Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

192 CHAPTER 6 Matrix Methods SubstitutingEq.(v)intoEq.(iii),weobtain { v 2 θ 2 L } = L^3 9 EI [ − 4 − 2 − 23 ]{ −W M/L } (vi) fr ...

6.8 Finite Element Method for Continuum Structures 193 6.8 FiniteElementMethodforContinuumStructures............................ ...

194 CHAPTER 6 Matrix Methods Thesolutionprocedureisidenticalinoutlinetothatdescribedintheprevioussectionsforskeletal structures; ...

6.8 Finite Element Method for Continuum Structures 195 FromEqs.(6.54)and(6.56),wecanwritedownexpressionsforthenodaldisplacements ...

196 CHAPTER 6 Matrix Methods whichwewriteas {ε}=[C]{α} (6.65) Substitutingfor{α}inEq.(6.65)fromEq.(6.60),wehave {ε}=[C][A−^1 ]{δ ...

6.8 Finite Element Method for Continuum Structures 197 Thetotalpotentialenergyofthebeamhasastationaryvaluewithrespecttothenodald ...

198 CHAPTER 6 Matrix Methods Hence, [Ke]= ∫L 0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ − 6 L^2 + 12 x L^3 − 4 L + 6 x L^2 6 L^2 − 12 x L^3 − 2 L ...

6.8 Finite Element Method for Continuum Structures 199 Fig.6.13 Triangular element for plane elasticity problems. thattheinverse ...

200 CHAPTER 6 Matrix Methods compatibilityofdisplacementalongtheedgesofadjacentelements.WritingEqs.(6.82)inmatrixform gives { u( ...

6.8 Finite Element Method for Continuum Structures 201 FromEqs.(1.18)and(1.20),weseethat εx= ∂u ∂x εy= ∂v ∂y γxy= ∂u ∂y + ∂v ∂x ...

202 CHAPTER 6 Matrix Methods Thus,inmatrixform, {ε}= ⎧ ⎨ ⎩ εx εy γxy ⎫ ⎬ ⎭ = 1 E ⎡ ⎣ 1 −ν 0 −ν 10 002 ( 1 +ν) ⎤ ⎦ ⎧ ⎨ ⎩ σx σy τx ...

6.8 Finite Element Method for Continuum Structures 203 asinEq.(6.74),fromwhich [Ke]= ⎡ ⎣ ∫ vol [B]T[D][B]d(vol) ⎤ ⎦ Inthisexpres ...

204 CHAPTER 6 Matrix Methods thatis, u 3 =α 1 + 2 α 2 + 2 α 3 (iii) FromEq.(i), α 1 =u 1 (iv) andfromEqs.(ii)and(iv), α 2 = u 2 ...

6.8 Finite Element Method for Continuum Structures 205 Also, [D]= ⎡ ⎣ ab 0 ba 0 00 c ⎤ ⎦ Hence, [D][B]= 1 4 ⎡ ⎣ −a −ba−b 02 b −b ...

206 CHAPTER 6 Matrix Methods Fig.6.14 Quadrilateral element subjected to nodal in-plane forces and displacements. Asinthecaseoft ...

6.8 Finite Element Method for Continuum Structures 207 or { u(x,y) v(x,y) } =[f(x,y)]{α} (6.98) Now,substitutingthecoordinatesan ...

208 CHAPTER 6 Matrix Methods IfYoung’smodulusE=200000N/mm^2 andPoisson’sratioν=0.3,calculatethestressesatthecenter oftheelement. ...

6.8 Finite Element Method for Continuum Structures 209 Now,substitutingforα 1 ,α 2 ,...,α 8 inEqs.(6.96), ui=0.00025−0.000125x−0 ...

210 CHAPTER 6 Matrix Methods Fig.6.15 Tetrahedron and rectangular prism finite elements for three-dimensional problems. thatis, ...

Problems 211 [5] Jenkins, W.M.,Matrix and Digital Computer Methods in Structural Analysis, McGraw-Hill Publishing Co. Ltd.,1969. ...