Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

232 CHAPTER 7 Bending of Thin Plates conditions for a free edge of (Mxy)x= 0 =0 and (Qx)x= 0 =0 are therefore replaced by the eq ...

7.3 Plates Subjected to a Distributed Transverse Load 233 inwhichmrepresentsthenumberofhalfwavesinthexdirectionandnrepresentsthe ...

234 CHAPTER 7 Bending of Thin Plates Thisequationisvalidforallvaluesofxandysothat Amn [( mπ a ) 4 + 2 (mπ a ) 2 (nπ b ) 2 + (nπ ...

7.3 Plates Subjected to a Distributed Transverse Load 235 SubstitutionforwfromEq.(i)intotheexpressionsforbendingmoment,Eqs.(7.7) ...

236 CHAPTER 7 Bending of Thin Plates Differentiatingthedeflectionfunctiongives ∂^4 w ∂x^4 = 0 ∂^4 w ∂y^4 = 0 ∂^4 w ∂x^2 ∂y^2 = 4 ...

7.4 Combined Bending and In-Plane Loading of a Thin Rectangular Plate 237 Fig.7.12 In-plane forces on plate element. onlytoinclu ...

238 CHAPTER 7 Bending of Thin Plates Fig.7.13 Component of shear loads in thezdirection. The determination of the contribution o ...

7.4 Combined Bending and In-Plane Loading of a Thin Rectangular Plate 239 Thetotalforceinthezdirectionisfoundfromthesummationoft ...

240 CHAPTER 7 Bending of Thin Plates Theseconditionsmaybesatisfiedbytheassumptionofadeflectedformoftheplategivenby w= ∑∞ m= 1 ∑∞ ...

7.6 Energy Method for the Bending of Thin Plates 241 Theeffectofaninitialcurvatureondeflectionisthereforeequivalenttotheapplicat ...

242 CHAPTER 7 Bending of Thin Plates Fig.7.14 (a) Strain energy of element due to bending; (b) strain energy due to twisting. we ...

7.6 Energy Method for the Bending of Thin Plates 243 Thetotalstrainenergyoftheelementfrombendingandtwistingisthus 1 2 ( −Mx ∂^2 ...

244 CHAPTER 7 Bending of Thin Plates 7.6.3 Potential Energy of In-Plane Loads We may consider each loadNx,Ny,andNxyin turn, and ...

7.6 Energy Method for the Bending of Thin Plates 245 andsince∂w/∂xissmall,then δa≈δx [ 1 + 1 2 ( ∂w ∂x ) 2 ] Hence, a= ∫a′ 0 [ 1 ...

246 CHAPTER 7 Bending of Thin Plates Fig.7.16 Calculation of shear strain corresponding to bending deflection. reduction in the ...

7.6 Energy Method for the Bending of Thin Plates 247 ItfollowsimmediatelythatthepotentialenergyoftheNxyloadsis Vxy=− 1 2 ∫a 0 ∫b ...

248 CHAPTER 7 Bending of Thin Plates Thetermmultipliedby2( 1 −ν)integratestozero,andthemeanvalueofsin^2 orcos^2 overacomplete nu ...

7.6 Energy Method for the Bending of Thin Plates 249 Example 7.4 Arectangularplatea×b,issimplysupportedalongeachedgeandcarriesau ...

250 CHAPTER 7 Bending of Thin Plates In this chapter, we dealt exclusively with small deflections of thin plates. For a plate su ...

Problems 251 whereDistheflexuralrigidity,νisPoisson’sratio,andAisaconstant. CalculatethevalueofAandhencethe centraldeflectionoft ...