DOCSLIB.ORG

Information to Users

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Information to Users INFORMATION TO USERS The most advanced technology has been used to photograph and reproduce this manuscript from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 313/761-4700 800/521-0600 Order Number 9105122 Thick plate theories, with applications to vibration Hanna, Nagy Fouad, Ph.D. The Ohio State University, 1990 Copyright ©1991 by Hanna, Nagy Fouad. All rights reserved. UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 Thick Plate Theories, with Applications to Vibration DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By N. F. HANNA, RSir. M.Sc. * » * * * The Ohio State University 1990 Dissertation Committee: Approved by Prof. O. G. McGee Prof. D. A. Mendelsohn ProLA. W.LE1SSA Department of Engineering Mechanics DEDICATION To my Lord Jesus Christ ACKNOWLEDGEMENTS I would like to thank most my adviser, Professor A. W. Leissa for suggesting the topic and advising me throughout its development Special gratitude and sincere regards are extended to Professor O. G. McGee and Professor D. Mendelsohn for being on the reading committee. Finally, greatest sincere thanks and appreciation go to my family for their love, support, understanding and encouragement without which this study would never have been completed. VITA August 16, 1957 Bom, Cairo, EGYPT June 1979 B.S. in Civil Engineering Ain Shams University, EGYPT August 1985 M.S. in Civil Engineering The Ohio State University Columbus, Ohio, USA 1981 - 1983 Teacher Assistant, Civil Engineering, Helwan Univer­ sity, Cairo, EGYPT 1981 - 1983 Structural engineer, Sabbour Consulting firm, Cairo, EGYPT 1985 - Present Graduate Teacher Assistant, Math Department, The Ohio State University, Columbus, Ohio, USA FIELDS OF STUDY Major Field : Engineering Mechanics. Mechanics : Strength of Materials, Continuum Mechanics, Composite Materials, Vibrations of Continuous media. Mathematics : Linear Algebra, Tensors, Calculus of Variations, Calculus. Numerical Analysis : Numerical Integration, Numerical Methods of Linear Algebra, Ritz Method, Finite Element Method. TABLE OF CONTENTS PAGE D e d ica tio n ......................................................................................................................... ii ACKNOWLEDGEMENTS.............................................................................................. iii VITA ............................................................................................................................... iv LIST OF FIGURES......................................................................................................... vii LIST OF TABLES .................................................................................. ix CHAPTER I. INTRODUCTION .............................................................................................1 II. LITERATUREREVIEW ....................................................................................3 2.1 Introduction ....................................................................................3 2.2 Classical Plate Theory (Thin Plates):- ......................................... 4 2.3 First Order Shear Deformation Theories ....................................7 2.3.1 Reissner's Approach ...............................................................8 2.3.2 Mindlin's A pproach ................................................. 11 2.3.3 Ambartsumyan's Approach ............................................ 18 2.4 Higher-Order Shear Deformation T heories ........................... 24 2.4.1 Higher-Order Theories using Reissner's Approach .. 25 2.4.2 Higher-Order Theories using Mindlin's Approach .. 32 2.4.3 Higher-Order Theories using Ambartsumyan's Approach ................................................................. 44 2.5 Discrete I.ayer T heories .............................................. 47 2.6 Two Dimensional Analysis using Finite Elements ................. 50 2.7 Three Dimensional Analysis ..................................................... 54 2.7.1 Exact Solutions ............................................................. 54 - v - 2.7.2 Approximate Solutions ................................................... 56 2.8 Other Reviews of the L iterature ............................................. 57 2.9 Other Ref erences ...................................................................... 58 m . ENERGY FUNCTIONALS FOR FREE VIBRATIONS OF RECTANGULAR PLATES ......................................................... 59 3.1 Classical Plate T h e o ry .............................................................. 60 3.2 Mindlin's T h e o ry ...................................................................... 61 3.3 Modified Kant T h eo ry .............................................................. 63 TV. DIFFERENTIAL EQUATIONS AND BOUNDARY CONDITIONS FOR MODIFIED KANT TH EO RY ...........................................68 V. FREE VIBRATION ANALYSIS OF ISOTROPIC HOMOGENEOUS RECTANGULAR PLATES USING THE RITZ M ETH O D .............................................................. 76 5.1 Mindlin Theory : ...................................................................... 76 5.2 Modified Kant T h eory .............................................................. 79 VI. RESULTS FOR FREE VIBRATIONS OF COMPLETELY FREE RECTANGULAR THICK PL A T E S....................................... 82 6.1 Introduction ................ 82 6.2 Symmetry Consideration ........................................................ 83 6.3 Convergencc Study ................................................................ 86 6.4 Comparison with Classical Plate Theory .............................. 92 6.5 Thick Plate Deformations ........................................................ 98 6.6 Effect of Varying Poisson's Ratio upon the Frequency Param eter............................................................................ I ll 6.7 Effect of Aspect Ratio and Thickness-to-Width Ratio .... 117 VII. DISCUSSION........................................................................................... 120 7.1 Summary and Conclusions .................................................. 120 7.2 Recommendations for Further Research ........................... 121 APPENDICES A. MODE SHAPES FOR POISSON'S RATIO 0 3 122 B. MODE SHAPES FOR OTHER POISSON'S RATIOS....................... 198 REFERENCES ...................................................................................................... 222 - vii - it LIST OF FIGURES Rectangular plate (cartesian coordinates). 6 General coordinates used in the analysis 77 SS-2 a/b=l, h/b=0.5 .......................... 109 SS-5 a/b=l, h/b=0.^ .......................... 110 Mode shape coordinates ........................... 122 Mode shape SS-1 , a/b =1 .................. 123 Mode shape SS-2 , a/b =1 ................. 124 Mode shape SS-3 , a/b = 1 .................... 125 Mode shape SS-4 , a/b =1 .................... 126 Mode shape SS-5 , a/b =1 .................... 127 Mode sha])e SS-6 , a/b =1 .................... 128 Mode shape SS-7 , a/b =1 .................... 129 Mode shape SS-8 , a/b =1 .................... 130 Mode shape SS-9 , a/b =1 .................... 131 Mode shape SS-10, a/b = 1 .................... 132 Mode shape SS-11 , a/b = 1 .................... 133 Mode shape SA-1 , a/b =1 .................... 134 Mode shape SA-2 , a/b =1 .................... 135 Mode shape SA-3 , a/b =1 .................... 136 Mode shape SA-4 , a/b =1 ........................ 137 A.17 Mode shape SA-5 , a/b -= ] ............................................................... 138 A.18 Mode shape SA-6 , a/b = 1 ............................................................... 139 A.l 9 Mode shape SA-7 , a/b = 1 ............................................................... 140 A.20 Mode shape SA-8 , a/b = 1 ............................................................... 141 A.21 Mode shape SA-9 , a/b =1 .............................................................. 142 A.22 Mode shape SA-10 , a/b =1 143 A.23 Mode shape SA-11 , a/b =1 144 A.24 Mode shape AA-1 , a/b = 1 ................................................................
Load more

Recommended publications
  • VIBRATIONS of THIN PLATE with PIEZOELECTRIC ACTUATOR: THEORY and EXPERIMENTS Parikshit Mehta Clemson University, [email protected]
    Clemson University TigerPrints All Theses Theses 12-2009 VIBRATIONS OF THIN PLATE WITH PIEZOELECTRIC ACTUATOR: THEORY AND EXPERIMENTS Parikshit Mehta Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_theses Part of the Engineering Mechanics Commons Recommended Citation Mehta, Parikshit, "VIBRATIONS OF THIN PLATE WITH PIEZOELECTRIC ACTUATOR: THEORY AND EXPERIMENTS" (2009). All Theses. 707. https://tigerprints.clemson.edu/all_theses/707 This Thesis is brought to you for free and open access by the Theses at TigerPrints. It has been accepted for inclusion in All Theses by an authorized administrator of TigerPrints. For more information, please contact [email protected]. VIBRATIONS OF THIN PLATE WITH PIEZOELECTRIC ACTUATOR: THEORY AND EXPERIMENTS A Thesis Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Master of Science Mechanical Engineering by Parikshit Mehta December 2009 Accepted by: Dr. Nader Jalili, Committee Chair Dr. Mohammed Daqaq Dr. Marian Kennedy ABSTRACT Vibrations of flexible structures have been an important engineering study owing to its both deprecating and complimentary traits. These flexible structures are generally modeled as strings, bars, shafts and beams (one dimensional), membranes and plate (two dimensional) or shell (three dimensional). Structures in many engineering applications, such as building floors, aircraft wings, automobile hoods or pressure vessel end-caps, can be modeled as plates. Undesirable vibrations of any of these engineering structures can lead to catastrophic results. It is important to know the fundamental frequencies of these structures in response to simple or complex excitations or boundary conditions. After their discovery in 1880, piezoelectric materials have made their mark in various engineering applications.
    [Show full text]
  • Buckling and Free Vibrations of Nanoplates – Comparison of Nonlocal Strain and Stress Approaches
    Article Buckling and Free Vibrations of Nanoplates – Comparison of Nonlocal Strain and Stress Approaches Małgorzata Chwał and Aleksander Muc * Institute of Machine Design, Cracow University of Technology, ul. Warszawska 24, 31-155 Kraków, Poland; [email protected] * Correspondence: [email protected] Received: 27 February 2019; Accepted: 1 April 2019; Published: 3 April 2019 Abstract: The buckling and free vibrations of rectangular nanoplates are considered in this paper. The refined continuum transverse shear deformation theory (third and first order) is introduced to formulate the fundamental equations of the nanoplate. In addition, the analysis involves the nonlocal strain and stress theories of elasticity to take into account the small-scale effects encountered in nanostructures/nanocomposites. Hamilton’s principle is used to establish the governing equations of the nanoplate. The Rayleigh-Ritz method is proposed to solve eigenvalue problems dealing with the buckling and free vibration analysis of the nanoplates considered. Some examples are presented to investigate and illustrate the effects of various formulations. Keywords: nonlocal stress theory; nonlocal strain theory; buckling; free vibrations; nanoplates; higher order shear deformation theories 1. Introduction Concerning nano-objects, it is observed that classical (local) continuum theories presenting scale-free relations are inadequate for the mechanical analysis because of the absence of any material length scale parameters, see e.g. works by Peddieson et al. [1], Lu et al. [2], and Barretta et al. [3]. At the nanometer scale, the size effects often become prominent. Some non-classical continuum approaches have been applied to describe the mechanical behavior of small-sized structures, such as couple-stress theory, e.g.
    [Show full text]
  • (TH) Institut Für Baustatik a Stabilized One–Point Integrated Quadrilateral
    Universit¨at Karlsruhe (TH) Institut fur¨ Baustatik A stabilized one–point integrated quadrilateral Reissner–Mindlin plate element F. Gruttmann, W. Wagner Mitteilung 7(2003) BAUSTATIK Universit¨at Karlsruhe (TH) Institut fur¨ Baustatik A stabilized one–point integrated quadrilateral Reissner–Mindlin plate element F. Gruttmann, W. Wagner Mitteilung 7(2003) BAUSTATIK Prof. Dr.–Ing. W. Wagner Telefon: (0721) 608–2280 Institut fur¨ Baustatik Telefax: (0721) 608–6015 c Universit¨at Karlsruhe E–mail: [email protected] Postfach 6980 Internet: http://www.bs.uni-karlsruhe.de 76128 Karlsruhe A stabilized one–point integrated quadrilateral Reissner–Mindlin plate element F. Gruttmann W. Wagner Institut f¨ur Werkstoffe und Mechanik im Bauwesen Institut f¨ur Baustatik Technische Universit¨at Darmstadt Universit¨at Karlsruhe (TH) Alexanderstraße 7 Kaiserstraße 12 64283 Darmstadt 76131 Karlsruhe Germany Germany Contents 1 Introduction 2 2 Basic Equations 3 2.1 Variational Formulation ............................. 3 2.2 Finite Element Equations ............................ 4 3 Examples 8 3.1 Constant bending patch test ........................... 9 3.2 Square plate, test of mesh distorsion ...................... 11 3.2.1 Clamped square plate subjected to a concentrated load ........ 11 3.2.2 Simply supported square plate subjected to uniform load ....... 13 3.3 Square plate subjected to uniform load, test of convergence behaviour .... 14 3.4 Corner supported square plate .......................... 14 3.4.1 Load case 1: uniform load ........................ 14 3.4.2 Load case 2: frequency analysis ..................... 16 3.5 Clamped circular thick plate subjected to a concentrated load ........ 17 4 Conclusions 18 A Appendix, The bending patch test 19 B Appendix, Explicit representation of the one-point integrated stiffness matrix 21 1 Abstract A new quadrilateral Reissner–Mindlin plate element with 12 element degrees of freedom is presented.
    [Show full text]
  • Scherbatiuk Vibration Of.Pdf
    VIBRATION OF SIMPLY SUPPORTED ORTHOTROPIC RECTANGULAR LAMINATED COMPOSITE CROSS-PLY PLATES BY KEVIN DANIEL SCHERBATIUK A thesis Submitted to the Faculty of Graduate Studies In Partial Fulfillment of the Requirements For the Deeree of MASTER OF SCIENCE Department of Civil Engineering University of Manitoba Winnipeg, Manitoba, Canada @ December.200l NationalLibrarY nationale l*l 3få".tg.:" Acquisitions and Acquisitions et Bibiiographicservices servicesbibliographiques 395 Wetlington Str€€t 395, rue Wellington OttawaON KlA0l{4 OttawaON K1AON4 Canacta Canada Your ñl€ Vote úlêrênca Ou frle Noüe rëlércnca The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettånt à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sell reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/filn, de reproduction sur papier ou sur format électronique. The author retains oumership of the L'auteur conserve la propriété du copynght in this thesis. Neither the droit d'autew qui protège cette thèse. thesis nor substantial exhacts from it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sâns son permission. aatorisation. 0-612-80016-4 Canad'ä THE UNIVERSITY OF MANITOBA FACULTY OF GRADUATE STUDIES COPYRIG"""*'SSION PAGE Vibration of simply supported
    [Show full text]
  • 6.1 Plate Theory
    Section 6.1 6.1 Plate Theory 6.1.1 Plates A plate is a flat structural element for which the thickness is small compared with the surface dimensions. The thickness is usually constant but may be variable and is measured normal to the middle surface of the plate, Fig. 6.1.1 lateral load middle surface of plate Fig. 6.1.1: A plate 6.1.2 Plate Theory Plates subjected only to in-plane loading can be solved using two-dimensional plane stress theory1 (see Book I, §3.5). On the other hand, plate theory is concerned mainly with lateral loading. One of the differences between plane stress and plate theory is that in the plate theory the stress components are allowed to vary through the thickness of the plate, so that there can be bending moments, Fig. 6.1.2. M Fig. 6.1.2: Stress distribution through the thickness of a plate and resultant bending moment Plate Theory and Beam Theory Plate theory is an approximate theory; assumptions are made and the general three dimensional equations of elasticity are reduced. It is very like the beam theory (see Book 1 although if the in-plane loads are compressive and sufficiently large, they can buckle (see §6.7) Solid Mechanics Part II 120 Kelly Section 6.1 I, §7.4) – only with an extra dimension. It turns out to be an accurate theory provided the plate is relatively thin (as in the beam theory) but also that the deflections are small relative to the thickness. This last point will be discussed further in §6.10.
    [Show full text]
  • Physically and Geometrically Non-Linear Vibrations of Thin Rectangular Plates Ivan Breslavsky, Marco Amabili, Mathias Legrand
    Physically and geometrically non-linear vibrations of thin rectangular plates Ivan Breslavsky, Marco Amabili, Mathias Legrand To cite this version: Ivan Breslavsky, Marco Amabili, Mathias Legrand. Physically and geometrically non-linear vibrations of thin rectangular plates. International Journal of Non-Linear Mechanics, Elsevier, 2013, 58, pp.30-40. 10.1016/j.ijnonlinmec.2013.08.009. hal-00864370 HAL Id: hal-00864370 https://hal.archives-ouvertes.fr/hal-00864370 Submitted on 26 Feb 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - ShareAlike| 4.0 International License Physically and Geometrically Non-linear Vibrations of Thin Rectangular Plates Ivan Breslavsky, Marco Amabili*, Mathias Legrand Abstract Static deflection as well as free and forced large-amplitude vibrations of thin rectangular rubber plates under uniformly distributed pressure are investigated. Both physical, through a neo-Hookean constitutive law to describe the non-linear elastic deformation of the material, and geometrical non-linearities are accounted for. The deflections of a thin initially flat plate are described by the von Karm´ an´ non-linear plate theory. A method for building a local model, which approximates the plate behavior around a deformed configuration, is proposed.
    [Show full text]
  • Note on Mathematical Development of Plate Theories
    Advanced Studies in Theoretical Physics Vol. 9, 2015, no. 1, 47 - 55 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2015.411150 Note on Mathematical Development of Plate Theories Patiphan Chantarawichit1, Parichat Kongtong2 Yos Sompornjaroensuk1,* and Jakarin Vibooljak3 1 Department of Civil Engineering Mahanakorn University of Technology Bangkok, 10530, Thailand * Correspondence author 2 Department of Automotive Engineering Thai-Nichi Institute of Technology Bangkok, 10250, Thailand 3 Vibool Choke Ltd., Part Bangkok, 10110, Thailand Copyright © 2014 P. Chantarawichit et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this article, the history of mathematical development for plate analysis is reviewed with emphasis on providing the plate’s governing differential equations in rectangular coordinates. Attention is mainly devoted to theories based on the Kirchhoff, Reissner, Mindlin, and Levinson theories for isotropic plates with uniform thickness. Therefore, it is expected that this review paper should be useful for scientists and researchers in assisting them to understand and classify relevant existing plate’s theories quickly. Mathematics Subject Classification: 35Q74, 74K20 Keywords: Partial differential equations, Thin plate, Thick plate, Kirchhoff’s plate, Reissner’s plate, Mindlin’s plate, Levinson’s plate. 48 Patiphan Chantarawichit et
    [Show full text]
  • Non-Linear Vibrations of Tow Placed Variable Stiffness Composite Laminates
    PROGRAMA DE DOUTORAMENTO EM ENGENHARIA MECÂNICA Non-linear Vibrations of Tow Placed Variable Stiffness Composite Laminates Hamed Akhavan Doctoral Program in Mechanical Engineering Supervisor: Pedro Manuel Leal Ribeiro October 2015 c Hamed Akhavan, 2015 Abstract Vibrations of structures and mechanical systems have plenty of advantages, but also disadvantages. Notably, vibrations can lead to failures or damage of equipment. Therefore, knowing more about the vibratory behaviour helps engineers to control damage and to take preventive measures in order to avoid damages or malfunctions. For a long time, optimising material properties was a goal in material design. An important aspect in the optimisation of fibrous composite materials is the fibre angles in different layers of the laminate. In traditional composite laminates, the fibre orientation is constant in a ply, but some researchers paid attention to the possibility of changing the fibre orientation angle inside a ply. A laminate with plies which have curvilinear fibre paths is considered as a variable stiffness composite laminate (VSCL). Due to the variation of in-plane stiffness in a VSCL, these laminates find special applications in naval, aerospace, or other structures (e.g. in solar array wings, aircraft wings, fuselages and tail sections). Vibrations of VSCL plates are analysed in this thesis. To define a curvilinear fibre path, it is assumed that the fibre angle varies linearly along one of the Cartesian axes, say x, from a value T0 at the centre of the panel to the value T1 at the panel edges, x = ±a=2, where the panel has length a. The fibre path defined in this fashion is called the reference fibre path; other fibre paths in the ply can be produced by shifting - or copying - this path with a constant distance through direction y.
    [Show full text]
  • Free and Forced Vibrations of Monolithic and Composite Rectangular Plates with Interior Constrained Points
    Free and Forced Vibrations of Monolithic and Composite Rectangular Plates With Interior Arka P. Chattopadhyay Department of Biomedical Engineering and Mechanics, Constrained Points M/C 0219 Virginia Polytechnic Institute and State University, The restriction of deformations to a subregion of a system undergoing either free or 495 Old Turner Street, forced vibration due to an irregularity or discontinuity in it is called mode localization. Blacksburg, VA 24061 Here, we study mode localization in free and forced vibration of monolithic and unidirec- e-mail: [email protected] tional fiber-reinforced rectangular linearly elastic plates with edges either simply sup- ported (SS) or clamped by using a third-order shear and normal deformable plate theory Romesh C. Batra1 (TSNDT) with points on either one or two normals to the plate midsurface constrained Honorary Fellow from translating in all three directions. The plates studied are symmetric about their mid- Department of Biomedical Engineering surfaces. The in-house developed software based on the finite element method (FEM) is and Mechanics, first verified by comparing predictions from it with either the literature results or those M/C 0219 Virginia Polytechnic computed by using the linear theory of elasticity and the commercial FE software ABAQUS. Institute and State University, New results include: (i) the localization of both in-plane and out-of-plane modes of vibra- 495 Old Turner Street, tion, (ii) increase in the mode localization intensity with an increase in the length/width Blacksburg, VA 24061 ratio of a rectangular plate, (iii) change in the mode localization characteristics with the e-mail: [email protected] fiber orientation angle in unidirectional fiber reinforced laminae, (iv) mode localization due to points on two normals constrained, and (iv) the exchange of energy during forced harmonic vibrations between two regions separated by the line of nearly stationary points that results in a beats-like phenomenon in a subregion of the plate.
    [Show full text]