Displacement FormulationExpressing the equilibrium equation with warping displacementhGu +(a+G)F=0satisfiedG+(a+Gsatisfied-0a?wa2OGV*w+(a+G)（器+%ax?ayBCs on the lateral surface:owowO=T"=txn.+tyn,+o.y.n,+GnoxOxOyowOwForarbitraryn:-αxαydydsdyaxdxawawdxdySorequivalentlyaxnEαl+αx=αVdydxdsdsoxdsdsdwQx2dn11
0 n T nn n z xz x yz y z z == + + ττσ ( ) 2 2 For arbitrary n: , or equivalently: 2 x y x y x y w w n n x y w w G y nG dy dx x n x y w dw d w y x x y yn y x ds ds d x y xn n ds α α α α α α α α α + ∂ ∂ = −+ + + ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ = = − ∂ ∂ =−= ⇒ = + Displacement Formulation • Expressing the equilibrium equation with warping displacement • BCs on the lateral surface: , x y dy dx n n ds ds = = − 2 G u ∇ ( ) u G x x λ ∂ ∂ + + ∂ ∂ v y ∂ + ∂ w z ∂ + ∂ F x + 2 0 satisfied G v = ∇ ( ) u G y x λ ∂ ∂ + + ∂ ∂ v y ∂ + ∂ w z ∂ + ∂ F y + ( ) 2 0 satisfied u Gw G z x λ = ∂ ∂ ∇+ + ∂ ∂ v y ∂ + ∂ w z ∂ + ∂ F z + 2 2 2 2 0 0 w w x y ∂ ∂ =⇒ + = ∂ ∂ 11
Displacement Formulation: BCs on the ends: 0 = P = (Ldxdy,: 0 = P, = [L tydxdy:0 = P, = L, 9, dxdy, 4: 0 = M,= [L, y9 dxdy:0 = M, = JL, xg dxdy, : T = M, = JL(xt, - y-)dxdy- are automatically satisfied.owow: T = M, = JL (xty - yx)dxdy= |dxdxXxaxdVowowT=(dxdy = αJaxdyyowx owJ=GIx+dxdy..Torsional Rigidityα dyα ax12
1 :0 , 2 :0 3 :0 x xz y yz A A z z P dxdy P dxdy P τ τ σ = = = = = = ∫∫ ∫∫ , 4 :0 x z A dxdy M y = = σ ∫∫ 5 :0 A y z dxdy = = M xσ ∫∫ ( ) ( ) 2 2 2 2 , 6: 1 5 are automatically satisfied. 6 : ( ) z yz xz A A z yz xz A A A dxdy T M x y dxdy w w T M x y dxdy xG x yG y dxdy y x w w T G x y x y dxdy J y x xw yw JG x y d y x τ τ τ τ α α α α α α = = − − ∂ ∂ = = − = + − −+ ∂ ∂ ∂ ∂ ⇒= + + − = ∂ ∂ ∂ ∂ = ++ − ∂ ∂ ∫∫ ∫∫ ∫∫ ∫∫ ∫∫ . . . Torsional Rigidity A xdy ∫∫ • BCs on the ends Displacement Formulation 12