3/29/2016 Gmail - RE: Permission to include published work in Dissertation Thesis

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Naveed Ahmad <stripling17@gmail.com>

RE: Permission to include published work in Dissertation Thesis 1 message

JINSUN LEE <aas@technopress.com> Mon, Mar 7, 2016 at 9:16 PMTo: navid@vt.edu

Dear Mr. Naveed Ahmad,

 

We are happy to grant the permission to use this paper below as a chapter in your Ph.D. Thesis:

 

Complex modes in damped sandwich beams using beam and elasticity theories

Authors: Naveed Ahmad and Rakesh K. Kapania

Advances in Aircraft and Spacecraft Science, Vol. 2, No. 1 (2015) 5776

DOI: http://dx.doi.org/10.12989/aas.2015.2.1.057

 

Sincerely yours,

 

ChangKoon Choi, Ph. D.

Managing Editor, Advances in Aircraft and Spacecraft Science

 

CKC/jsl

 

From: TechnoPress [mailto:info@technopress.com]  

Ihavenottransferredthecopyrightstothefollowingjournalandconferencepublications.OnthenextpageIhaveattachedascreenshotfromAIAAaboutthecopyrightsandonthefollowingpages,firstpageofeverypublicationisattached,clearlyspecifyingthatIretainthecopyrightsforthesepublications.

JournalArticle

1. NaveedAhmadandRakeshK.Kapania."FreeVibrationAnalysisofIntegrallyStiffened

PlateswithPlate-StripStiffeners",AIAAJournal,2016.http://arc.aiaa.org/doi/abs/10.2514/1.J054372

ConferenceProceedings

1. Naveed AhmadandRakesh K. Kapania. "Damped Free Vibration Response of anAdhesivelyBondedStiffenedPlatewithPlate-StripStiffeners",57thAIAA/ASCE/AHS/ASCStructures, Structural Dynamics, andMaterials Conference, AIAA SciTech, (AIAA2016-0471).http://dx.doi.org/10.2514/6.2016-0471

2. NaveedAhmadandRakeshK.Kapania."FreeVibrationAnalysisofanIntegrallyStiffened

PlatewithPlate-StripStiffenersusingaSetofStaticTimoshenkoBeamFunctions",56thAIAA/ASCE/AHS/ASC Structures, Structural Dynamics, andMaterials Conference, AIAASciTech,(AIAA2015-1167).http://dx.doi.org/10.2514/6.2015-1167

3. NaveedAhmadandRakeshK. Kapania. "ComplexModes inDamped SandwichBeams

Using Beam and Elasticity Theories", 54th AIAA/ASME/ASCE/AHS/ASC Structures,Structural Dynamics, and Materials Conference, (AIAA2013-1641).http://dx.doi.org/10.2514/6.2013-1641

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Free Vibration Analysis of Integrally Stiffened Plates withPlate-Strip Stiffeners

Naveed Ahmad∗ and Rakesh K. Kapania†

Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061

DOI: 10.2514/1.J054372

We investigated the natural frequencies and mode shapes of a freely vibrating, integrally stiffened and/or steppedplate. The stiffeners used here were plate-strip stiffeners as opposed to the normally employed rib stiffeners. Both theplate and stiffeners were analyzed using the first-order shear deformation theory. The deflections and rotations wereassumed as a tensor product of Timoshenko beam functions, chosen appropriately according to the given boundaryconditions. Unlike Navier and Levy solution techniques, the approach used in this paper can also be applied to fullyclamped, free, and cantilever supported stiffened plates. The governing differential equations were solved using theRayleigh–Ritz method. The development of the stiffness and the mass matrices in the Ritz analysis was found toconsume a huge amount of CPU time due to recursive integration of Timoshenko beam functions. An approach issuggested to greatly decrease this amount ofCPU time, by replacing the recursive integration in a loop structure in thecomputer program, with the analytical integration of the integrand in the loop. The numerical results were comparedwith the solutions available in the literature as well as the commercially available finite-element software ABAQUS®,and the results were found to be highly satisfactory.

Nomenclaturea = length of plate, mb = width of plate, mE = elastic modulus of plate and stiffeners, PaG = shear modulus of plate and stiffeners, Pahi = thickness of ith layer, mK = stiffness matrixM = mass matrixu!i" = displacement of ith layer in x direction, mv!i" = displacement of ith layer in y direction, mw = displacement in z direction, mκ = shear correction factorρ = density of plate and stiffeners, kg∕m3

ϕx = rotation of normal to plate midsurface about y axisϕy = rotation of normal to plate midsurface about x axisωn = nth natural frequency of stiffened structure, rad∕s

I. Introduction

S TIFFENED structures have been studied extensively in thepublished literature and are vastly used in various industries to

avoid critical frequencies and provide appropriate stiffness to thestructure. The main objective of this paper is to investigate theresponse of a structure employing plate-strip stiffeners and/orstepped plates. These type of structures are used in the aerospaceindustry where weight, local stiffness, natural frequencies, and thecorresponding mode shapes are of great importance [1,2]. Thesestructural elements have also been effectively used in certain parts ofthe vehicles in the automotive industry because of their lightweightand high-stiffness structural properties [3].

Chopra [4] provided an exact solution for the vibration of thinisotropic rectangular stepped plates using the classical plate theory(CPT). This studywas among the first few investigations of a steppedplate; however, only simply supported boundary conditions wereconsidered. The procedure involveddividing the nonuniform steppedplate into uniform plates of different thicknesses and then applyingthe continuity conditions at the interface between two plates. Thismethod is commonly referred to as the classical method in thepublished literature.Guo et al. [5] pointed out that the continuity conditions used by

Chopra [4] must be changed before being used for stepped plateshaving different materials. They also noted the complications insolving the eigenvalue problem for higher modes using the classicalmethod. They made use of the improved classical finite strip method(FSM), which proved to be more accurate and efficient, particularlyfor the case of higher modes. The improvements in FSM wereintroduced by using dynamic shape functions instead of normallyused static functions. Again, only simply supported boundaryconditions were considered in the analysis.Boscolo and Banerjee [6] developed dynamic stiffness elements

for plates with uniform/nonuniform thicknesses that can be used toprovide exact analytical solutions. They reported results using bothCPT as well as first-order shear deformation theory (FSDT). Thedynamic stiffness method (DSM)makes use of the exact closed-formsolution of the structure’s differential equations, to obtain a dynamicfrequency dependent stiffness matrix, that contains both mass andstiffness properties. This approach can only solve for plates, whichhave two opposite edges as simply supported. The complications indeveloping the dynamic stiffness elements for nonuniform plates, aswell as the method leading to the nonlinear eigenvalue problempresents some disadvantages of the DSM.Hull and Buchanan [1] used FSDT and finite-element analysis to

study the vibration of thick orthotropic stepped plates. Duan andWang [2] used a discrete singular convolution algorithm for thevibration analysis of thin rectangular stepped plates usinginterpolation polynomials on each step. Two jump and continuityconditions at the interface were used to connect the two steps inthe plate. Demasi and Yu [7] used an invariant axiomatic schemecalled a generalized unified formulation to assess the variationalasymptotic plate and shell analysis (VAPAS). They consideredsandwich structures with high face-to-core-stiffness ratios and foundout that VAPAS with one equivalent single-layer theory attainscomparable accuracy to various fourth-order zigzag and layerwisetheories.Kapania and Liu [8] developed an equivalent plate model for

handling discontinuities in general builtup wing structures. The

Presented as Paper 2015-1167 at the 56th AIAA/ASCE/AHS/ASCStructures, Structural Dynamics, and Materials Conference, Kissimmee, FL,5–9 January 2015; received 29 March 2015; revision received 18 August2015; accepted for publication 10 October 2015; published online 6 January2016. Copyright © 2015 by Naveed Ahmad and Rakesh K. Kapania.Published by the American Institute of Aeronautics and Astronautics, Inc.,with permission. Copies of this paper may be made for personal or internaluse, on condition that the copier pay the $10.00 per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 1533-385X/15 and $10.00 in correspondence with the CCC.

*Ph.D. Candidate, Department of Biomedical Engineering andMechanics.Student Member AIAA.

†Mitchell Professor of Aerospace and Ocean Engineering. AssociateFellow AIAA.

Article in Advance / 1

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Damped Free Vibration Response of An Adhesively

Bonded Sti↵ened Plate with Plate-Strip Sti↵eners

Naveed Ahmad⇤ and Rakesh K. Kapania†

Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, USA

We investigated the response of an adhesively bonded plate employing plate-strip sti↵-

eners, unlike the rib sti↵eners often investigated by researchers. Both plate and sti↵eners

were analyzed using the first-order shear deformation theory (FSDT). The deflections and

rotations were assumed as a tensor product of Timoshenko beam functions, chosen appro-

priately according to the given boundary conditions. Unlike Navier and Levy’s solution

techniques, the approach used in this paper can also be applied to fully clamped, free and

cantilever supported sti↵ened plates. The problem was analyzed using the principle of vir-

tual work (PVW). At first, we did not consider damping in the adhesive in order to validate

our code, by comparing our results with those available in the literature as well as with the

results obtained using ABAQUS

R�. The results were found to be highly satisfactory. We

also considered the e↵ect of changing the sti↵ness of the adhesive layer on the vibration

of the bonded system. As a second step, we included damping in the sti↵ened structure

using complex modulus approach, a widely used technique to represent the rheology of

the viscoelastic material. We observed an overall increase in the natural frequencies of

the system, due to the damping provided by the viscoelastic material. Moreover, it was

noticed that when the thickness of the adhesive layer is increased, the natural frequencies

and loss factor of the sti↵ened structure decrease. A viscoelastic material with high loss

factor and small thickness, will be a perfect design variable to obtain overall high damping

in the structure.

Nomenclature

u

(i) displacement of the i-th layer in the x-directionv

(i) displacement of the i-th layer in the y-directionw displacement in the z-direction�

x

rotation of normal to the plate mid surface about y-axis�

y

rotation of normal to the plate mid surface about x-axish

i

thickness of i-th layera length of the plateb width of the plateE elastic modulus of the plate and the sti↵enersG shear modulus of the plate and the sti↵eners shear correction factorK sti↵ness matrixM mass matrix⇢ density of the plate and the sti↵eners⇢

a

density of the adhesive layer!

n

n

th natural frequency of the sti↵ened structure⌘

n

n

th loss factor of the sti↵ened structure⌘

c

loss factor of the viscoelastic core layer

⇤Ph.D. Candidate, Department of Biomedical Engineering and Mechanics, AIAA Student Member.†Mitchell Professor of Aerospace and Ocean Engineering, Associate Fellow, AIAA.

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57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 4-8 January 2016, San Diego, California, USA

AIAA 2016-0471

Copyright © 2015 by Naveed Ahmad & Rakesh K. Kapania. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

AIAA SciTech

Free Vibration Analysis of an Integrally Sti↵ened Plate

with Plate-Strip Sti↵eners using a Set of Static

Timoshenko Beam Functions

Naveed Ahmad ⇤

Virginia Tech., Blacksburg, VA 24061, USA

Rakesh K. Kapania†

Virginia Tech., Blacksburg, VA 24061, USA

We investigated the natural frequencies and mode shapes of a freely vibrating, integrally

sti↵ened and/or stepped plate. The sti↵eners used here, were plate-strip sti↵eners as op-

posed to rib sti↵eners. Both plate and sti↵eners were analyzed using the first order shear

deformation theory (FSDT), which employs the transverse shear deformation and rotary

inertias as well as the longitudinal displacements. The deflections and rotations were as-

sumed as a product of Timoshenko beam functions, chosen appropriately according to the

given boundary conditions. Unlike Navier and Levy solution techniques, the approach used

in this paper can also solve for fully clamped and free sti↵ened plates. The governing di↵er-

ential equations were solved using Rayleigh-Ritz method. Sti↵ened plates were considered

in the study with simply-supported, fixed and free boundary conditions. The procedure

adopted in this paper can analyze plates with centrically and non-centrically placed sti↵en-

ers. Moreover, the development of the sti↵ness and the mass matrices in the Ritz analysis

consumes huge amount of CPU time due to recursive integration of Timoshenko beam

functions, which include sets of trigonometric functions. A technique is used to greatly

decrease the amount of CPU time, by replacing the recursive symbolic integration in a

loop structure in the computer program, with the solution of the integrand in the loop.

The numerical results were compared with the exact solutions available in the literature

and the commercially available finite element software ABAQUS

R�, and the results were

found to be highly satisfactory. Some parametric studies were carried out to exhibit the

influence of certain important parameters on the natural frequencies of the sti↵ened plate.

Finally, a plate with end viscous dampers was studied and some conclusions were drawn

about the mode shapes of the plate.

I. Introduction

Sti↵ened structures have been studied extensively in the published literature and are vastly used invarious industries to avoid critical frequencies and provide appropriate sti↵ness to the structure. The mainobjective of this paper is to investigate the response of a structure employing plate-strip sti↵eners and/orstepped plates. These type of structures can be used in the aerospace industry where weight, structuralfrequency and local sti↵ness are of great importance.1,2These structural elements have also been e↵ectivelyused in certain parts of the vehicles in the automotive industry, because of their light weight and high sti↵nessstructural peoperties.3

Chopra4 provided an exact solution for the vibration of thin isotropic rectangular stepped plates usingthe classical plate theory (CPT). This study was amongst the first few investigations of a stepped plate;however, only simply-supported boundary conditions were considered. The procedure involved separatingthe non-uniform stepped plate into uniform plates of di↵erent thicknesses, and then applying the continuityconditions at the interface between two plates. This method is commonly referred to as the classical methodin the published literature.

Guo, Keane and Mosherefi-Torbati,5 pointed out that the continuity conditions used by Chopra4 must bechanged before being used for stepped plates having di↵erent materials. They also noted the complications

⇤Graduate Student, Department of Engineering Science and Mechanics, AIAA Student Member.†Mitchell Professor of Aerospace and Ocean Engineering, Lifetime Associate Fellow, AIAA.

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56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 5-9 January 2015, Kissimmee, Florida

AIAA 2015-1167

Copyright © 2015 by Naveed Ahmad & Rakesh K. Kapania. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

AIAA SciTech

Complex Modes in Damped Sandwich Beams usingBeam and Elasticity Theories

Naveed Ahmad ∗

Department of Engineering Science and Mechanics, Virginia Tech., Blacksburg, VA 24061, USA

Rakesh K. Kapania†

Department of Aerospace and Ocean Engineering, Virginia Tech., Blacksburg, VA 24061, USA

We investigate complex damped modes in beams in the presence of a viscoelastic layersandwiched between two elastic layers. The key idea behind this type of structure is to in-crease damping in a beam affected by the presence of a highly-damped core layer vibratingmainly in shear. Simply supported and cantilever boundary conditions were considered inthis study. The problem was solved using two approaches, (1) Euler-Bernoulli beam theoryand analyzed using the Rayleigh-Ritz method, and (2) by using plane stress elasticity basedfinite element method. The damping in the layers was modeled using complex modulus.Numerical examples were studied with and without complex modulus in the elastic layers,and complex shear modulus in the core layer. The loss factor was calculated by modalstrain energy method, and by solving a complex eigenvalue problem. The numerical re-sults were compared with those available in literature and the results were found to besatisfactory.

I. Introduction

Extensive literature is available on damping of structural vibrations and noise by employing a viscoelasticlayer sandwiched between two elastic layers. These types of sandwich structures are frequently used in theaerospace industry,1,2 and provide an effective mean of dissipating the noise and vibrational energy by usinga soft and heavily damped viscoelastic material.3,4 Kerwin5 introduced the theory for damping of flexuralwaves in a constrained viscoelastic layer structure. Mead and Marcus,6 and DiTaranto7 followed the work byKerwin to investigate damping in structures utilizing viscoelastic core layers. Mead and Marcus presentedan analytical solution for determining the loss factor of such structures and derived a sixth order differen-tial equation in terms of the transverse displacement; whereas, DiTaranto derived a sixth order differentialequation in terms of the longitudinal displacement. The main idea behind this type of structures is thatdamping can be obtained due to highly damped shear vibrations in the viscoelastic core layer, sandwichedbetween two elastic layers. Transverse shear strain is not considered in the elastic layers.

Most of the previous work did not consider longitudinal and rotary inertias in the core. Rao and Nakra8

considered these inertias in their work for both plates and beams having constrained layers. But they re-ported that while these inertias do not play a vital role when we consider homogeneous beams vibrating atlow frequencies, these can be of significant importance if interest is in very high frequencies. For unsymmet-ric beams, these inertias have to be considered even for low frequencies due to inherent coupling betweenin-plane and transverse vibrations. Rao9 reported the optimal design of a three layered sandwich beamwhich can lead to the maximum loss factor by using different shear parameters and geometrical properties.Lall, Asnani and Nakra10 worked on partially covered sandwich beams.

∗Graduate Student, AIAA Student Member.†Mitchell Professor of Aerospace Engineering, Lifetime Associate Fellow, AIAA.

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54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference April 8-11, 2013, Boston, Massachusetts

AIAA 2013-1641

Copyright © 2013 by Authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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