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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

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