Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

172 CHAPTER 6 Matrix Methods Bysuperpositionofthesetwoconditions,weobtainrelationshipsbetweentheappliedforcesandthe nodaldisplac ...

6.3Stiffness Matrix for Two Elastic Springs in Line 173 Finally,wesetu 1 =u 2 =0,u 3 =u 3 andobtain Fx,3=kbu 3 =−Fx,2 Fx,1= 0 } ...

174 CHAPTER 6 Matrix Methods andfortheelementconnectingnodes2and3as [K 23 ]= [ k 22 k 23 k 32 k 33 ] (6.16) Inourtwo-springsyste ...

6.3Stiffness Matrix for Two Elastic Springs in Line 175 ThefirststepistorewriteEq.(6.13)inpartitionedformas ⎧ ⎨ ⎩ Fx,1 Fx,2 Fx,3 ...

176 CHAPTER 6 Matrix Methods 6.4 MatrixAnalysisofPin-jointedFrameworks...................................................... The ...

6.4Matrix Analysis of Pin-jointed Frameworks 177 arbitraryreferenceaxesx,y.Weshallrefereverymemberoftheframeworktothisglobalcoor ...

178 CHAPTER 6 Matrix Methods where[T]isknownasthetransformationmatrix.Asimilarrelationshipexistsbetweenthesetsofnodal displaceme ...

6.4Matrix Analysis of Pin-jointed Frameworks 179 Hence, uj−ui=λ(uj−ui)+μ(vj−vi) SubstitutinginEq.(6.31)andrewritinginmatrixform, ...

180 CHAPTER 6 Matrix Methods directioncosinesλandμtakedifferentvaluesforeachofthethreemembers,sorememberingthatthe angleθismeasu ...

6.4Matrix Analysis of Pin-jointed Frameworks 181 wheretheyoverlap;forexample,the[k 11 ]submatrixinEq.(ii)receivescontributionsfr ...

182 CHAPTER 6 Matrix Methods Ifwenowdeleterowsandcolumnsinthestiffnessmatrixcorrespondingtozerodisplacements,we obtain the unkno ...

6.6 Matrix Analysis of Space Frames 183 Finally,theforcesinthemembersarefoundfromEqs.(6.32),(vii),and(viii) S 12 = AE L [1 0] { ...

184 CHAPTER 6 Matrix Methods componentsFx,i,Fy,i,Fz,iandFx,j,Fy,j,Fz,j,respectively.Thememberstiffnessmatrixreferredtoglobal coo ...

6.7Stiffness Matrix for a Uniform Beam 185 orinabbreviatedform {F}=[T]{F} Thederivationof[Kij]foramemberofaspaceframeproceedsoni ...

186 CHAPTER 6 Matrix Methods Fig.6.6 Forces and moments on a beam element. vi,vj,andθi,θj.Wedonotincludeaxialforceshere,sincethe ...

6.7Stiffness Matrix for a Uniform Beam 187 whichisoftheform {F}=[Kij]{δ} where[Kij]isthestiffnessmatrixforthebeam. It is possibl ...

188 CHAPTER 6 Matrix Methods wehave,fromEqs.(6.45)and(6.46), [Kij]=EI ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 12 μ^2 /L^3 SYM − 12 λμ/L^312 λ^2 /L ...

6.7Stiffness Matrix for a Uniform Beam 189 Fig.6.7 Idealization of a beam into beam elements. Fig.6.8 Idealization of a beam sup ...

190 CHAPTER 6 Matrix Methods Fig.6.10 Assemblage of two beam elements. fromEq.(6.44);thus, [K 12 ]=EIa v 1 θ 1 v 2 θ 2 ⎡ ⎢ ⎢ ⎢ ⎢ ...

6.7Stiffness Matrix for a Uniform Beam 191 Example 6.2 DeterminetheunknownnodaldisplacementsandforcesinthebeamshowninFig.6.11.Th ...