Aircraft Structures
Second Edition
Ashish Garg
(M.E. IISc, Bangalore)
GATE Aerospace Forum Educational Services
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Aircraft Structures
Ashish Garg
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No part of this publication may be reproduced or transmitted in any form or by any means
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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
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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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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 :
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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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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
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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
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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
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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
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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
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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
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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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