Aircraft Structures Second Edition

Transcription

1 Aircraft Structures Second Edition Ashish Garg (M.E. IISc, Bangalore) GATE Aerospace Forum Educational Services

2 Aircraft Structures Ashish Garg All right reserved. No part of this publication may be reproduced or transmitted in any form or by any means electronic or mechanical, including photocopy, recording or any information storage and retrieval system- without permission in writing from GATE Aerospace Forum Educational Services by Author This book is manufactured in India and is authorized for sale only by GATE Aerospace Forum Educational Services. First Edition : 2014 Second Edition : 2015

3 Dedicated to My Grandparents (whose blessings and love have always been with me) My Teachers and Parents (who have made me capable of what I am today) Scientists (who work throughout their lives in contributing theories to mankind worth applying, feeling, reading and writing again and again) My Friends and Colleagues (whose support enabled me to complete it) My Students (who have been my teachers and suggested me to start the project)

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5 Preface This book is an attempt to explain fundamental laws of Solid Mechnics, Structure Dynamics, and Vibrations theory in a simple language as well as to create a compendium of the concepts of Aircraft Structures. It has been principally designed and documented keeping in mind the course structure and understanding level of students majoring Aeronautical or Aerospace Engineering in India. This book will definitely complement their specialist reading of Aircraft Structures. Salient features of the book are as follows : Includes all important definitions and detailed mathematical explanations. Derivations are followed by various numerical examples for better understanding. Covers GATE (AE) Aircraft Structures Syllabus. Covers previous year's questions of GATE (AE) Aircraft Structures along with their solutions. This Second edition have few additional topics, with more solved Examples, and extended exercises questions compare to last edition. I would like to express my deepest gratitude towards Prof. Ranjan Ganguli, Prof. Dinesh Kumar Harursampath and Prof. H.S.N. Murthy for teaching me Structures during my masters at IISc. I would like to thank IISc, Bangalore, TIFR-CAM and Sathyabama University for providing me an excellent working environment, as well as to my seniors who have been a constant source of inspiration and support for me. I express sincere and heartfelt thanks to my students Sonakshi Shyam Arora, Ravi Shakya, Pranav Jindal, Abhay Gupta, Abhinav Sharma and Sandeep Tripathi for helping me in preparing this edition. Thank you Sakshi and Smriti for the cover page design for this edition. I would express my deepest gratitude for my parents and other family members for their whole-hearted affection, inspiration, constant encouragement and support. Last but not the least, I am very much thankful to some of the greatest prodigies in the field of Aerospace, Shri J.R.D. Tata (Founder - IISc), Dr. Satish Dawan (Father of Experimental Fluid Dynamics), Dr. A.P.J. Abdul Kalam (Former President of India), Kalpana Chawla (Indian Astronaut - NASA), Prof. Roddam Narasimha (Director of JNCASR) for being an unmatched inspiration for thousands of scholars like me in Aerospace Engineering

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7 Table of Contents Chapter 1: Aircraft Materials & Elasticity...1 Necessary properties of Materials...1 Basic Aircraft Structural Elements...2 Axial Members...2 Shear Member...3 Bending Member...4 Torsion Member...6 Aircraft Materials and its properties...7 Metal Alloys...7 Composites...8 Stresses and associated strains...9 How does an axial stress affect the component? From fracture point of view From molecular point of view Rigid Body Rigid Body Motion Stress Strain Tensile Strain Compressive Strain Shear Strain Equilibrium Equation for non-uniform Stress Variation For 2D -Plain stress equilibrium equation Determination of Stresses in Inclined Plane under equilibrium Tractions or total forces on any surfaces Principal Stresses Shear stress Transformation of Stress on rotating plane Stress Strain characteristics curve True Stress and True Strain Comparison of Engineering and True stress-strain curves... 35

8 Stress - Strain Relations Strains induced by Tangential Stress Stress Strain Relations in three dimension Orthotropic Materials Isotropic Materials Poisson s Ratio Thermal Stresses and Strains Elastic Strain Energy Plane Stress strain problems Plane Strain Problem Plane Stress Problem Relation between Bulk Modulus (K) and Young s Modulus (E) Governing Equations for elasticity problem Equilibrium Equations Boundary Conditions Compatibility Equations Compatibility equation for Plane Strain Compatibility equation for Plane Stress Airy Stress Function Mohr Circle Equation of Mohr Circle for strain Exercises Ques 9:- In the case of pure shear, the principal stresses are Chapter 2: Torsion Torsion Torsion of arbitrary cross- section Rods having circular cross section Circular Rods in Series Circular Rods in Parallel Torsion in thin sheets Displacement along z-direction in thin sheets Torsional Constant for arbitrary thin sections T Section Open curved thin sections... 94

9 Torsion in thin closed arbitrary shells Shear flow Breth- Bratho theorem Multicell thin sheet closed sections Exercises Chapter 3: Shear Forces & Bending Moments Simply Supported Beam carrying a Concentrated Load at mid span Simply Supported Beam with Concentrated Load at an arbitrary point Simply Supported Beam carrying various Concentrated Load Simply Supported Beam carrying Uniformly Distributed Load Simply Supported Beam subjected to a moment Cantilever Beam having Concentrated Load at its Free End Cantilever Beam carrying Uniformly Distributed Load over the span Simply Supported Beam Carrying a Uniformly Varying Load Trusses and Frames Perfect Frame and assumptions in analysis Determination of internal forces in Trusses using method of joints Exercises Chapter 4 : Buckling of Vertical Members Definitions Elastic Stability Boundary Conditions Euler s Theory of Buckling of Columns Column Hinged at both Ends Column Fixed at One End and Hinged from Other end Column Fixed at Both Ends Column Fixed at One End and Free at the Other Limitations of Euler s Formula Rankine Formula Gordon s Formula Straight Line Formula Eccentric loading on a hinged long Columns Secant formula Perry s Formula

10 Short Column with initial curvature Eccentric loading in short Columns Combined Effect of Eccentric Loading and Initial Curvature Plastic Buckling of Columns Euler Engesser theory of Plastic Buckling Assumptions Exercises Chapter 5: Beam Theory Centroid Moment of Inertia Parallel Axis Theorem Perpendicular Axis Theorem Simple Bending Pure Bending Theory of Simple Bending Saint Venant s Principle Bending stresses Neutral Axis Section Modulus Flitched Beam Deflection of Beam under bending Double Integration Method Macaulay s Method Shear Forces and Shear Stress distribution Rectangular cross section Rectangular section tilted at Circular cross section I section Shear stress distribution for following geometries are For H section For Hollow circular section For Plus section For inverted channel section

11 For Triangular section Product of inertia Parallel Axis Theorem for Product Moment of Inertia Principal Axes Bi-Directional Bending Unsymmetrical Bending Stresses due to Unsymmetrical Bending Shear Centre Shear centre in thin member closed sections For semi - circular closed section For rectangular/square closed section Shear centre in thin member open sections Circular shape thin wall open section Converging section 'T', ' > ', 'L', and '+' sections Symmetrical - I section Channel open section Shear centre in open finite thickness sections For Channel section For equal leg angle section For unsymmetrical I- section Castigliano's Theorem Uses of Castigliano's theorem Deflection under axial load Deflection under torsion Rotation under torsion Deflection under bending Rotation under bending Deflection under shear Deflection under horizontal shear Exercises Chapter 6 Failure Theory Total Strain Energy Volumetric Strain Energy

12 Shear Strain Energy Maximum Principal Stress(Rankine Theory) The Maximum Principal Strain Theory (St. Venant s theory) Maximum Shear stress theory or Guest Coulomb s theory or Tresca s theory Maximum Strain Energy Theory or Beltrami Haigh s Theory Distortion Energy Theory or Von Mises Theory Exercises Chapter 7 Theory of Vibrations Degree of Freedom Natural Frequency Classification of Vibrations Free Vibration Forced Vibration Undamped and Damped Oscillations Linear and Non linear Vibrations Deterministic and Non Deterministic Vibration Spring Element Combination of spring Springs in Parallel Springs in Series Harmonic Motion Spring- Mass system in Single degree of freedom Vibration of Un-damped System Vibration with Viscous Damping Critical Damping Constant & Damping Ratio Logarithmic Decrement Energy stored in damper Free vibration with Coulomb Damping Response of an (SOF) Undamped System under Harmonic Force Response at Resonance Total Response of Undamped System Response of a Damped SOF system under Harmonic Force Two degree of Freedom System Equations of motion for Forced Vibration

13 Free Vibration analysis of an Two DOF Undamped System Finding the natural frequency and amplitude ratios for few cases of Two DOF system Two DOF system with viscous damping Continuous Vibration Transverse Vibration of a String or Cable Free vibration of uniform string String having both ends fixed For Cantilever( one end fixed and other free) Longitudinal Vibrations in a Bar Boundary conditions for rod in longitudinal motion Torsional Vibration in Shaft Lateral Vibration in Beam Exercises Bibliography