Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

72 CHAPTER 3 Torsion of Solid Sections whichmaybewritten,fromFig.3.5,as τzx=− ∂φ ∂x dx dn − ∂φ ∂y dy dn =− ∂φ ∂n (3.16) where, i ...

3.1Prandtl Stress Function Solution 73 whichgives T= πR^4 2 G dθ dz thatis, T=GJ dθ dz (iii) inwhichJ=πR^4 / 2 =πD^4 /32(Disthed ...

74 CHAPTER 3 Torsion of Solid Sections Fig.3.6 Torsion of a bar of elliptical cross section. Thesemimajorandsemiminoraxesareaand ...

3.2St. Venant Warping Function Solution 75 bythesevaluesgives T=G dθ dz πa^3 b^3 (a^2 +b^2 ) (iv) fromwhich(seeEq.(3.12)) J= πa^ ...

76 CHAPTER 3 Torsion of Solid Sections theiroriginalunloadedshape,althoughtheymaysufferwarpingdisplacementsnormaltotheirplane. T ...

3.3 The Membrane Analogy 77 simplifiesto ( ∂ψ ∂y +x ) m+ ( ∂ψ ∂x −y ) l= 0 (3.21) Itmaybeshown,butnotaseasilyasinthestressfuncti ...

78 CHAPTER 3 Torsion of Solid Sections Fig.3.7 Membrane analogy: in-plane and transverse loading. Fig.3.8 Equilibrium of element ...

3.4 Torsion of a Narrow Rectangular Strip 79 and q N =−F= 2 G dθ dz Theanalogynowbeingestablished,wemaymakeseveralusefuldeductio ...

80 CHAPTER 3 Torsion of Solid Sections Fig.3.9 Torsion of a narrow rectangular strip. Althoughφdoesnotdisappearalongtheshortedge ...

3.4 Torsion of a Narrow Rectangular Strip 81 solutionincreasesass/tdecreases.Therefore,inordertoretaintheusefulnessoftheanalysis ...

82 CHAPTER 3 Torsion of Solid Sections Weshouldnotclosethischapterwithoutmentioningalternativemethodsofsolutionofthetorsion prob ...

Problems 83 Fig. P.3.4 τmax(atcenterofeachside)=− a 2 G dθ dz dθ dz = 15 √ 3 T Ga^4 w= 1 2 a dθ dz (y^3 − 3 x^2 y). P.3.5 Determ ...

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CHAPTER 4 Virtual Work and Energy Methods..................................................... Manystructuralproblemsarestatical ...

86 CHAPTER 4 Virtual Work and Energy Methods Fig.4.1 Work done by a force and a moment. Therefore,weseethattheworkdonebytheforce ...

4.2 Principle of Virtual Work 87 4.2.1 Principle of Virtual Work for a Particle InFig.4.2,aparticle,A,isactedonbyanumberofconcur ...

88 CHAPTER 4 Virtual Work and Energy Methods Iftheparticle,A,isinequilibriumundertheactionoftheforces,F 1 ,F 2 ,...,Fk,...,Fr,th ...

4.2 Principle of Virtual Work 89 internalforceswillbeequalandopposite,inotherwordsself-equilibrating.Supposenowthattherigid body ...

90 CHAPTER 4 Virtual Work and Energy Methods Fig.4.4 Use of the principle of virtual work to calculate support reactions. ...

4.2 Principle of Virtual Work 91 whichgives RC=W a L whichistheresultthatwouldhavebeenobtainedfromaconsiderationofthemomentequil ...