Aircraft Structures Second Edition Ashish Garg (M.E. IISc, Bangalore) GATE Aerospace Forum Educational Services
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. 2015 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
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)
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
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?... 10 From fracture point of view... 11 From molecular point of view... 11 Rigid Body... 12 Rigid Body Motion... 12 Stress... 14 Strain... 15 Tensile Strain... 15 Compressive Strain... 15 Shear Strain... 16 Equilibrium Equation for non-uniform Stress Variation... 17 For 2D -Plain stress equilibrium equation... 18 Determination of Stresses in Inclined Plane under equilibrium... 19 Tractions or total forces on any surfaces... 22 Principal Stresses... 24 Shear stress... 29 Transformation of Stress on rotating plane... 31 Stress Strain characteristics curve... 34 True Stress and True Strain... 35 Comparison of Engineering and True stress-strain curves... 35
Stress - Strain Relations... 35 Strains induced by Tangential Stress... 38 Stress Strain Relations in three dimension... 39 Orthotropic Materials... 40 Isotropic Materials... 41 Poisson s Ratio... 41 Thermal Stresses and Strains... 41 Elastic Strain Energy... 43 Plane Stress strain problems... 45 Plane Strain Problem... 45 Plane Stress Problem... 46 Relation between Bulk Modulus (K) and Young s Modulus (E)... 49 Governing Equations for elasticity problem... 50 Equilibrium Equations... 50 Boundary Conditions... 51 Compatibility Equations... 51 Compatibility equation for Plane Strain... 54 Compatibility equation for Plane Stress... 54 Airy Stress Function... 56 Mohr Circle... 67 Equation of Mohr Circle for strain... 67 Exercises... 73 Ques 9:- In the case of pure shear, the principal stresses are... 74 Chapter 2: Torsion... 77 Torsion... 77 Torsion of arbitrary cross- section... 77 Rods having circular cross section... 83 Circular Rods in Series... 87 Circular Rods in Parallel... 90 Torsion in thin sheets... 90 Displacement along z-direction in thin sheets... 93 Torsional Constant for arbitrary thin sections... 93 T Section... 93 Open curved thin sections... 94
Torsion in thin closed arbitrary shells... 96 Shear flow... 96 Breth- Bratho theorem... 96 Multicell thin sheet closed sections... 102 Exercises... 110 Chapter 3: Shear Forces & Bending Moments... 112 Simply Supported Beam carrying a Concentrated Load at mid span... 112 Simply Supported Beam with Concentrated Load at an arbitrary point... 113 Simply Supported Beam carrying various Concentrated Load... 115 Simply Supported Beam carrying Uniformly Distributed Load... 117 Simply Supported Beam subjected to a moment... 119 Cantilever Beam having Concentrated Load at its Free End... 120 Cantilever Beam carrying Uniformly Distributed Load over the span... 121 Simply Supported Beam Carrying a Uniformly Varying Load... 123 Trusses and Frames... 133 Perfect Frame and assumptions in analysis... 135 Determination of internal forces in Trusses using method of joints... 135 Exercises... 138 Chapter 4 : Buckling of Vertical Members... 195 Definitions... 195 Elastic Stability... 195 Boundary Conditions... 196 Euler s Theory of Buckling of Columns... 196 Column Hinged at both Ends... 196 Column Fixed at One End and Hinged from Other end... 199 Column Fixed at Both Ends... 201 Column Fixed at One End and Free at the Other... 203 Limitations of Euler s Formula... 205 Rankine Formula... 208 Gordon s Formula... 209 Straight Line Formula... 209 Eccentric loading on a hinged long Columns... 209 Secant formula... 209 Perry s Formula... 211
Short Column with initial curvature... 213 Eccentric loading in short Columns... 215 Combined Effect of Eccentric Loading and Initial Curvature... 215 Plastic Buckling of Columns... 216 Euler Engesser theory of Plastic Buckling... 217 Assumptions... 217 Exercises... 219 Chapter 5: Beam Theory... 221 Centroid... 221 Moment of Inertia... 222 Parallel Axis Theorem... 223 Perpendicular Axis Theorem... 223 Simple Bending... 223 Pure Bending... 223 Theory of Simple Bending... 224 Saint Venant s Principle... 224 Bending stresses... 225 Neutral Axis... 226 Section Modulus... 235 Flitched Beam... 237 Deflection of Beam under bending... 239 Double Integration Method... 239 Macaulay s Method... 241 Shear Forces and Shear Stress distribution... 244 Rectangular cross section... 245 Rectangular section tilted at... 247 Circular cross section... 248 I section... 250 Shear stress distribution for following geometries are... 254 For H section... 254 For Hollow circular section... 254 For Plus section... 255 For inverted channel section... 255
For Triangular section... 255 Product of inertia... 256 Parallel Axis Theorem for Product Moment of Inertia... 256 Principal Axes... 257 Bi-Directional Bending... 257 Unsymmetrical Bending... 261 Stresses due to Unsymmetrical Bending... 261 Shear Centre... 264 Shear centre in thin member closed sections... 264 For semi - circular closed section... 264 For rectangular/square closed section... 265 Shear centre in thin member open sections... 266 Circular shape thin wall open section... 266 Converging section... 267 'T', ' > ', 'L', and '+' sections... 267 Symmetrical - I section... 268 Channel open section... 268 Shear centre in open finite thickness sections... 270 For Channel section... 270 For equal leg angle section... 271 For unsymmetrical I- section... 273 Castigliano's Theorem... 277 Uses of Castigliano's theorem... 279 Deflection under axial load... 279 Deflection under torsion... 280 Rotation under torsion... 280 Deflection under bending... 280 Rotation under bending... 280 Deflection under shear... 280 Deflection under horizontal shear... 280 Exercises... 282 Chapter 6 Failure Theory... 284 Total Strain Energy... 284 Volumetric Strain Energy... 285
Shear Strain Energy... 286 Maximum Principal Stress(Rankine Theory)... 286 The Maximum Principal Strain Theory (St. Venant s theory)... 288 Maximum Shear stress theory or Guest Coulomb s theory or Tresca s theory... 290 Maximum Strain Energy Theory or Beltrami Haigh s Theory... 290 Distortion Energy Theory or Von Mises Theory... 291 Exercises... 302 Chapter 7 Theory of Vibrations... 305 Degree of Freedom... 305 Natural Frequency... 307 Classification of Vibrations... 307 Free Vibration... 307 Forced Vibration... 307 Undamped and Damped Oscillations... 307 Linear and Non linear Vibrations... 307 Deterministic and Non Deterministic Vibration... 308 Spring Element... 308 Combination of spring... 309 Springs in Parallel... 309 Springs in Series... 310 Harmonic Motion... 312 Spring- Mass system in Single degree of freedom... 313 Vibration of Un-damped System... 316 Vibration with Viscous Damping... 318 Critical Damping Constant & Damping Ratio... 319 Logarithmic Decrement... 322 Energy stored in damper... 325 Free vibration with Coulomb Damping... 326 Response of an (SOF) Undamped System under Harmonic Force... 330 Response at Resonance... 331 Total Response of Undamped System... 332 Response of a Damped SOF system under Harmonic Force... 332 Two degree of Freedom System... 336 Equations of motion for Forced Vibration... 337
Free Vibration analysis of an Two DOF Undamped System... 338 Finding the natural frequency and amplitude ratios for few cases of Two DOF system... 340 Two DOF system with viscous damping... 343 Continuous Vibration... 345 Transverse Vibration of a String or Cable... 345 Free vibration of uniform string... 348 String having both ends fixed... 349 For Cantilever( one end fixed and other free)... 349 Longitudinal Vibrations in a Bar... 350 Boundary conditions for rod in longitudinal motion... 351 Torsional Vibration in Shaft... 352 Lateral Vibration in Beam... 354 Exercises... 362 Bibliography... 367