VIRTUAL PRINCIPLES IN AIRCRAFT STRUCTURES

Volume 1: Analysis

MECHANICS OF STRUCTURAL SYSTEMS

Editors: J.S. Przemieniecki and G.E. Orat/as

L. Fryba, Vibration of solids and structures under moving loads. 1973 ISBN 90-01-32420-7

K. Marguerre and H. Wolfel, Mechanics of vibrations. 1979 ISBN 90-286-0086-6

E.B. Magrab, Vibrations of elastic structural members. 1979 ISBN 90-286-0207-0

R.T. Haftka and M.P. Kamat, Elements of structural optimization. 1985 ISBl'l90-247-2950-5{hardbound) ISBN 90-247-3062-7{paperback)

J.R. Vinson and R.L. Sierakowski, The behaviQr of structures composed of composite materials. 1986 ISBN90-247-3125-9{hardbound) ISBN 9o.:247-3578-5{paperback)

B.E. Gatewood, Virtual Principles in Aircraft Structures Volume 1. 1989. ISBN 90-247-3754-0

B.E. Gatewood, Virtual Principles in Aircraft Structures Volume 2. 1989. ISBN 90-247-3755-9 ISBN 90-247-3753-2 (set).

Volume 6

B.E. GATEWOOD

Profe$$or Emeritw, Dept. of Aeronautical and Astronautical Engineering, The Ohio State University, Columbw, Ohio, U.S.A.

Virtual Principles in Aircraft Structures Volume 1: Analysis

Kluwer Academic Publishers Dordrecht / Boston / London

Library of Congress Cataloging.m-Pub/ication Data

Gatewood, Burford Echols, 1913-Virtual principles of aircraft structures.

(Mechanics of structural systems) Includes bibliographies and indexes. Contents: v. 1. Analysis -- v. 2. Design, plates,

finite elements. 1. Airframes. 2. Structures, Theory of.

3. Strength of materials. 1. Title. II. Series. TL671.6.G37 1989 629.134'1 88-13303

ISBN-13: 978-94-010-7018-8 e-ISBN-13: 978-94-009-1165-9 DOl: 10.1007/978-94-009-1165-9

Published by Kluwer Academic Publishers, P.O. Box 17, 8800 AA Dordrecht, The Netherlands

Kluwer Academic Publishers incorporates the pUblishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk, and MTP Press.

Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A.

In all other countries, sold and distributed 'by Kluwer Academic Publishers Group, P.O. Box 822,8800 AH Dordrecht, The Netherlands

printed on acidfi"ee paper

All Rights Reserved (91989 by Kluwer Academic Publishers Softcover reprint of the hardcover 1st edition 1989

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

Content8 Volume 1: A nalY8i8

]Preface xill

Chapter 1 / The basic three, two, and one dimensional equations .in structural analysis 1

1.1 Introduction 1.2 Three diIJ1ensional equations 1.3 The displacement method of solution 1.4 The stress method of solution 1.5 The combined method of solution

1 2 6 7 8

1.6 Two dimensional equations 8 1. 7 Saint Venant's principle 10 1.8 One dimensional beam equations 11 1.9 No shear stresses in the beam 13 1.10 Beam cross section of a thin plate with one shear stress 13 1.11 Thin web beams with, large flange areas and one shear stress 17 1.12 Torsion of circular cross section and thin wall closed box 18 1.13 Thin web box beam with general loading 20 1.14 Inelastic effects in beams with temperature 21 1.15 Example of inelastic axial stresses and strains with temperature 25 1.16 Sequence loading and thermal cycling in beams 27 1.17 Load-strain design curves for beams 31 1.18 Problems 33

References 35

Chapter 2 / Virtual displacement and virtual force methods in structural analysis 37

2.1 Introduction 37 2.2 The principle of virtual displacements 2.3 The unit displacement theorem 2.4 The principle of virtual forces

39 41 43

vi Oontents of Volume 1: Analysis

2.5 The unit load theorem 45 2.6 The principle of mixed virtual stresses and virtual displacements 46 2.7 The mixed unit displacement and unit load theorem 47 2.8 Two dimensional form of the virtual principles 48 2.9 One dimensional forms of the virtual principles 49 2.10 The one dimensional virtual principles with temperature, inelastic

and large displacement effects 54 2.11 Matrix forms of the virtual principles 56 2.12 Problems 59

References 61

Ch~pter 3 / The virtual principles for pin-jointed trusses 62

3.1 Introduction 62 3.2 The unit displacement theorem for trusses 64 3.3 The unit load theorem for trusses 70 3.4 Inelastic effects with temperature changes in trusses 74 3.5 Matrix equations for trusses. from the unit displacement theorem 81 3.6 Matrix equations for trusses from the unit load theorem 87 3.7 Matrix equations for trusses from the mixed unit displacement and

unit load theorem 96 3.8 Problems 98

References 102

Chapter 4 / The virtual principles for simple beams 103

4.1 Introduction 103 4.2 Principle of virtual displacements for beams 104 4.3 Point values for beam elements by the principle of virtual displace-

ments 108 4.4 Principle of virtual forces for beams 114 4.5 Point values for beam elements by the unit load theorem 119 4.6 ,Principle of mixed virtual stresses and virtual displacements for

beams 122 4.7 Inelastic and temperature effects in simple beams 123 4.8 Matrix equations for beams from the unit displacement theorem 130 4.9 Matrix equations for beams from the unit load theorem 136 4.10 Matrix equations for beams from the mixed unit displacement and

unit load theorem 145 4.11 The beam colump equations 147 4.12 Problems 149

References 152

Chapter 5 / Box beam shear stresses and deflections 153

5.1 Introduction 153 5.2 Shear stresses in beams 154

Oontent! of Volume 1: Analysis vii

5.3 Torsional shear stresses in beams 158 5.4 Shear flows in open box beams 160 5.5 Shear flows in single cell box beams 163 5.6 Shear flows in multi-cell box beams 165 5.7 Shear center for closed box beams 168 5.8 Shear flows in tapered box beams 170 5.9 Inelastic and buckling shear stresses in beams 173 5.10 Axial and bending deflections of box beams with inelastic and

temperature effects 174 5.11 Shear deflections of beams 177 5.12 Torsional rotation of beams 183 5.13 Rota$ion of swept wings 189 5.14 Span~ise airload distribution and static wing divergence under

rotation 190 5.15 Static aileron effectiveness and reversal speed under wing rotation 195 5.16 Problems 199

References 201

Chapter 6 / Shear lag in thin web structures 202

6.1 Introduction 202 Part 1. Solutions for determinate cases 203

6.2 Shear flows due to concentrated loads into thin webs 203 6.3 Shear flows around cut-outs in thin web beams 206 6.4 Cut-outs in box beams 208 6.5 Shear flows in ribs and bulkheads 210 6.6 Forces on ribs due to airloads and taper effects 214

Part B. Solutions for redundant cases 216 6.7 Restraint effects in thin web structures 216 6.8 Shear flows in redundant beams in one plane 216 6.9 Deflections of thin w:eb structures 226 6.10 Flexibility matrices for shear web elements and stiffener elements 231 6.11 Matm solutions for thin web beams in one plane 233 6.12 Matrix solutions for box beams 241 6.13 Load redistribution in swept back wings 248 6.14 Problems 250

References 252

Appendix A / Notes on matrix algebra 254

A.1 Definition of matrices 254 A.2 Addition, subtraction, multiplication of matrices 255 A.3 Determinants 256 A.4 Matrix inversion 257 A.5 Solution of systems of simultaneous equations by matrices 260 A.6 Solution of systems of simultaneous equation,s by tri-diagonal

matrices 261

viii Oontents of Volume 1: Analysis

A.7 Solution of systems of equations by Jordan successive transformations

A.8 Matrix representations A.9 Orthogonal matrices A.10 Eigenvalues and eigenvectors of matrices A.11 Note on matrix notation

References

Appendix B / External forces on Hight vehicles

265 267 270 275 276 277

279

B.1 Introduction 279 B.2 \ Inertial forces for rigid body translation and rotation in a vertical

plane 281 B.3 Air forces on airplane wing 285 B.4 Airplane equilibrium equations in flight. Load factors 287 B.5 Velocity-load factor (V -n) diagram for design 289 B.6 Wing spanwise lift coefficient distribution 292 B.7 Spanwise lift coefficient distribution on twisted wings 295 B.8 Spanwise airload, shear, and moment distributions on wing 297 B.9 Distribution of inertia forces on wing and fuselage 301 B.10 Forces and moments on landing gear structures 304 B.11 Thermal forces 308 B.12 Miscellaneous forces 310 B.13 Deflection effects on the external forces 310 B.14 Criteria for the structure to support the external forces 312 B.15 Problems 314

References 316

Appendix C / Derivation of the strain energy theorems from the virtual principles 317

C.1 Work and strain energy 317 C.2 Maximum and minimum strain energy and total potential energy 319 C.3 Theorem of minimum total potential energy 321 C.4 Theorem of minimum strain energy 322 C.5 Castigliano's theorem (Part J) 323 C.6 Hamilton's principle 323 C.7 Theorem of minimum total complementary potential theory 324 C.8 Theorem of minimum complementary strain energy 325 C.9 Castigliano's theorem (Part II) 326 C.10 Reissner's variational principle. 326 C.11 Comparison of the virtual principles and the strain energy theorems 327

References 328

Index 329

Contents Volume 2: Design, Plates, Finite Elements

Preface

List of selected equations in Volume 1 referred to in Volume 2

Chapter 1 / Allowable stresses of Hight vehicle materials

1.1 Introduction 1.2 Tension, shear, and bearing allowable stresses 1.3 Temperature effects on allowable stresses 1.4 Allowable compression stresses 1.5 Allowable combined stresses 1.6 Creep effects on allowable stresses 1. 7 Room temperature fatigue effects upon allowable stresses 1.8 Temperature effects upon allowable fatigue stresses 1.9 Crack effects upon allowable fatigue stresses 1.10 Problems

References

Chapter 2 / Analysis and design of joints and splices

2.1 Introduction 2.2 Analysis of plate splices with axial tension forces 2.3 Multi-row tension splices 2.4 Joints with eccentric loading 2.5 Minimum weight design of splice for beam with rectangular cross section 2.6 Design of splices for I-beams and thin shear webs 2.7 Deflection effects on load distribution in splices 2.8 Temperature and inelastic effects on load distribution in splices 2.9 Welded joints 2.10 Bonded joints 2.11 Problems

References

ix

x Oontents of Volume 1!: Design, Plates, Finite Elements

Chapter 3 / Structural design of aircraft components

3.1 Introduction 3.2 Design of minimum weight columns without local buckling 3.3 Design of minimum weight sections with local buckling and crippling 3.4 Design of minimum weight columns with local buckling 3.5 Minimum weight design for stiffened panels in compression 3.6 Effective areas for stiffened panels 3.7 Effect of load intensity on wing design 3.8 Design of box beam cross sections with four spar caps 3.9 Analysis of diagonal tension beams 3.10 Problems

References

Chapter " / Analysis and design of pressurized structures

4.1 Introduction 4.2 Membrane stresses in thin shells 4.3 Cut-outs in thin shells with membrane stresses 4.4 Bending in circular cylindrical shells with axially symmetric loading 4.5 Bending in pressurized aircraft fuselages from stringers and frames 4.6 Bending of non-circular cross sections with internal pressure 4.7 Bending of non-circular fuselage rings with internal pressure 4.8 Bending of non-circular fuselage rings with point loads 4.9 Effect of internal pressure on buckling of cylindrical shells 4.10 Pressure stabilized structures 4.11 Problems

References

Chapter 5 / Approximate solutions using the virtual principles

5.1 Introduction 5.2 Approximate solutions for beams using the principle of virtual displace-

ments 5.3 Approximate solutions for columns 5.4 The tapered cantilever beam with numerical integration 5.5 Tapered beam finite element matrices for columns 5.6 The unit load theorem and numerical integration 5.7 Approximate solutions for beams using the mixed virtual principle 5.8 Problems

References

Chapter 6 / Dynamics of simple beams

6.1 Introduction 6.2 Bending vibrations of simple beams

Oontents of Volume 1!: Design, Plates, Finite Elements

6.3 Forced motion of uniform beam 6.4 Approximate solutions for frequencies and mode shapes 6.5 Torsional vibrations of simple beams 6.6 Finite element matrices for beam frequencies 6.7 Flutter of wing segment with one degree of freedom 6.8 Flutter of wing segment with two degrees of freedom 6.9 Dynamic loads on beams 6.10 Problems

References

Chapter 1 / The plate equations

7.1 Introduction 7.2 The plate inplane case using the principle of virtual displacements 7.3 The plate inplane case using the principle of virtual forces 7.4 The plate inplane case using the mixed virtual principle 7.5 The plate bending case using the principle of virtual displacements 7.6 The plate bending case using the principle of virtual forces 7.7 The plate bending case using the mixed virtual principle 7.8 Combined inplane and lateral forces 7.9 Combined forces with large bending deflections 7.10 Buckling of plates 7.11 Plate vibrations 7.12 Problems

References

Xl

Chapter 8 / Approximate matrix equations for plate finite elements

8.1 Introduction 8.2 The point unknowns for the matrices 8.3 The methods to obtain the matrix equations 8.4 Inplane plate element matrices from the principle of virtual displacements 8.5 Inplane plate element matrices from the principle of virtual forces 8.6 Inplane plate element matrices from the mixed virtual principle 8.7 Bending plate element matrices from the principle of virtual displacements 8.8 Bending plate element matrices from the principle of virtual forces 8.9 Bending plate element matrices from the mixed virtual principle 8.10 Matrices for constant stress triangular elements 8.11 Problems

Chapter 9 / Matrix structural analysis using finite elements

9.1 Introduction 9.2 General beam elements in local coordinates 9.3 General beam elements in datum coordinat~es 9.4 Triangular plate elements with inplane forces

xii OontentlJ of Volume t: DelJign, Platu, Finite Elements

9.5 Assembly of finite elements by the virtual principles References

Chapter 10 / Composite Materials

10.1 Introduction 10.2 Stress-strain equations for nonisotropic materials 10.3 Stress-strain equations for plane stress in an orthotropic material 10.4 Forces and moments in laminated plates 10.5 Stresses in laminated plates 10.6 Allowable stresses for laminated plates 10.7 Interlamina stresses 10.8 \ Joints in laminated plates 10.9 Bending deflections of laminated plates 10.10 Buckling loads for laminated plates 10.11 Vibrations of laminated plates 10.12 Problems

References

Preface

The basic partial differential equations for the stresses and displacements in clas­sical three dimensional elasticity theory can be set up in three ways: (1) to solve for the displacements first and then the stresses; (2) to solve for the stresses first and then the displacements; and (3) to solve for both stresses and displacements simultaneously. These three methods are identified in the literature as (1) the displacement method, (2) the stress or force method, and (3) the combined or mixed method. Closed form solutions of the partial differential equations with their complicated boundary conditions for any of these three methods have been obtained only in special cases.

In order to obtain solutions, various special methods have been developed to determine the stresses and displacements in structures. The equations have been reduced to two and one dimensional forms for plates, beams, and trusses. By neglecting the local effects at the edges and ends, satisfactory solutions can be obtained for many case~. The procedures for reducing the three dimensional equations to two and one dimensional equations are described in Chapter 1, Volume 1, where the various approximations are pointed out.

Integral transform methods, energy methods, Rayleigh-Ritz and Galerkin ap­proximation methods, virtual principles, and finite element methods have been developed to aid in solving more complicated structural problems. In recent years (see Chapter 2, Volume 1, References 1 and 2 and Introduction) it has been shown that three virtual principles, which correspond to the three meth­ods of solution described above, give a rational basis for the energy methods, Rayleigh-Ritz methods, and finite element methods.

By using integral transform methods, it is possible to convert the three sets of three dimensional elasticity equations described above to integral forms involv­ing stresses, strains, and/or displacements directly. These integral forms can be identified as (1) the principle of virtual dispiac~ments, (2) the principle of virtual forces, and (3) the principle of mixed virtual stresses and virtual displacements. The principles can be used directly to obtain exact solutions for simple struc­tures, or they can be put into matrix forms and used directly for finite elements. Assumed functions can be put into the virtual principles to obtain a system of simultaneous equations, which correspond exactly; to the Rayleigh-Ritz system of simultaneous equations.

xiii

xiv Preface

The three virtual principles are derived in Chapter 2, Volume 1, and used throughout the text. Temperature and inelastic effects are included in the one dimensional forms. Comparisons of the three methods are made to show that for a given structure one method may be much simpler to use than the others.

The virtual principles are used to obtain differential equations and boundary conditions for beams and beam columns in Chapter 4, Volume 1, for vibration of beams in Chapter 6, Volume 2, and for plates in Chapter 7, Volume 2. They are used to obtain stresses and displacements in determinate and redundant trusses and beams in Chapter 3 and 4, Volume 1, respectively; to get deflections of box beam aircraft type structures in Chapter 5, Volume 1; to solve redundant shear lag problems in Chapter 6, Volume 1, and Chapter 2, Volume 2; to ob­tain approximate solutions using assumed functions in Chapters 5, 6, 7, and 8, Volufne 2; to solve complex structures with finite elements and matrix equations in Chapters 3, 4, and 6, in Volume 1, and Chapters 5, 6, 8, and 9 in Volume 2; to assemble finite element matrices in Chapters 3, 4, and 6 in Volume 1 and Chapter 9 in Volume 2.

All the energy theorems are derived from the virtual principles in Appendix C, Volume 1.

Although the external forces on airplanes are described in Appendix B, Volume 1, and aircraft type structures are emphasized in the text, the virtual principles and the procedures given apply to all types of structures.

There are 230 solved examples in the two volumes so that they can be used not only as textbooks but also as reference books by practicing engineers in structural analysis. A pocket calculator was used to solve the examples and only a pocket calculator is needed to solve the 540 problems in the two volumes. Of course, computers can be used to solve the matrix equations for more complex structures with assumed functions or finite elements.

The material in Volume 1 together with Chapters 1 to 6 in Volume 2 origi­nated from lecture notes used by the author in aircraft structures courses at the Ohio State University. The more advanced Chapters 7-10 (Volume 2) on plates, finite elements, and· composite materials have been added. It is assumed that students have a knowledge of differential equations and mechanics of materials. Although not necessary, some knowledge of integral transforms (such as Laplace Transforms or Fourier Transforms) would help in understanding the derivation of the virtual principles in Chapter 2, Volume 1. The necessary matrix algebra is given in the Appendix A in the same volume.

As may be gathered from the above, the book is published in two volumes. Volume 1 contains subject matter usually covered in the two undergraduate courses in analysis of aircraft structures. Volume 2 contains the subject matter for the usual undergraduate design course as well as chapters on plates, dynamics, finite elements, and composite materials for a more advanced course.

B.E. GATEWOOD

List of Selected Equations 'tn Volume 1 Referred to 'tn Volume 2

In Volume 2, references are made to chapters, sections, figures, and equations in Volume 1. For the convenience of those using Volume 2, a number of the referenced equations least affected by being out of context are listed here.

Uzz,z + Uzy,y + Uzz,z + Xz = 0,

Uzy,z + Uyy,y + Uyz,z + Xy = 0,

Uzz,z + Uyz,y + Uzz,z + X z = 0, auzz

Uzz,z = ax' etc. for notation,

UZZI + Uzym + Uzz" = sz,

Uzyl + Uyym + Uyz" = Sy,

Uzzi + Uyzm + u zz ,,: = Sz,

ezz = uz,z, eyy = UY,Y' ezz = uz,z,

ezy = uy,z + Uz,Y' ezz = uz,z + uz,z,

eyz = Uz,y + uy,z·

Uz =u.zO - yOz + zOy,

Uz = UzO - xOy + yOz.

ezz = (1/ E) [uzz - II(Uyy + uzz )] + aT, eyy = (1/ E) [uyy - II(Uzz + uzz )] + aT,

ezz = (1/ E)[uzz - II(Uzz + U yy )] + aT,

G- E - 2(1+ II)"

0' zz,z + 0' zz,Z + Xz = 0,

UZZI + Uzz" = Sz,

Uzz = V + I/I,zz,

xv

U:z;z = -</>,:£z.

(1.4)

(1.5)

(1.7)

(1.10)

(1.11)

(1.12)

(1.21) (1.22) (1.24)

XVi Lvt of Selected EquatioRJ

P" (2 2) ( )3/ ClJI" = -1 h - z, Iu = t 2h 12, 2 .,," (1.44)

ClJIJI = Co + C3 z + (P,,/ Iu)yz - EaT(z),

ClJIJI = (E/ER)[-ERf!E + (PJI + PJlE)/AE - (ZIuE - zInE) X

X (Ms + MzE)/H + (xIssE - ZInE)(M" + M"E)/H]. (1.90)

CI"" = (E/ER)[ - EReE + (P" + PJlE)/AE - (Ms + MsE)(z/IszE )+

+ (M" + M"E)(X/InE)]. (1.91)

ClJIJI = (E/ER)[-EReE + (PJI + P"E)/AB - (Ms + MzE)(z/IszE)]. (1.92)

ClJIJI = -Ms(z/ Iu).

EJlJle:m/ FJI = (ClJIJI/ FJI) + (3/7}(ClJIJI/ F,,)"

= (CI JIJI / FJI) + (EJlJle,. / F,,), f!,. = (3FJI/7 EJIJI}(ClJIJI/ FII )",

[(R,sS,s + R,IIS,II + R, .. S, .. )dV = Is R(IS,s + mS,JI + nS,,,)dS-

-[ R(S,u + S,JIJI + S,n) dV.

() = dU"G = _ dUll

s dy· dz ' () __ dUsG _ dUJI

,,- dy - dx·

1 -II -II 0 0 0 -II 1 -II 0 0 0

{e} = IJ) + {eEl, 1 -II -II 1 0 0 0

IJI=- 0 0 0 §. 0 0 E G

0 0 0 0 B 0 G 0 0 0 0 0 ~

{P} = Ikl{u} - {PB},

Ik) = [le1llCld dV, {PEl = [1f!111J1-1{eE } dV.

Ik) = Iv ICllflJIIClli dV.

(1.93)

(1.94)

(2.2)

(2.31)

(2.53a)

(2.68)

(2.69)

(2.70)

(2.71)

List of Selected Equations

{u} = [C]{P} + {UE},

[C] = [[u1Ile1]dV, {ue} = [[u1]{eE}dV.

[M] {S} = [-[CF] [s]] {S} = {VE } . u [sIT [0] u P

{P} = [sf {S}.

[M]-l = [-ks + ksspk-1s'f,ks ksspk- 1] k- 1 s'f,ks k- 1 ,

[k] =l[s]'f,[kslls]p, [ks] = [CFrl,

d2 uV eV =-z~

!IY dy2 ' Uv - u1 uV

zb - zm zm'

v M'f z u =---

!IY Izz ' v _ v; MV _ 1 pV vV _ 1 pV

Uyz - Aa' z - mzm zm' z - Pzm zm·

- MzL6~L - PzOu~o - Mz06~0 - LPzmU~m = o. m

d2 ( d2 UZb ) dy2 Elzz dy2 - pz = 0,

( EI d~~b - MZL)L = 0, ( EI d:;~b + Mzo)o = 0,

[:y ( EI d~;~b ) + PzL L = 0, [ d~ ( EI d~;~b ) - PzO L = 0,

U z = (3UzL - LOzL)(fV + (-2UzL + L6zL)(fV+ + (3uzO + LOzo}(l - f)2 + (-2uzO - LOzo)( 1 - f)3.

xvii

(2.76)

(2.77)

(2.86)

(3.38b)

(3.108)

(4.3)

(4.4)

(4.5)

(4.6)

(4.7)

(4.15)

(4.18)

(4.39)

XVlll List of Selected Equations

{P} = [k]{u} - {PE},

[k] = [sjT[kslls], {PE} = [sIT {SE},

M",c = - Pc Uzb, columns.

Pc,cr = 1("2 EI",,,,/ L2 = 1("2 EA/(L/ p)2, simple support.

Pc,cr = 1("2 E I",,,, / 4L 2 , cantilever beam,

Pc,cr = 4~ EI",,,,/L2, ends clamped.

- p. v - - cUzb,yuzb,y

(EIuzb,yy) ,yy + PcUzb,yy - pz = o.

N

dOy = M /GJ dy y , 1/GJ = (1/4A~) L)s/Gt)k.

k=l

The two basic criterions for the material in aircraft design are:

(4.62b)

(4.90)

(4.136)

(4.141)

(4.143)

(4.147)

(4.150)

(5.83)

(5.106)

(A) The stress kL(jL, where (jL is produced by the largest external forces expected during the life of the vehicle, shall not exceed the yield stress Fy of the material. kL is a nondimensional factor to be specified for the design of a particular vehicle (kL = 1 in many cases). This condition is expressed as a margin of safety (M.S.),

(B.74)

The condition (A) is a condition to prevent permanent set of the structure during its life, where the stress (jL is denoted as a limit stress and may be produced by the worst points on the V -n diagram. '

List of Selected Equations xix

(B) The stress kuUL shall not exceed the ultimate failing stress Fu of the material during the life of the vehicle, where ku is a specified factor for the design. The stress UL is the same as in condition (A) and ku = 1.50 for many airplanes, but ku may vary from 1.20 for some guided missiles to ku = 2.00 for some critical types of external forces. The condition is expressed as

M.S. = (Fu/kuUL) - 1 ~ 0.00, (B). (B.75)

The condition (B) is a condition to prevent failure during the life of the airplane. Both conditions (A) and (B) must be satisfied at every point in the structure

of the vehicle. For any particular material with Fy and Fu known and kL and ku specified, it is evident that one of the conditions will be more critical than the other, ~iving a smaller M.S., or

(Fu/Fy) < (ku/kL)' (Fu/Fy) > (ku/kL)'

use (B),

use (A). (B.76)