Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

52 CHAPTER 2 Two-Dimensional Problems in Elasticity Further, ∂^4 φ ∂x^4 = 0 ∂^4 φ ∂y^4 =− 120 Dy ∂^4 φ ∂x^2 ∂y^2 = 60 Dy Substit ...

2.4 St. Venant’s Principle 53 FromEq.(vi) B+ 15 Dh^2 =0(ix) sothat,subtractingEq.(viii)fromEq.(ix) D=− q 40 h^3 Then B= 3 q 8 h ...

54 CHAPTER 2 Two-Dimensional Problems in Elasticity Fig.2.4 Stress distributions illustrating St. Venant’s principle. We may, th ...

2.6 Bending of an End-Loaded Cantilever 55 Fig.2.5 Displacements produced by rigid body rotation. 2.6 BendingofanEnd-LoadedCanti ...

56 CHAPTER 2 Two-Dimensional Problems in Elasticity Fig.2.6 Bending of an end-loaded cantilever. giving A=− Bb^2 8 Fromthesecond ...

2.6 Bending of an End-Loaded Cantilever 57 (2) the distribution of shear and direct stresses at the built-in end is the same as ...

58 CHAPTER 2 Two-Dimensional Problems in Elasticity and ∂f 2 (x) ∂x = Px^2 2 EI +C ∂f 1 (y) ∂y = Py^2 2 IG − νPy^2 2 EI +D sotha ...

2.6 Bending of an End-Loaded Cantilever 59 Thedeflectioncurvefortheneutralplaneis (v)y= 0 = Px^3 6 EI − Pl^2 x 2 EI + Pl^3 3 EI ...

60 CHAPTER 2 Two-Dimensional Problems in Elasticity Fig.2.8 (a) Distortion of cross section due to shear; (b) effect on distorti ...

Problems 61 Fig. P.2.2 asanAirystressfunctionanddeterminethecoefficientsA,B,andC. Ans. A= 2 Pl/td^3 , B=− 2 P/td^3 , C= 3 P/ 2 t ...

62 CHAPTER 2 Two-Dimensional Problems in Elasticity Fig. P.2.4 AsasolutiontothestressanalysisproblemanAirystressfunctionφispropo ...

Problems 63 Ans. Thestressfunctionsatisfiesthebiharmonicequation.Thebeamisacantileverunderauniformlydis- tributedloadofintensity ...

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CHAPTER 3 Torsion of Solid Sections............................................................... Theelasticitysolutionofthetor ...

66 CHAPTER 3 Torsion of Solid Sections Fig.3.1 Torsion of a bar of uniform, arbitrary cross section. which identically satisfies ...

3.1Prandtl Stress Function Solution 67 where∇^2 isthetwo-dimensionalLaplacianoperator ( ∂^2 ∂x^2 + ∂^2 ∂y^2 ) Therefore,theparam ...

68 CHAPTER 3 Torsion of Solid Sections Thus,φisconstantonthesurfaceofthebar,andsincetheactualvalueofthisconstantdoesnotaffect th ...

3.1Prandtl Stress Function Solution 69 Fig.3.3 Derivation of torque on cross section of bar. Therefore,weareinapositiontoobtaina ...

70 CHAPTER 3 Torsion of Solid Sections Fig.3.4 Rigid body displacement in the cross section of the bar. or u=−θyv=θx (3.9) Refer ...

3.1Prandtl Stress Function Solution 71 ItisconvenienttointroduceatorsionconstantJdefinedbythegeneraltorsionequation T=GJ dθ dz ( ...