Theory and Analysis of Flight Structures

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Preface vii

CHAPTER INTRODUCTIONSteps in Structural Designpplied Loads and Temperahtres 2

Actual Stresses and Deflections 41 1.1 2.1 3.1 4. Allowable Stresses r Deflections 51 5.1 6.

Comparison of pplied and Allowable Stresses and DeflectionsSttmma y 7References 7Problems 8 \

CHAPTER 2 STRESS AND STRAIN 12 1. Introduction 102 2. Stress: Definitions and Notations 102 3. Equations of Equilibnum 12

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2 4. Stress Ttansfonnations for Rotation of A x s 62 5. Principal Stresses and aximum Shem Stres es2 6. Dejlections and Strains 202 7. Strain-transfonnation Equations 242 8. Compatibility Equations 262 9. Summary 27References 28

Problems 28

CHAPTER 3 MECHANICAL BEHAVIOR OF MATERIALS 33 1. Introduction 313 2. The Tensile Test 323 3. Compression and Shear Tests 353 4. I dealizations of the Stress-Strain Curve 363 5. Three-parameter Representations of Stress-Strain Curves 383 6. Effect of Temperature upon Short-lime Static Properties 43 7. Creep 423 8. Fatigue 443 9. Allowable M echanical Properties 47

3 10. M aterial Selection 493 11. Three-dimensional Linearly Elastic Stress-Strain

Relationships 51References 56Problems 57

CHAPTER 4 INTRODUCTION TO THE THEORY OF ELASTICITY 594 1. Introduction 594 2. Displacement Formulation 604 3. Stress Formulation 614 4. Two-dimensional Problems 624 5. Stress-function Formulation 644 6. The Inverse M ethod 704 7. The Semi-inverse M ethod 754 8. St. Venant s Princip e 75

References 76Problems 77

CHAPTER 5 FINITE DIFFERENCE METHODS 795 1. Introduction 795 2. Finite-difference Operators 85 3. Application to EqU? librium Boundary-value Problems 855 4. Application to Eigenvalue P oblems 905 5. Solution of M atrix Eigenvalue Equations 93



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R 6 INTRODUCTION TO WORK AND ENERGY PRINCIPLES 90 1 ntroductton 902. Work and Energy 996:3 Virtual Work and Equilibrium 1016 4. Coordinates and Degrees o Freedom 10465. Stability 10566.6 7.

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6 10.

6 11.

6 12.6 13.614.

Small Displacements o a Conservat1've System 107Strain Energy and Complementary Strain Energy 110Potential and Complementary Potential o External ForcesThe Principle o the Stationary Value o the Total PotentialThe Principle o the Stationary Value o the TotalComplementary Potential 118De1 ivation o Equilibrium and Compatibility Equationsby Va1 iational M ethods 120The Rayleigh-Ritz M ethod 122The Recip1 ocal Theorems o Betti and MaxwellThe Use o Virtual Work to Compute DejlectionsRefe? ences 131Problems 131

127128

A TER 7 BENDING AND EXTENSION OF BEAMS 33

7 1.7 2.7 3.7 4.7 5.7 6.7 7.7 8.7 9.

7 10.

I ntroducUonSt1ess Resultants 133 133Stresses Due to Extension and Bending 135M odulus-weighted Section Propertt es 139Accuracy o Beam-stress Equation 141I dealization o Stiffened-shell St1uctures 143Equilibrium Equations 152Beam Dejlections 157The Dijferential Equations o Beams, Ba1's, and GablesEnergy Expressions for Beams 168References 173P1oblems 17 4

UAPTER 8 THE TORSION OF SLENDER BODIES 788 1.8 2.8 3.8 4.8 5.8 6.8 7.

l nlroduction 178Prandtl Stress-}unction FormulationThe M embrar/e Analogy 185W arping-.function Formulation 188

179

Analytical Methods for Approximate SolutionsThin-walled Open Sections 198Thin-walled Closed Sections 202

8 8. Accuracy o Torsion Theory 213

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0. Differ ntial JJiqHa.tiontJ fm Minl lt l 'orqrw 21References 220Problems 220

CHAPTER 9 STRESSES DUE TO SHEAR N THIN WALLEDSLENDER BEAMS 2249 1. Introduction 2249 2. Open Sections 2259 3. Fluid-flow nalogy 2369 4. Shear Center 2399 5. Closed Sections 2409 6. Effects of Taper 2479 7. Transverse M ember Loads 253


CHAPTER 1 DEFLECTION ANALYSIS OF STRUCTURES 25810 1. Introduction 25810 2. The Method of Virtual Work 25810 3. Equations for U of Simple Elements 26310 4. Relative Displacements 27310 5. Flexibility and Stiffness Matrices 27510 6. Distributed Loads and Weighting Matrices 280


CHAPTER 11 STATICALLY INDETERMINATE STRUCTURES 28611 1. Introduction 28611 2. pplication of the Principle of the Stationary Value of

the Total Potentia l 28711 3. pplication of the Princip e of the Stationary Value of

the Total Complementary Potential 29011 4. Equations for U of Simple Elements 29411 5. Notes on a s ~ c and Redundant-force Systems 29511 6. Elastic-center and Column-analogy M ethods 304


CHAPTER 12 INTRODUCTION TO MATRIX METHODSOF STRUCTURAL ANALYSIS 31712 1. Introductwn 31712 2. The Force M ethod 31812 3. Discussion of the Force Method 323


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Jlpliln twn lo SlijJmNl h ll :l2l lt1 n ; p l a c l l l t < r t t M thod a : 3 : ~

/ ) i M I I t ~ s i o n of the D1spla em nt M ethodConr ludin l Jlemarks 344Nef r n e e . ~ 4 5J>robl ms 346

R 13 THE BENDING ND EXTENSION OF T IN PL TES 348J31. Introduction 34132. Geometry of the Reference Su1jace 349133. Stress Resultants 35413 4. Equilibrium Equations 354135 S t r a ~ n d i s p l a c e m e n t and Compatibility Equations 357136 Stress-Strain Equations 36013 7 Formulations of the Plate Equations 36113 8. Boundary Conditions 36613 9. The Differential Equations for Plates and Membranes 371

1310. The Navier Solution 37713 11. Strain Energy of Plates 37913 12. A pproximate M ethods 384


UA TER 14 PRIM RY BENDING INST BILITYND F ILURE OF COLUMNS 391

14 1. Introduction 39114 2. Small Dejlections of Lineady Elastic Perfect Columns 39214 3. App1oximate Methods 39914 4. Small Dejlections of Imperfect Elastic Columns 40714 5. Large Dejlections of Columns 41114 6. Inelastic Columns 41714 7. Empirical Column Equations 426


II PTER 15 INST BILITY ND F ILURE OF PL TES 4315 1 . Introduction 43015 2. F o r m u l a t i o ~ f the Buckling Problem 43015 3. Elastic uc ing of a Simply Supported Plate inUn iaxial Co pression 43 215 4. Buckling of Uniform Rectangular Plates with Simple

Edge Loadings 43415 5. Approximaie Methods 43915 6. Combined Loads and Interaction Curves 445

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157. Effecls of Large Dejlections and l n1.tial Imprf ections 44915 8. Inelastic Bucklin g of Plates 45515 9. The Failure of Plates 462


CHAPTER 6 INSTABILITY AND FAILURE OF THIN WALLED COLUMNSAND STIFFENED PLATES 46716 1. I nt1oduction 46716 2. S ondary Instability of Columns 46816 3. Crippling of Columns 47416 4. Failure of Th in walled Columns 47916 5. Compressive uckling of St iffened Panels 4816 6. Crippling of Stiffened Panels 48816 7. I nte1jastener uckling and W rinkling 49016 8. Failure of Stiffened Panels 497


Index 5 7


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Theory and nalyof Flight Structur


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tr duction

t T PS IN STRUCTURAL DESIGN' l 'lit pi OC s of design and analysis of flight structures may be divided into1 e following steps:1, ' l'IH d termination of the critical combinations of applied loads and

1 mperatures to which the structure is subjected.'1he htyout of the design in vvhich the arrangement, size, and materialsof th component parts of the structure are tentatively decided upon.,1 ' l'he d termination of the actual stresses and deformations in theHLructure due to the applied 1oads and temperatures.

1, ' l'he d termination of the allowable stresses or deformations of theHLructure.1, ' l ' li< omparison of steps 3 and 4 to determine whether the design of

Hl, 1 2 is adequate and efficient. f the design is either inadequateor v rdesigned (and therefore ine:fficient), steps 2 tt> 5 must bercp ated until a satisfactory design is obtained.Th steps form a successive-approximation procedure, for the loads

d 1 1 r p ratures of step 1 are functions of the details of the structuralcl 1 , which in turn depend upon the loads and temperatures. In the '' HLnp; . of the design process weights, loads, and temperatures areu llc ht H d upon crude estimates. These are continuously refined as thed 1 n pmgrosses and more accurate information becomes available.l11rin p; t.h arly phases, the methods of structural analysis are usuallyl 1 1d pon m p l i f i e ~ theories, as the expense and time necessary for more hd 'nLc m thods r o t justified until the loads and temperatures are1 n mor a curately. The structural analyst must therefore be capabl e111 c ll l ' i np; th range from educated guesses to sophisticated analyses.ll11 d nH ip;n Lhat finally evolves is a compromise involving structural aerotl\ ll nanio fnbrication, maintenance, and operational considerations.


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THEORY ND N LYSIS OF FLIGHT STRUCTURES

The substantiation of the final design is usually documented by thefollowing comprehensive reports which are submitted to the agencywhich is procuring or certifying the vehicle:1. A weight nd balance report, which gives the weights centers of gravitymass moments of inertia and weight distributions of the vehicle andeach of its major components.2. A loads report, which contains the aerodynamic weight and inertialforce distributions for each of the critical load conditions. Shearbending-moment torque and axial-load curves are also given for

major components.3 A structural-temperature report, which gives the temperature distribu

tions that occur simultaneously with the critical load conditions.4. A stress-analysis report, which substantiates the actual and allowablestresses and defiections for each of the critical load temperatureconditions for all components of the structure.5. An aeroelastic report, which gives the predicted speeds at which fiutterdivergence and control reversai will occur. The effects that struc-

tural deformations have upon air loads and control effectiveness arealso contained in this report.The structural-analysis group usually prepares the stress-analysis reportand assists in the preparation of the other reports.

1 2 PPLIED LO DS ND TEMPER TURESThe loads imposed upon the structure may be divided into two classesthose encountered on the ground and those in fiight. Ground loads arethose loads imposed during fabrication assembly shipping storage andhandling. In the case of missiles they include launch operations whilefor aircraft they involve the loads imposed by taxiing and landing. Flightloads are those loads applied to the structure during its fiight phase andinclude the lo ads imposed by maneuver gusts and wind shear. In missilesthey also involve the forces encountered during boost and staging operations. Temperatures are usually not significant in the ground-operationsphase but during the fiight phase they are often of equal or greater importance than the loads. This is especially true for fiight in the supersonic orhypersonic regimes. In sorne cases the structure may have to withstandthe aerodynamic loads imposed by passing through the subsonic tran-sonic supersonic and hypersonic phases of fiight. At the same time itmay be subjected to temperatures ranging from the extreme lows ofcryogenie fuels and radiation to space to the highs associated with aerodynamic heating hcat from the propulsion unit and radiation from thesun.


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1 n r ~ d mny nl o be cat gorized according to how they act up n th11111 l.tll'c. Surface j01ces are those forces which act upon the surfac f

1111 LttwLw , '.g., aerodynamic or hydrodynamic pressures, aerostatic orl 1hn t.n.t.i 6.ln cll'tl< t L provide for a separation between the limit loads and the1 . tt 1 L \\Il ieh the st"ucture fails, a factor of sajety is specified. This1 l1 " \dioh mu.y vary a_ccording to the mission of the vehicle, is usually1 l111 ' inllnl>iL d craft and may be as low as 1.25 for missiles. Therably lower than those used in civil or machin truc-' ' ' ' I,IH u H f such low factors of safety requires consid rabl ub-t fit il od .ion hy 1tnu.ly i and test.

'liu ltltimate load (sometimc n wn as the design load i d fin u 1:1


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4 THEORY ND N LYSIS OF FLIGHT STRUCTURES

the product of the limit load and the factor of safety. The failing loadultimate strength) of the structure should be only slightly greater thanthe ultimate load. t should be noted that in fiight structures the limitload is conventionally multiplied by the factor of safety. On the otherhand, in civil and machine structures the ultimate strength is usuallydivided by the factor of safety to give a working strength. Both methods,of course, give the same result. The ultimate load is often specified bygiving an ultimate-load factor nua which is equal to the product of thelimit-load factor and the factor of safety. The ultimate loads are thenobtained by multiplying the basic loads by the ultimate-load factor.

1-3 CTU L STRESSES ND DEFLECTIONSThe major portion of this book is devoted to methods of analysis for predicting the stresses and deflections of structural components underapplied loads and temperatures. In the mechanics of deformable bodiesit is usually necessary to introduce simplifying assumptions to arrive at asolution to the problem. The results achieved by using these assumptions must therefore be regarded as approximate, and it is possible toassess the degree of approximation only by knowing the nature andsignificance of the assumptions. Considerable effort is made in this textto underscore the assumptions and limitations of the theories discussed.In practice it is seldom that all the assumptions will be fulfilled, but it isonly by an intimate knowledge of the development of the theories thatthe equations can be intelligently applied to situations which do not precisely follow the conditions of the theory.

The approximations can be divided into physical and mathematicalcategories. Physical approximations are simplifying assumptions regarding the mechanical behavior of the material, the shape and proportions ofthe body, the manner in which it deforms or how the stresses are distributed , and the nature of the loading. :Yiathematical approximationsare often necessary in order to arrive at simple solutions or, in sorne cases,to obtain a solution at all. In many cases these mathematical approximations will also imply physicallimitations. For instance, if it is assumedfor mathematical convenience that the sine of an angle may be replaced bythe angle, the results will be acceptable only for small angles.In the physical approximations we usually replace the real deformable body and loads by a simple conceptional modeZ which embodies thesignificant characteristics of behavior of the real system. For instance,we may idealize the force-di placement behavio r of the material by one qfthe methods discu scd in hap. 3. We may also make assumptions ont he mode of d f rmati 11 of th o y. 1 or xamples in Chap. 7 weassum that }lan tOHH H Li ns f u b am romain plane and normal to


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Il t IIHIIHJC 1ON 51111 . iHof a am as it bends, and in Chap. 13 we assume that normal to1111 t n i d : - ~ u r f n r main normal to that surface as a plate deforms. In thelttd , or stiff ncd-shell structures we shall find it convenient to replace

1111 not. ual Hiructure by an idealized one having longitudinal stiffenerlltth t oHH, nly axial forces and thin webs which resist only shear forces.ltt' nluai the accuracy of these assumptions it is necessary to comparelitt t'tM tdL:; with those of more accurate theories or \vith experiments.

11 eomplex structures we shall subdivide the structure into simpler1 ' ' rd H for which methods of analysis exist. We view the compositeltttfLun 1 H nn assemblage of beams, shear webs, plates, shells, etc., and

d, 1 lop m il d of analysis for these simpler structural shapes.W H 11 v r possible we shaH treat the stresses and deformations

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mission of the vehicle, e.g. , those which prevent the free motion of movingparts or produce adverse dynamic or aeroelastic effects.It is usually specified that the material properties used in determining the allowable stresses and defiections be taken from Ref. 19 or that theproperties be substantiated by tests made by the materials manufactureror the contracter. These properties should refiect the temperature of thestructure and the duration of the load.

15 COMPARISON O APPLIED AND ALLOWABLE STRESSES AND DEFLECTIONSAs mentioned earlier, it is necessary to compare the applied and allowablestresses and defiections to determine whether the structure is effi.cientlydesigned. This is done by computing the margin of safety defined as

l\ :S = allowable load _ 1app ied loadFor the limit-load condition this becomes

L .t MS = yield load _ 1Inu applied limit loadand for the ultimate-load condition

Ultimate l\18 = ~ o l l a p s e load - 1apphed ult1mate loadThe smaller of these t o margins of safety con trois the design.

1-1)

(1-2)

(1-3)

In many cases the stresses are directly proportional to the loads (orare assumed soin the linear theories), and the word load can thereforebe replaced by stress in these equations. It is seen from Eq. (1-1) thatan effi.ciently designed structure vvoul have a very small positive marginof safety. Hm:vever, in sorne cases it may be desirable to have relativelylarge positive margins of safety to provide a growth potential for the craftso that increased performance or payload could be accommodated withoutredesigning the structure.

t has been pointed out that the design process is one of successiveapproximations until a satisfactory margin of safety is reached. In mostcases, and especially if the structure is unconventional, tests are performedto substantiate the analysis and prove the strength and stiffness of thestructure. A reduction in the structural weight of a fiight vehicle permits an increase in payload or performance. I t is therefore economicallyfeasible to use expensive materials and fabrication methods and to expendmany manhours of analy is and testing if it results in a decrease in structural weight.


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IMMA Y1111 1 1di K d H us i n giv only a cur ory introduction to the consid-

lthnL ont, r into the evolution of a structural design. 1 ro' ' ' ' ' L de HoripLions of th process may be found in Refs. 7 to 11. Tho

, l t l l t 11d1 l ' of LhiH L xt will addre s itself to the third and fourth t ps, 1 l11 11 i 1 H


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14. Glaser, P. E.: Aerodynamically Heated Structures, Prentice-Hall, loc., Englewood Cliffs, N.J., 1962.15. Bisplinghoff, R. L., H. Ashley, and R. L. Halfman: Aeroelasticity, AddisonWesley Publishing Company, Inc., Reading, Mass., 1955.16. Fung, Y. C.: An Introduction to the Theory of Aeroelasticity,'' John WileySons, Inc., New York, 1955.17. Manual on Aeroelasticity, NATO Advisory Group for Aeronautical Research

and Development, 1959.18. Bisplinghoff, R. L., and H. A. Ashley: Principles of Aeroelasticity, John WileySons, Inc., New York, 1962.19. Metallic Materials and Elements for Flight Vehicle Structures, Military and-book MIL-HDBK-5A, Feb. 8, 1966.

PRO LEMS1 1. A 600-lb satellite is mounted in the upper stage of a launch vehicle. During theboosted vertical-flight phase, a peak acceleration of 9g is reached. The satellite ismated to the booster by four bolts loaded in shear, each of which has an ultimateshear strength of 2126 lb. The specified factor of safety is 1.25. Determine a) thelimit load per bolt, b) the ultimate load per bolt, and c) the ultimate margin ofsafety. [Ans (a) 1500lb; b) 1875lb; c) 0.135.]1 2. The fuel tank of a vertically launched rocket contains kerosene (specifie gravity0.8) and is pressurized to 100 psig at a sea-level pressure of 14.7 psia. The peak boostacceleration of 9g occurs at an altitude where the ambient pressure is 5 psia and ata time when the depth of the unexpended fuel is 100 in. Determine the ultimatebursting pressure at the bottom of the tank at this time assuming an ultimate factorof safety of 1.25. [Ans Putt = 173 psi.]1-3. The nose of a cargo airplane is at a body station (BS) of 0 in. The loaded planeweighs 150,000 lb, and its center of gravity is at BS 250. The centers of pressure ofthe aerodynamic forces on the wing and tail are respectively at BS 200 and 550 Thefuselage is 600 in. long and together with its contents weighs a constant 150 lb/in.The tail weighs 2000 lb and has a center of gravity at BS 560. Determine the ultimate shear and bending moment in the fuselage at BS 200 for a limit trimmed (nopitching acceleration) maneuver load factor of n = 3g including gravity Assumea 1.5 factor of safety. [Ans Vult = 115,000 lb, ult = 8.7 X 106 in.-lb.]

B S 0Fig. P1 3

1wn

200 250 550 600

1 4. A 96,600-lb transport airplane has a mass moment of inertia of 48,300,000 lb-in.sec2 about a pitch axis passing through its center of gravity. During landing, when


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flllllllllf(, JI Nl 111d ynu rni e lift li O. tim s th w ight, it is uhj t t th groun l 1 ads sh wn.

f , 1 ' ' ' ' ' 11) t.ho limit-1 ad fa t r in th v rti al dir Lion at th c nt r f gravity11d /1) t,l lirnit pitcbing accel rati n in radians p r second per s ond. [Ans:1./i l; (b) = O 89 rad/ ec 2 .]

250,000 lbFig. Pl 4 and 15

11 1 Ir Ptro ni s package weighing 100 lb is located 400 in. aft of the center of gravity11 litt rt p nn f Prob. 1-4. Determine the ultimate verticalload that the packagetlfiJIIII 1 ltrJI. I ts are subjected to during landing. [Ans 664lb.]

Theory and Analysis of Flight Structures

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