Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

452 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams FollowingthemethodofExample15.9,wedeterminethesupportreactionsandfi ...

15.3 Deflections due to Bending 453 fromwhich C 1 =− 27 wL^3 2048 Equations(iv)and(v)thenbecome EIv′= 3 64 wLz^2 − w 6 [ z− L 2 ...

454 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams Inthisproblem,anexternalmomentM 0 isappliedtothebeamatB.Thesupportr ...

15.3 Deflections due to Bending 455 Fig.15.24 Determination of the deflection of a cantilever. Example 15.12 Determine the horiz ...

456 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams Atthefreeendofthecantilever(z=L),thehorizontalcomponentofdeflection ...

15.4 Calculation of Section Properties 457 15.4.2 Theorem of Perpendicular Axes InFig.15.26,thesecondmomentsofarea,IxxandIyy,oft ...

458 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams whichgives Ixx= bd^3 12 (15.35) Similarly, Iyy= db^3 12 (15.36) Fre ...

15.4 Calculation of Section Properties 459 Fig.15.29 Second moments of area of a circular section. Theequivalenceofthesetwoexpre ...

460 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams whichgives Ixx= πd^4 64 (15.39) Clearlyfromsymmetry Iyy= πd^4 64 (1 ...

15.4 Calculation of Section Properties 461 entiresectionoftheproductsecondmomentofareaofallsuchpairsofelementsresultsinazerovalu ...

462 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams Fig.15.31 (a) Actual thin-walled channel section; (b) approximate r ...

15.4 Calculation of Section Properties 463 ahorizontalaxisthroughitscentroidisgivenby Ixx= 2 ∫a/^2 0 ty^2 ds= 2 ∫a/^2 0 t(ssinβ) ...

464 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams Fig.15.33 Second moment of area of a semicircular section. fromwhic ...

15.4 Calculation of Section Properties 465 FromEq.(15.19), σz= Mx(Iyyy−Ixyx) IxxIyy−Ixy^2 (i) Thesectionpropertiesarecalculateda ...

466 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams Fig.15.35 Distribution of direct stress inZ-section beam of Example ...

15.6 Temperature Effects 467 Fig.15.36 Beam section subjected to a temperature rise. Wehaveseenthatthestrainproducedbyatemperatu ...

468 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams We have noted that beam sections in aircraft structures are general ...

15.6 Temperature Effects 469 Fig.15.37 Beam section of Example 15.15. Fig.15.38 Thin-walled beam section subjected to a varying ...

470 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams Thedirectstressonanelementδsinthewallofthesectionisthen,fromEq.(15. ...

Problems 471 FromEqs.(15.50)and(15.53), MxT=Eα 2 T 0 at(a)+ ∫a −a Eα T 0 a (a+y)ytds thatis, MxT=EαT 0 { 2 a^2 t+ 1 a [ ay^2 2 + ...