Aircraft structures for engineering students - part 1 pps

- tor E N G I N E E RrN G STU D E NTS THIRD EDITION T.H.G. MEGSON Aircraft Structures for engineering students To The Memory of My Father Aircraft Structures for engineering students Third Edition T. H. G. Megson i EINEMANN OXFORD AMSTERDAM BOSTON LONDON NEWYORK PARIS SANDIEGO SANFRANCISCO SINGAPORE SYDNEY TOKYO Butterworth-Heinemann An imprint of Elsevier Science Linacre House, Jordan Hill, Oxford OX2 8DP 200 Wheeler Road, Burlington, MA 01803 First published by Arnold 1972 First published as paperback 1977 Second edition published by Arnold 1990 Third edition published by Arnold 1999 Reprinted by Butterworth-Heinemann 2001 (twice), 2002,2003 Copyright Q 1999, T H G Megson. All rights reserved. The right of T H G Megson to be identified as the authors of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England WIT 4LP. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to the publishers. Permissions may be sought directly from Elsevier’s Science and Technology Rights Department in Oxfod, UK: phone: (+44) (0) 1865 843830; fax: (+44) (0) 1865 853333; email: permissions@elsevier.co.uk. You may also complete your request on-line via the Elsevier Science homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalogue record for this book is available from the Library of Congress ISBN 0 340 70588 4 For information on all Butterworth-Heinemann publications please visit our website at www.bh.com Typeset in 10112 Times by Academic & Technical vpesetting, Bristol Printed and bound in Great Britain by MPG Books Ltd, Bodmin, Cornwall Contents Preface Preface to Second Edition Preface to Third Edition Part I Elasticity 1 Basic elasticity 1.1 Stress 1.2 1.3 Equations of equilibrium 1.4 Plane stress 1.5 Boundary conditions 1.6 1.7 Principal stresses 1.8 Mohr’s circle of stress 1.9 Strain 1.10 Compatibility equations 1.11 Plane strain 1.12 Determination of strains on inclined planes 1.13 Principal strains 1.14 Mohr’s circle of strain 1.15 Stress-strain relationships 1.16 Experimental measurement of surface strains Notation for forces and stresses Determination of stresses on inclined planes References Problems 2 Two-dimensional problems in elasticity 2.1 Two-dimensional problems 2.2 Stress functions 2.3 Inverse and semi-inverse methods 2.4 St. Venant’s principle 2.5 Displacements 2.6 Bending of an end-loaded cantilever ix xi Xlll 1 3 3 5 7 8 9 10 11 12 16 19 20 21 23 23 24 28 32 32 36 37 38 39 42 43 43 vi Contents Reference Problems 48 48 3 Torsion of solid sections 3.1 Prandtl stress function solution 3.2 3.3 The membrane analogy 3.4 St. Venant warping function solution Torsion of a narrow rectangular strip References Problems 4 Energy methods of structural analysis 4.1 4.2 Total potential energy 4.3 Principle of virtual work 4.4 4.5 4.6 Application to deflection problems 4.7 4.8 Unit load method 4.9 Principle of superposition 4.10 The reciprocal theorem 4.11 Temperature effects Strain energy and complementary energy The principle of the stationary value of the total potential energy The principle of the stationary value of the total complementary energy Application to the solution of statically indetenninate systems References Further reading Problems 5 Bending of thin plates 5.1 5.2 5.3 5.4 5.5 5.6 Pure bending of thin plates Plates subjected to bending and twisting Plates subjected to a distributed transverse load Combined bending and in-plane loading of a thin rectangular plate Bending of thin plates having a small initial curvature Energy method for the bending of thin plates Further reading Problems 6 Structural instability 6.1 Euler buckling of columns 6.2 Inelastic buckling 6.3 Effect of initial imperfections 6.4 6.5 6.6 Buckling of thin plates 6.7 Inelastic buckling of plates 6.8 Stability of beams under transverse and axial loads Energy method for the calculation of buckling loads in columns Experimental determination of critical load for a flat plate 51 51 59 61 63 65 65 68 68 70 71 73 76 77 85 100 103 103 107 109 110 110 122 122 125 129 137 141 142 149 149 152 152 156 160 162 165 169 173 174 6.9 Local instability 6.10 Instability of stiffened panels 6.11 Failure stress in plates and stiffened panels 6.12 Flexural-torsional buckling of thin-walled columns 6.13 Tension field beams References Problems Part I1 Aircraft Structures 7 Principles of stressed skin construction 7.1 Materials of aircraft construction 7.2 Loads on structural components 7.3 Function of structural components 7.4 Fabrication of structural components Problems 8 Airworthiness and airframe loads 8.1 8.2 Load factor determination 8.3 Aircraft inertia loads 8.4 Symmetric manoeuvre loads 8.5 8.6 Gust loads 8.7 Fatigue References Further reading Problems Factors of safety - flight envelope Normal accelerations associated with various types of manoeuvre 9 Bending, shear and torsion of open and closed, thin-walled beams 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 Structural idealization 9.9 9.10 Deflection of open and closed section beams Bending of open and closed section beams General stress, strain and displacement relationships for open and single cell closed section thin-walled beams Shear of open section beams Shear of closed section beams Torsion of closed section beams Torsion of open section beams Analysis of combined open and closed sections Effect of idealization on the analysis of open and closed section beams Problems 10 Stress analysis of aircraft components 10.1 Tapered beams 10.2 Fuselages Contents vii 174 175 i77 180 188 197 197 209 211 21 1 220 223 225 232 233 233 235 238 244 248 25 1 257 27 1 272 272 276 276 29 1 295 300 307 316 322 327 331 342 345 362 3 62 3 74 viii Contents 10.3 Wings 10.4 Fuselage frames and wing ribs 10.5 Cut-outs in wings and fuselages 10.6 Laminated composite structures Reference Further reading Problems 11 Structural constraint 1 1.1 General aspects of structural constraint 11.2 Shear stress distribution at a built-in end of a closed section beam 1 1.3 Thin-walled rectangular section beam subjected to torsion 11.4 Shear lag 11.5 Constraint of open section beams References Problems 12 Matrix methods of structural analysis 12.1 Notation 12.2 Stiffness matrix for an elastic spring 12.3 Stiffness matrix for two elastic springs in line 12.4 Matrix analysis of pin-jointed frameworks 12.5 Application to statically indeterminate frameworks 12.6 Matrix analysis of space frames 12.7 Stiffness matrix for a uniform beam 12.8 Finite element method for continuum structures References Further reading Problems 13 Elementary aeroelasticity 13.1 Load distribution and divergence 13.2 Control effectiveness and reversal 13.3 Structural vibration 13.4 Introduction to ‘flutter’ References Problems 380 406 415 425 432 432 432 443 443 445 449 455 465 485 486 494 495 496 497 500 507 507 509 516 533 533 533 540 54 1 546 55 1 568 576 577 Index 582 Preface During my experience of teaching aircraft structures I have felt the need for a text- book written specifically for students of aeronautical engineering. Although there have been a number of excellent books written on the subject they are now either out of date or too specialist in content to fulfil the requirements of an undergraduate textbook. My aim, therefore, has been to fill this gap and provide a completely self- contained course in aircraft structures which contains not only the fundamentals of elasticity and aircraft structural analysis but also the associated topics of airworthi- ness and aeroelasticity. The book is intended for students studying for degrees, Higher National Diplomas and Higher National Certificates in aeronautical engineering and will be found of value to those students in related courses who specialize in structures. The subject matter has been chosen to provide the student with a textbook which will take him from the beginning of the second year of his course, when specialization usually begins, up to and including his final examination. I have arranged the topics so that they may be studied to an appropriate level in, say, the second year and then resumed at a more advanced stage in the final year; for example, the instability of columns and beams may be studied as examples of structural instability at second year level while the instability of plates and stiffened panels could be studied in the final year. In addition, I have grouped some subjects under unifying headings to emphasize their interrelationship; thus, bending, shear and torsion of open and closed tubes are treated in a single chapter to underline the fact that they are just different loading cases of basic structural components rather than isolated topics. I realize however that the modern trend is to present methods of analysis in general terms and then consider specific applications. Nevertheless, I feel that in cases such as those described above it is beneficial for the student’s understanding of the subject to see the close relationships and similarities amongst the different portions of theory. Part I of the book, ‘Fundamentals of Elasticity’, Chapters 1-6, includes sufficient elasticity theory to provide the student with the basic tools of structural analysis. The work is standard but the presentation in some instances is original. In Chapter 4 I have endeavoured to clarify the use of energy methods of analysis and present a consistent, but general, approach to the various types of structural problem for which energy methods are employed. Thus, although a variety of methods are dis- cussed, emphasis is placed on the methods of complementary and potential energy. [...]... O‘A‘ - OA - O’A’ - S OA SX (1. 16) Now - or u>’ + (V dV -v + dw ) ~ ( W + ~ S X- w 17 18 Basic elasticity which may be written when second-order terms are neglected O’A’ = Sx( 1 +2 g ) ’ Applying the binomial expansion to this expression we have O’A’ = Sx (1 +E) (1. 17) in which squares and higher powers of &/ax are ignored Substituting for O‘A’ in Eq (1. 16) we have i1 E, & It follows that =- =- (1. 18)... 2(0 +~ / 2 = ) 2X Ty -( ax - ay) ( , 0 a) , , C O q e JK7-X ax -ay) + +) = 7 +4T?y -2 rXy JiL7Gi Substituting these values in Eq (1. 9) gives Tmax,min = *t d ~ a - ay)2 x + 4~;y (1. 14) Here, as in the case of principal stresses, we take the maximum value as being the greater algebraic value Comparing Eq (1. 14) with Eqs (1. 11) and (1. 12) we see that (1. 15) Equations (1. 14) and (1. 15) give the maximum shear... aJr - 16 0 N/mm 2 = (iii) and 74,; 16 Basic elasticity Fig 1. 11 Solution of Example 1 I using Mohr's circle of stress Having obtained the principal stresses we now use Eq (1. 15) to find the maximum shear stress, thus The solution is rapidly verified from Mohr's circle of stress (Fig 1. 11) From the = arbitrary origin 0, OP1 and 0 P 2 are drawn to represent o ~ 16 0N/mm2 and ay = -1 20N/mm2 The mid-point... and x, we obtain differ- or Substituting from Eqs (1. 18) and (1. 21) and rearranging (1. 24) Similarly (1. 25) and (1. 26) Equations ( I 21 )-( 1. 26) are the six equations of strain compatibility which must be satisfied in the solution of three-dimensional problems in elasticity Although we have derived the compatibility equations and the expressions for strain for the general three-dimensional state of... J dv, + + ‘Os and - -2 ?xy (0 ,- c ) y 28= cos 2(8 4?xy n/2) = 4 U Y - Uy) J W Rewriting Eq (1. 8) as U an = % ( l + c o s 2 0 ) + - ~ ( 1 -cos28)+rx,sin28 2 2 ahd substituting for (sin28, cos 28} and {sin2(8 + 7r/2), cos 2(8 + n/2)} in turn gives 01 = 0.Y ~ + 0.v (1. 11) and where aIis the maximum or major principal stress and olI is the minimum or minor principal stress Note that a1 is algebraically... are found as follows Differentiating r from Eqs (1. 20) with respect , , to x and y gives a2y,yy- dv a2 axay axay ax ~ au + a2 ay axay or since the functions of u and v are continuous which may be written, using Eq (1. 18) a2y,, axay - a*&, d2E, +ax* ay2 (1. 21) 19 20 Basic elasticity In a similar manner 2 r, , d - - d2E, +L - a2E- ayaz ay2 az2 (1. 22) (1. 23) , : If we now differentiate rx,with respect... point 11 12 Basic elasticity while aI1 algebraically the least Therefore, when aIIis negative, i.e compressive, it is is possible for oII to be numerically greater than oI The maximum shear stress at this point in the body may be determined in an identical manner From Eq (1. 9) dr de - - ( ,- cy) 28 - a cos + 2rxysin 28 = 0 giving (1. 13) It follows that sin 20 = Jm'4 c0s28 = sin 2(0 +~ / 2 = ) 2X Ty -( ax... to rXy 8 = u - C tan J ~ Considering vertical equilibrium gives + uAB sin e = uyAC T,,BC or Txy cot e = - (ii) Hence from the product of Eqs (i) and (ii) 2 Txy = - Sx)(U ( (4 Now substituting the values nX = 16 0N/mm2, gy = -1 20N/mm 200 N/mm2 we have 2 and u = c1= T~~= f 113 N/mm2 Replacing cot B in Eq (ii) by 1/ tan 8 from Eq (i) yields a quadratic equation in u 2 - U(Ux - U y ) + UxUy - Try = 0 2 The... point Therefore, if the shear strain in the x-7 plane is T,~ then the angle between the displaced line elements O’A’ and O’C‘ in Fig 1. 12 is 7r/2 - yxzradians Now cos A’O’C’ = cos(7r/2 - yxz)= sin yxzand as yxz is small then cos A10 C = “ T From the trigonometrical relationships for a triangle = , COS A’O’C’ = + (O’A’)2 (O‘C’)2 - (A’C’)2 2(0’ ) (O’C’) A’ (1. 19) We have previously shown, in Eq (1. 17), that... ED and simplifying 7 = - - (sx - Cy) sin 28 - rXy 28 cos (1. 9) 2 1. 7 Priricipal stresses For given values of a , ay and T,, in other words given loading conditions, anvaries , with the angle 8 and will attain a maximum or minimum value when dan/d8 = 0 From Eq (1. 8) 5 = -2 a, do cos 8 sin 8 + 20, sin 8 cos 8 + 2ryycos 28 = 0 Hence - ( u ~ u,,) 28 sin + 2TsJ cos 28 = 0 or (1. 10) Two solutions, 8 and . 10 3 10 3 10 7 10 9 11 0 11 0 12 2 12 2 12 5 12 9 13 7 14 1 14 2 14 9 14 9 15 2 15 2 15 6 16 0 16 2 16 5 16 9 17 3 17 4 6.9 Local instability 6 .10 Instability of stiffened panels 6 .11 Failure. stress 1. 9 Strain 1. 10 Compatibility equations 1. 11 Plane strain 1. 12 Determination of strains on inclined planes 1. 13 Principal strains 1. 14 Mohr’s circle of strain 1. 15 Stress-strain. section beams Problems 10 Stress analysis of aircraft components 10 .1 Tapered beams 10 .2 Fuselages Contents vii 17 4 17 5 i77 18 0 18 8 19 7 19 7 209 211 21 1 220 223 225 232 233