 m, L = 1.00 m; boundary conditions in y-directionCS.

ener, ASME international Mech. Engineering Congress andExposition on Advanced Structures and Materials for Light-Weight Design, Seattle, Washington, (2007).

23 Yuceoglu, U., Javanshir J., and Eyi S. Free Vi-bration analysis of Integrally-Stiffened and/or Stepped-Thickness Plates or Panels with Two Side Stiffeners, 49th

AIAA/ASME/ASCE/AHS/ASC Structures, Struc. Dynamicsand Materials (SDM) Conference and Exhibit, Schaum-burg, Illinois, (2008).

24 Yuceoglu, U., Guvendiko O, Gemalmayan, N., andSunar, O. Effects of Central and Non-Central PlateStiffeners on Free Vibrations Response of Integrally-Stiffened and/or Stepped-Thickness Plates or Panels, 49th

AIAA/ASME/ASCE/ASC/AHS Structural, Struct. Dynamicsand Materials (SDM) Conference and Exibit, Schaumburg,Illinois, (2008).

25 Yuceoglu, U. and Ozerciyes, V. Free Bending Vibrations ofPartially-Stiffened, Stepped-Thickness Composite Plates,

Advanced Materials for Vibro-Acoustic Applications, 23,191–202, (1996).

26 Yuceoglu, U. and Ozerciyes, V. Natural Frequencies andMode Shapes of Composite Plates or Panels with a Cen-tral Stiffening Plate Strip, Vibroacoustic Methods in Pro-cessing and Characterization of Advanced Materials andStructures, 24, 185–196, (1997).

27 Yuceoglu, U. and Ozerciyes, V. Free Bending Vibrations ofComposite Base Plates or Panels Reinforced with a Non-Central Stiffening Plate Strip, Vibroacoustic Characteriza-tion of Advanced Materials and Structures, 25, 233–243,(1998).

28 Yuceoglu, U. and Ozerciyes, V. Sudden Drop Phe-nomena in Natural Frequencies of Partially Stiffened,Stepped-Thickness, Composite Plates or Panels, 40th

AIAA/ASME/ASCE/AHS/ASC Structures, Struct. Dynamics,and Materials (SDM) Conference and Exhibit, 2336–2347,(1999).



Figure 9. Dimensionless natural frequencies Ω versus aspect ratio a/L inintegrally-stiffened and/or stepped thickness plate or panel system with twostiffeners (orthotropic case). Plate 1 = Plate 2 = Graphite-Epoxy, Plate 3 =Plate 4 = Plate 5 = Kevlar-Epoxy; lI = 0.2 m, lII = 0.2 m, lIII = 0.2 m,lIV = 0.2 m, lV = 0.2 m; a varies, h1 = h2 = 0.04 m, h3 = h4 =h5 = 0.02 m, L = 1.00 m; (a) boundary conditions in y-directions CC, (b)boundary conditions in y-directions CS.

29 Yuceoglu, U. and Ozerciyes, V. Natural Frequenciesand Modes in Free Transverse Vibrations of Stepped-Thickness and/or Stiffened Plates and Panels, 41st

AIAA/ASME/ASCE/AHS/ASC Structures, Struct. Dynamics,and Materials (SDM) Conference and Exhibit, ???, (2000).

30 Yuceoglu, U. and Ozerciyes, V. Sudden Drop Phenomenain Natural Frequencies of Composite Plates or Panels witha Central Stiffening Plate Strip, International Journal ofComputers and Structures, 76 (1–3), 247–262, (2000).

31 Yuceoglu, U. and Ozerciyes, V. Orthotropic CompositeBase Plates or Panels with a Bonded Non-Central (or Ec-centric) Stiffening Plate Strip, ASME Journal of Vibrationand Acoustics, 125, 228–243, (2003).

32 Yuceoglu, U., Ozerciyes, V., and Cil, K. Free Flexural Vi-brations of Bonded Centrally Doubly Stiffened CompositeBase Plates or Panels, ASME International Mechanical En-gineering Congress and Exposition, Anaheim, California,(2004).

33 Yuceoglu, U., Guvendik, O., and Ozerciyes, V. Free Flex-ural Vibrations Response of Composite Mindlin Plates orPanels with a Centrally Bonded Symmetric Double LapJoint (or Symmetric Double Doubler Joint), ASME Interna-tional Mechanical Engineering Congress and Exposition,Orlando, Florida, doi: 10.2514/6.2000-1348, (2005).

Figure 10. Dimensionless natural frequencies Ω versus stiffeners length ra-tio lI(= lII)/L in integrally-stiffened and/or stepped thickness plate or panelsystem with two stiffeners (orthotropic case). Plate 1 = Plate 2 = Graphite-Epoxy, Plate 3 = Plate 4 = Plate 5 = Kevlar-Epoxy; lI = lII varies ⇒ lIII =lIV = lV varies; a = 0.5 m, h1 = h2 = 0.04 m, h3 = h4 = h5 = 0.02 m,L = 1.00 m; (a) boundary conditions in y-directions CC, (b) boundary con-ditions in y-directions CS.

34 Mindlin, R. D. Infuence of Rotatory Inertia and Shear onFlexural Motions of Isotropic, Elastic Plates, ASME Journalof Applied Mechanics, 18, 31–38, (1951).

35 Reissner, E. The Effect of Transverse Shear Deformationson the Bending of Elastic Plates, ASME Journal of AppliedMechanics, 12 (2), 69–77, (1945).

36 Whitney, J. M. and Pagano, N. J. Shear Deformation in Het-erogeneous Anisotropic Plates, ASME Journal of AppliedMechanics, 37, 1031–1036, (1970).

37 Khdeir, A. A., Librescu, L., and Reddy, J. N. Analysisof Solutions of a Refined Shear Deformation Plate Theoryfor Rectangular Composite Plates, International Journal ofSolids and Structures, 23 (10), 1447–1463, (1987).

38 Subramanian, P. Flexural Analysis of Laminated CompositePlates, Composite Structures, 45, 51–69, (1999).

39 Yuceoglu, U. and Ozerciyes, V. Free Vibrations of BoundedSingle Lap Joints in Composite Shallow Cylindrical ShellPanels, AIAA Journal, 43 (12), 2537–2548, (2005).

40 Yuceoglu, U., Toghi, F., and Tekinalp, O. Free Bending Vi-brations of Adhesively-Bonded, Orthotropic Plates with aSingle Lap Joint, ASME Journal of Vibration and Acous-tics, 118, 122–134, (1996).



Figure 11. Dimensionless natural frequencies Ω versus thickness ratio h3(=h4 = h5)/h1(= h2 = 0.04) in integrally-stiffened and/or stepped thicknessplate or panel system with two stiffeners (orthotropic case). Plate 1 = Plate 2= Graphite-Epoxy, Plate 3 = Plate 4 = Plate 5 = Kevlar-Epoxy; lI = 0.2 m,lII = 0.2 m, lIII = 0.2 m, lIV = 0.2 m, lV = 0.2 m; a = 0.50 m, h1 =h2 = 0.04 m, h3 = h4 = h5 varies, L = 1.00 m; (a) boundary conditionsin y-directions CC, (b) boundary conditions in y-directions CS.

APPENDIX A: MINDLIN PLATE THEORY ASAPPLIED TO ORTHOTROPIC PLATES

Equations of Motion of Mindlin Plates

∂Mx

∂x+∂Myx

∂y−Qx +

h

2

(q+zx + q−zx

)=ρh3

12

∂2ψx∂t2

;

∂Myx

∂x+∂My

∂y−Qy +

h

2

(q+zy + q−zy

)=ρh3

12

∂2ψy∂t2

;

∂Qx∂x

+∂Qy∂y

+(q+z + q−z

)= ρh

∂2w

∂t2; (A.1)

where q-s are upper and lower loads or surface stresses, respec-tively.

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

Mx =h3

12

(B11

∂ψx∂x

+B12∂ψy∂y

);

Qx = κ2xhB55

(ψx +

∂w

∂x

);

My =h3

12

(B21

∂ψx∂x

+B22∂ψy∂y

);

Figure 12. Dimensionless natural frequencies Ω versus bending stiffness ra-tio in integrally-stiffened and/or stepped thickness plate or panel system withtwo stiffeners (orthotropic case). Plate 1 = Plate 2 = Graphite-Epoxy, Plate3 = Plate 4 = Plate 5 varies; lI = 0.2 m, lII = 0.2 m, lIII = 0.2 m,lIV = 0.2 m, lV = 0.2 m; E(1)

y = E(2)y constant, E(3)

y = E(4)y = E

(5)y

varies; a = 0.50 m, h1 = h2 = 0.04 m, h3 = h4 = h5 varies,L = 1.00 m;(a) boundary conditions in y-directions CC, (b) boundary conditions in y-directions CS.

Qy = κ2yhB44

(ψy +

∂w

∂y

);

Myx =h3

12B66

(∂ψx∂y

+∂ψy∂x

); (A.2)

where κ2 terms are the Shear Correction Factors of the MindlinPlate Theory, and B-s are material coefficients in the or-thotropic stress-strain relations (Hooke’s Law) such that

B11 =Ex

1− νxyνyx; B44 = Gyz;

B22 =Ey

1− νyxνxy; B55 = Gxz;

B12 = B21 = νyxB11 = νxyB22; B66 = Gxy. (A.3)

Stress Resultant-Displacement Relations(in terms of Stiffnesses)

Mx = D11∂ψx∂x

+D12∂ψy∂y

;

Qx = A55

(ψx +

∂w

∂x

);

My = D21∂ψx∂x

+D22∂ψy∂y

;



Qy = A44

(ψy +

∂w

∂y

);

Myx = D66

(∂ψx∂y

+∂ψy∂x

);

D21 = D12; (A.4)

where the bending stiffness D-s and the shear stiffness A-s are

Dik =h3Bik

12; (i, k = 1, 2);

D66 =h3B66

12;

A44 = κ2yhB44; A55 = κ2xhB55. (A.5)

Mindlin Boundary Conditions

F (free) : Mnt = Mn = Qn = 0;

S (simply supported) : w = ψt = Mn = 0;

F (clamped) : w = ψn = ψt = 0; (A.6)

where n and t are normal and tangential directions.

APPENDIX B: SPECIAL FORM OF GOVERN-ING PDES FOR PARTS I, II, III, IV, V (OR-THOTROPIC MINDLIN PLATE THEORY)

1

l′k

∂ψ(j)y

∂ξk=

1

B(j)22

(12

h3jM (j)y −B

(j)12

1

a

∂ψ(j)x

∂η

);

1

l′k

∂ψ(j)x

∂ξk=

12

h3jB(j)66

M (j)yx −

1

a

∂ψ(j)y

∂η;

1

l′k

∂w(j)

∂ξk=

1

κ2yhjB(j)44

Q(j)y − ∂ψ(j)

y ;

1

l′k

∂M(j)yx

∂ξk=ρjh

3j

12

∂ψ(j)x

∂t2− 1

a

∂M(j)x

∂η+

Q(j)x −

hj2

(q(+)zx + q(−)zx

);

1

l′k

∂M(j)y

∂ξk=ρjh

3j

12

∂ψ(j)y

∂t2− 1

a

∂M(j)yx

∂η+

Q(j)y −

hj2

(q(+)zy + q(−)zy

);

1

l′k

∂Q(j)y

∂ξk= ρjhj

∂2w(j)

∂t2− 1

a

∂Q(j)x

∂η−(q(+)zy − q(−)zy

);

(B.1)

where

j = 1; k = I; (in Part I)j = 2; k = II; (in Part II)j = 3; k = III; (in Part III)j = 4; k = IV; (in Part IV)j = 5; k = V; (in Part V) (B.2)

and where q-s are the surface loads (or stresses) which are iden-tically zero in this case. In the above system of equations, the

state vectors of the problem are given as column vectors forParts I, II, III, . . . ,V, respectively

Y(j) (ξk, η)

=ψ(j)x , ψ(j)

y ,W (j),M(j)yk ,M

(j)y , Q(j)

y

T;

(j = 1, 2, 3, 4, 5; k = I, II, III, IV,V) . (B.3)

APPENDIX C: CLASSICAL LEVY’SSOLUTIONS

Following the Domain Decompositions Technique, theClassical Levy’s Solutions in the x-direction, corresponding,to each part or region are given below.

Displacement and Angles of Rotation

w(j) (η, ξk, t) = h1

∞∑m=1

∞∑n=1

W(j)

mn (ξk) sin(mπη)eiωmnt;

ψ(j)x (η, ξk, t) =

∞∑m=1

∞∑n=1

ψ(j)

mnx (ξk) cos(mπη)eiωmnt;

ψ(j)y (η, ξk, t) =

∞∑m=1

∞∑n=1

ψ(j)

mny (ξk) sin(mπη)eiωmnt; (C.1)

(0 < ξI < 1); (0 < ξII < 1);

(0 < ξIII < 1); (0 < ξIV < 1);

(0 < ξV < 1);

where j = 1 for Part I, j = 2 for Part II, j = 3 for Part III,j = 4 for Part IV, j = 5 for Part V, k = I for Part I, k = IIfor Part II, k = III for Part III, k = IV for Part IV, k = V forPart V.

Stress Resultants

M (j)x (η, ξk, t) =

h51B(1)11

a3

∞∑m=1

∞∑n=1

M(j)

mnx (ξk) sin(mπη)eiωmnt;

M (j)yx (η, ξk, t) =

h51B(1)11

a3

∞∑m=1

∞∑n=1

M(j)

mnyx (ξk) cos(mπη)eiωmnt;

M (j)y (η, ξk, t) =

h51B(1)11

a3

∞∑m=1

∞∑n=1

M(j)

mny (ξk) sin(mπη)eiωmnt;

Q(j)y (η, ξk, t) =

h41B(1)11

a3

∞∑m=1

∞∑n=1

Q(j)

mny (ξk) sin(mπη)eiωmnt;

Q(j)x (η, ξk, t) =

h41B(1)11

a3

∞∑m=1

∞∑n=1

Q(j)

mnx (ξk) cos(mπη)eiωmnt;

(C.2)

(0 < ξI < 1); (0 < ξII < 1);

(0 < ξIII < 1); (0 < ξIV < 1);

(0 < ξV < 1);

where j = 1 for Part I, j = 2 for Part II, j = 3 for Part III,j = 4 for Part IV, j = 5 for Part V, k = I for Part I, k = IIfor Part II, k = III for Part III, k = IV for Part IV, k = V forPart V.


free flexural vibration response of integrally- stiffened and ......vibration response of...

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