Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

272 CHAPTER 8 Columns WehaveseeninChapter7thatexactsolutionsofplatebendingproblemsareobtainablebyenergy methodswhenthedeflecteds ...

8.5 Energy Method for the Calculation of Buckling Loads in Columns 273 stiffer,andoffersgreaterresistancetobuckling,asweobservef ...

274 CHAPTER 8 Columns Integratingandsubstitutingthelimits,wehave U+V= 17 35 PCR^2 v^20 l 2 EI − 3 5 PCR v^20 l Hence, ∂(U+V) ∂v ...

8.6 Flexural–Torsional Buckling of Thin-Walled Columns 275 DifferentiatingEq.(8.53)twicewithrespecttozgives EIxx d^4 v dz^4 =−PC ...

276 CHAPTER 8 Columns Fig.8.16 Flexural–torsional buckling of a thin-walled column. Fromsimplebeamtheory(Chapter15) EIxx d^2 v d ...

8.6 Flexural–Torsional Buckling of Thin-Walled Columns 277 pointBwillbedisplacedtoB′′.ThehorizontalmovementofBinthexdirectionist ...

278 CHAPTER 8 Columns ExpandingEq.(8.66)andnotingthatσisconstantoverthecrosssection,weobtain T(z)=−σ d^2 u dz^2 yS ∫ Sect tds+σ ...

8.6 Flexural–Torsional Buckling of Thin-Walled Columns 279 Anassumedbuckledshapegivenby u=A 1 sin πz L v=A 2 sin πz L θ=A 3 sin ...

280 CHAPTER 8 Columns Example 8.3 Athin-walledpin-endedcolumnis2mlongandhasthecrosssectionshowninFig.8.17.Iftheends ofthecolumna ...

8.6 Flexural–Torsional Buckling of Thin-Walled Columns 281 thatis, I 0 =3.52× 105 +0.22× 105 =3.74× 105 mm^4 ThetorsionconstantJ ...

282 CHAPTER 8 Columns Fig.8.18 Column section of Example 8.4. The position of the shear center S is found using the method of Ex ...

8.6 Flexural–Torsional Buckling of Thin-Walled Columns 283 Itcanbeseenthatthisvalueofflexural–torsionalbucklingloadislowerthanan ...

284 CHAPTER 8 Columns Fig.8.20 Determination of torsion bending constant for column section of Example 8.5. Thetorsionbendingcon ...

8.6 Flexural–Torsional Buckling of Thin-Walled Columns 285 Theboundaryconditionsareθ=0whenz=0andz=L,andsincethewarpingissuppress ...

286 CHAPTER 8 Columns Sinceθ=0atz=l, cosμL= 1 or μL= 2 nπ Therefore, μ^2 L^2 = 4 n^2 π^2 or σI 0 −GJ E = 4 n^2 π^2 L^2 Thelowes ...

Problems 287 Problems.............................................................................................. P.8.1 Thesys ...

288 CHAPTER 8 Columns P.8.4 A uniform, pin-ended column of lengthland bending stiffnessEIhas an initial curvature such that the ...

Problems 289 Fig. P.8.6 P.8.7 Therelationbetweenstressσandstrainεincompressionforacertainmaterialis 10.5× 106 ε=σ+ 21000 ( σ 490 ...

290 CHAPTER 8 Columns Fig. P.8.9 where α=P/Pe, Pe=π^2 EI/l^2 Assumetissmallcomparedwithdsothatthefollowingrelationshipsareapplic ...

Problems 291 P.8.12 Atubularcolumnoflengthlistaperedinwall-thicknesssothattheareaandthesecondmomentofarea ofitscrosssectiondecre ...