Torsion of CylindersGuided by Observationsfrom Strength of Materials:. Projection of each section on x-yplane rotates as rigid-body aboutcentral axis: Amount of projected sectionrotation is linear function of axialcoordinate: Plane cross-sections will not remainplane after deformation thus leadingto a warping displacement6
x y z T ℓ S R Guided by Observations from Strength of Materials: Torsion of Cylinders 6 • Projection of each section on x-y plane rotates as rigid-body about central axis • Amount of projected section rotation is linear function of axial coordinate • Plane cross-sections will not remain plane after deformation thus leading to a warping displacement
Torsional Deformation. In-plane / projected displacementsu=-rβsin=-βy, =rβcos=βxS· Angleoftwist: β=αzOX. The warping displacement is assumed to beRa function of only the in-plane coordinates=u=-αyz, v=αxz, w=w(x,y): Now must show assumed displacement formwill satisfy all elasticity field equations7
u r y vr x =− =− = = β θ β β θβ sin , cos β α= z. ⇒ =− = = u yz v xz w w x y α α , , ( , ). Torsional Deformation • In-plane / projected displacements • Angle of twist: • The warping displacement is assumed to be a function of only the in-plane coordinates 7 • Now must show assumed displacement form will satisfy all elasticity field equations
Stress Function Formulation. Strain and stress field=0=6,=8,,=0Ta8,=8,=0axyVLu=-αyzowow1.V=αxzL2axOxw=w(x,y)ow1ow=G8ax+Tax+V2V22ayay: Two stress components satisfy. Equilibrium equations result inatyeotyagatot,-2GαVY=0ayaxaxOzdyayayayay2GαUTax?ay2Xzdyax: Stress compatibility isy=y(x,y) Prandtl Stress Functionautomatically satisfied8
0 0 1 2 (, ) 1 2 x y z xy x y z xy xz xz yz yz u yz w w v xz y G y x x w wxy w w x G x y y εεεε σσστ α α εα τα εα τα = = = = = = = = = − ∂ ∂ = ⇒ =−+ ⇒ = −+ ∂ ∂ = ∂ ∂ =+ =+ ∂ ∂ Stress Function Formulation • Strain and stress field • Equilibrium equations result in yz z xz x y ∂τ ∂τ ∂σ + + ∂ ∂ F z z + ∂ 0 , xz yz y x ψ ψ τ τ = ∂ ∂ ⇒ = =− ∂ ∂ • Two stress components satisfy 2 2 2 2 2 2 2 xz yz G y x G x y τ τ α ψ ψ ψ α ∂ ∂ − =− ∂ ∂ ∂ ∂ ⇒ ∇ = + =− ∂ ∂ • ψ = ψ(x,y) • Prandtl Stress Function 8 • Stress compatibility is automatically satisfied
Stress Function Formulation: BCs.On lateral surfaceT"=gin,+Van,+t=0=0=0T=n,+,n,+t=0=0=0=0=T"=tn+tyn,+o.aydxdyay dy0=0=i.e. set. y = 0axdsdsOy ds On ends: T" =±tx, T" =±ty, T" =o, =0 More interested in satisfying the resultantend-loadingsdyds: 0 = P, = [L, trdxdy,: 0= P, = [, TydxdydxSdydx:0= P,= J,.dxdy,:0=M,=JJ,y9.dxdy1dsds: 0 = M, = [L, xg dxdy, :T = M, = [L(xt,y- - yTx-)dxdyx
n T x x = σ x yx n + τ y zx z n n +τ 0 00 n T y xy τ =⇒= = x y n + σ y zy z n n +τ 0 00 n T nn n z xz x yz y z z ττσ =⇒= =++ i.e. 0 0 0, set: 0. dy dx d y ds x ds ds ψ ψ ψ ψ = ⇒ ∂ ∂ +− − = ⇒ = ∂ ∂ = 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 ∫∫ , 6: z ( yz xz ) A A dxdy T M x y dxdy = = − τ τ ∫∫ ∫∫ Stress Function Formulation: BCs • On lateral surface • On ends: • More interested in satisfying the resultant end-loadings , 0 nnn TTT x xz y yz z z =± =± = = τ τσ , x y dy dx n n ds ds = = − x y z T ℓ S R n
Stress Function Formulation: BCs. On ends (with a big help from the Green's theorem)- are automatically satisfied.: J, tdxdy= J,% dxdy =-],wdx= J,wn,ds = 0satisfied J,,dxdy = -J, % dxdy = - ], vdy =-J, yn,ds = 0 satisfieda(xy)a(yyayay: T = J,(x - ) dxdy = -JJdxdy :dxdyOxayayaxT=2], ddy= -[(-μdx+ xydy) +2[[ y dxdy =The assumed stress function yields a governing Poisson equationThe stress function vanishes on the lateral boundary.The overall torque is related to the integral of the stress Function. The remaining end conditions are automatically satisfied10
( ) ( ) 3 5 are automatically satisfied. 1 : 0 satisfied 2 : 0 satisfied 6 : xz A A SS yz A A SS yz xz y A x A dxdy dxdy y d dx n ds xdy dxdy d x x T x y dxdy x y dxdy y n ds xy x ψ τ ψ ψ ψ τ ψ ψ ψ ψ ψ τ τ − ∂ = = = = ∂ ∂ =− =− =− = ∂ ∂ ∂ ∂ = − =− + =− + ∂ ∂ − ∂ ∫∫ ∫∫ ∫ ∫ ∫∫ ∫∫ ∫ ∫ ∫∫ ∫∫ ( ) ( ) 2 A S y dxdy y y dx x dy ψ ψ ψ ψ ∂ − ∂ =− − + ∫∫ ∫ 2 2 A A + ⇒= ψ ψ dxdy T dxdy ∫∫ ∫∫ Stress Function Formulation: BCs • On ends (with a big help from the Green’s theorem) • The assumed stress function yields a governing Poisson equation. • The stress function vanishes on the lateral boundary. • The overall torque is related to the integral of the stress Function. • The remaining end conditions are automatically satisfied. 10