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A STUDY ON THE DYNAMIC INSTABILITY OF CYLINDRICAL SHELL DUE TO PARAMETRIC EXCITATION By DEBABRATA PODDER ROLL NO.: 000810402005 REGN. NO.: 81964 of 2001 - 2002 EXAM. ROLL NO.: M4CIV 10-05 Under the Guidance of DR. PARTHA BHATTACHARYA A Thesis Paper to be submitted in Partial Fulfillment of the Requirements for the Degree of Master of Engineering in Civil Engineering (Specialization: Structural Engineering) At the Department of Civil Engineering Faculty of Engineering and Technology Jadavpur University Kolkata 700 032 (ii) DEPARTMENT OF CIVIL ENGINEERING FACULTY OF ENGINEERING AND TECHNOLOGY JADAVPUR UNIVERSITY KOLKATA 700 032 CERTIFICATE OF RECOMMENDATION This is to certify that the thesis titled, A Study on the Dynamic Instability of Cylindrical Shell due to Parametric Excitation, that is being submitted by Debabrata Podder (Roll no. 000810402005) to Jadavpur University for the partial fulfillment of the requirements for awarding the degree of Master of Civil Engineering (Structural Engineering) is a record of bona fide research work carried out by him under my direct supervision & guidance. The work contained in the thesis has not been submitted in part or full to any other university or institution or professional body for the award of any degree or diploma. Dr. Partha Bhattacharya Reader Department of Civil Engineering Jadavpur University Kolkata 700032 Countersigned by _______________________ ________________________ Prof. S. Chakrabarti Prof. N. Chakraborti Head of the Department Dean, FET Department of Civil Engineering Jadavpur University Jadavpur University Kolkata 700032 Kolkata 700032 (iii) DEPARTMENT OF CIVIL ENGINEERING FACULTY OF ENGINEERING AND TECHNOLOGY JADAVPUR UNIVERSITY KOLKATA 700 032 CERTIFICATE OF APPROVAL This thesis paper is hereby approved as a credible study of an engineering subject carried out and presented in a manner satisfactorily to warrant its acceptance as a pre-requisite for the degree for which it has been submitted. It is understood that, by this approval the undersigned do not necessarily endorse or approve any statement made, opinion expressed or conclusion drawn therein but approved the thesis paper only for the purpose for which it is submitted. Board of Thesis Paper Examiners: 1. 2. (iv) ACKNOWLEDGEMENT I gratefully acknowledge the resourceful guidance, active supervision and constant encouragement of my supervisor, Dr. Partha Bhattacharya Dr. Partha Bhattacharya Dr. Partha Bhattacharya Dr. Partha Bhattacharya of the Department of Civil Engineering, Jadavpur University, Kolkata, who despite his other commitments could find time to help me in bringing this Thesis to its present shape. I do convey my sincere thanks and gratitude to him. I also acknowledge my gratefulness to all Professors and staffs of Civil Engineering Department, Jadavpur University, Kolkata, for extending all facilities to carry out the present study. I also thankfully acknowledge the assistance and encouragement received from my family members, friends and others during the preparation of this Thesis. _______________________________ Debabrata Podder Jadavpur University, Kolkata M.C.E. (Structural Engineering) Date: Roll No.:- 000810402005 Regn. No.:- 81964 of 2001-02 Exam. Roll No.:- M4CIV 10-05 (v) ABSTRACT Oscillatory instability of shell structures has been a major cause of concern in many branches of engineering. The dynamic instabilities are the result of pulsating time varying loads mainly inertial or thermal in origin. The greatest danger posed by such instabilities is that the failure is very quick and abrupt. In parametric instability the rate of increase in amplitude is generally exponential and thus potentially dangerous, while in typical resonance due to external excitation the rate of increase in response is linear. More over damping reduces the severity of typical resonance, but may only reduce the rate of increase during parametric resonance. Parametric instability occurs over a region of parameter space and not at discrete points. It may occur due to excitation at frequencies remote from the natural frequencies. Researchers have worked in understanding the behavior of such instabilities. With this particular concept in mind, a theoretical formulation has been developed in the present work for the analysis of singly curved surfaces subjected to in plane periodic loading and undergoing parametric excitation. A four noded iso-parametric shell element having five mechanical degrees of freedom per node, using Mindlin and Reisseners shallow shell theory has been developed in MATLAB platform. The first order shear deformation and effect of rotary inertia has been considered. The results obtained by the present FE code for static, free vibration and buckling analysis are verified with the ANSYS finite element software. Parametric instability studies have been carried out for cylindrical shells having different fibre orientations, various geometric properties with different R/a ratio. A generalized Rayleigh proportional damping has been considered for all the cases to study the shift of the stability point with respect to frequency ratio in various cases. The obtained results are discussed in detail and conclusions highlighting the important findings are made. (vi) CONTENTS CERTIFICATE OF RECOMMENDATION II CERTIFICATE OF APPROVAL III ACKNOWLEDGEMENT IV ABSTRACT V SYMBOLS IX-X LIST OF FIGURES XI-XII LIST OF TABLES XIII CHAPTER 1: INTRODUCTION 1-13 1.1 GENERAL INTRODUCTION 1 1.2 TYPES OF DYNAMIC INSTABILITY 2-3 1.2.1 Impulsive loading 3 1.2.2 Circulatory loads 3 1.2.3 Aero elastic problems 3 1.2.4 Buckling in the testing machine 3 1.3 PARAMETRIC EXCITATION 4-6 1.4 SHELLS 7 1.4.1 Shell as a Structural Form 7 1.4.2 Parametric Excitation on Shell Structure 7 1.5 LITERATURE REVIEW 8-11 1.5.1 Literature review on parametric excitation 8-9 1.5.2 Literature review on shell 9-10 1.5.3 Literature review on shell structures under parametric- instability or dynamic instability 10-11 1.6 OBJECTIVE AND SCOPE OF THE PRESENT WORK 12 1.7 ORGANIZATION OF REPORT 12-13 (vii) CHAPTER-2: CONSTITUTIVE EQUATIONS 14-25 2.1 INTRODUCTION 14 2.2 COMPOSITE MATERIALS 14 2.3 LAMINA AND LAMINATE 14-15 2.4 ASSUMPTIONS REGARDING THE BEHAVIOR OF A LAMINATE 15 2.5 MACRO MECHANICAL BEHAVIOR OF COMPOSITE LAMINATES 15-18 2.6 DISPLACEMENT MODELLING 18-20 2.7 STRESS -STRAIN RELATIONS FOR A LAMINATE 21-24 2.8 ENERGY FORMULATION 24-25 CHAPTER3: FINITE ELEMENT FORMULATION 26-43 3.1 INTRODUCTION 26 3.2 FORMULATION 27-43 3.2.1 Selection of element 27-28 3.2.2 Strain-Displacement relations 29-31 3.2.3 Structural stiffness matrix 31-32 3.2.4 Element mass matrix 32-33 3.2.5 Geometric stiffness matrix 33-38 3.2.6 Element load vector 39 3.4.7 Governing equations of motion 39-40 3.4.8 Stability equations 41-43 Calculation of damping for the present case (C) 42 CHAPTER-4: RESULTS AND DISCUSSION 44-66 4.1 INTRODUCTION 44 4.2 STATIC ANALYSIS 44-47 4.2.1 Isotropic cantilever shell 45-46 4.2.2 Composite cantilever shell 46-47 (viii) 4.3 FREE VIBRATION ANALYSIS 47-50 4.3.1 Isotropic cantilever shell 47-48 4.3.2 Composite cantilever shell 48-50 4.4 BUCKLING ANALYSIS 50-53 4.4.1 Isotropic cantilever shell 50-51 4.4.2 Composite cantilever shell 51-53 4.4 PARAMETRIC INSTABILITY STUDY 53-66 4.4.1 Isotropic cantilever shell 54-55 4.4.2 Composite ca