Material and Computational Mechanics Group 1 L4. Torsion of beams: St. Venant (uniform) torsion, CCSM: chap 10.1, 10.2.1-2 à Characteristics of torsion problem ü Angle of twist - warping ü Kinematics of torsion à St. Venant and Vlasov torsion à St. Venant torsion ü Kinematics ü Stress - strain relation (Hooke's law) ü Equilibrium ü Prandtl's stress function ü Governing equations
35
Embed
L4. Torsion of beams: St. Venant (uniform) torsion, CCSM: chap …ragnar/ship_structures_home/lectures/... · 2005. 9. 16. · Material and Computational Mechanics Group 9 L5. Torsion
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Material and Computational Mechanics Group 1
L4. Torsion of beams: St. Venant (uniform) torsion, CCSM: chap 10.1, 10.2.1-2
àCharacteristics of torsion problem
üAngle of twist - warping
üKinematics of torsion
à St. Venant and Vlasov torsion
à St. Venant torsion
üKinematics
ü Stress - strain relation (Hooke's law)
ü Equilibrium
ü Prandtl's stress function
üGoverning equations
Material and Computational Mechanics Group 2
àCharacteristics of torsion problem
üAngle of twist - warping
üKinematics of torsion
Assume rigid cross section and twist-rotation about the SC (TC): From geometry
ur=rθ@xD, w =ur cos@αDv = −ur sin@αD, y =rcos@αD
z =rsin@αD=⇒w@x, yD=y θ@xD, v@x, zD= −zθ@xDNote!
x =JyzN; x̂ =J−z
yNwithx⋅x̂ =0 ⇒u=Jv
wN= θ@xD J−z
yN= θ@xD x̂
The induced strain state now becomes:
εx= ∂ux∂x
; εy = ∂v∂y
:=0; εz = ∂w∂z
:=0
γxy =∂ux∂y
+∂v∂x
=∂ux∂y
−zθ
γxz = ∂ux∂z
+ ∂w∂x
= ∂ux∂z
+y θ
γyz =∂v∂z
+∂w∂y
= − θ@xD+θ@xD:=0
24
25
26
Material and Computational Mechanics Group 3
à St. Venant and Vlasov torsion
Consider characteristics of torsion depending on the kinematics:
1) St. Venant (uniform) torsion is obtained for cross sections who preserve their shapealong the beam during torsion
γyz = εy = εz = εx = ∂ ux∂x
= 0
γxy = ∂ux∂y − z θ ≠ 0
γxz = ∂ux∂z + y θ ≠ 0
= or J γxy
γxzN =
ikjjjj ∂ux∂y
∂ux∂z
y{zzzz + θ J −zy
N or γ = ∇ ux + θ x̂
Note! the gradient operator ∇ is associated with postion x, i.e.
∇ ⋅ x = 2 and ∇ ⋅ x̂ = 0
2) Vlasov (nonform) torsion is obtained for cross sections who change their shape alongthe beam during torsion
γyz = εy = εz = 0 ; γ = ∇ ux + θ x̂ := 0
εx = ∂ ux∂ x
≠ 0
Note! Vlasov torsion will be considered later on. Cf. examples of cross sections
26
Material and Computational Mechanics Group 4
à St. Venant torsion
üKinematics
Let us restate the kinematics as: only out-of-plane shear deformations are assumed tooccur
γxy = ∂ ux∂y
− z θ ≠ 0, γxz = ∂ ux∂ z
+ y θ ≠ 0 or γ = ∇ux + θ x̂
ü Stress - strain relation (Hooke's law)
Note! due to kinematics
γyz = εy = εz = εx = 0 ⇒ τyz = σy = σz = σx = 0
From Hooke's law we have
τ = G γ or J τxy
τxzN = G J γxy
γxzN with G =
iso
E2 H1 + νL = Shear modulus
Stress tensor w.r.t Cartesian basis, cf. fig. x,
σ =ikjjjj 0
τxy
τxz
τxy
00
τxz
00
y{zzzz 27
Material and Computational Mechanics Group 5
ü Equilibrium
Equilibrium requires
∇¯
⋅ σ + q = 0 ⇒ 9 ∂τxy∂y + ∂τxz
∂z + qx = ∇ ⋅ τ + qx = 0
∂τxy∂x + qy = 0
∂τxz∂x + qz = 0
Assume: qx = 0fl Equilibrium condition
∇ ⋅ τ = 0
ü Prandtl's stress function
Introduce Prandtl's stress function φ = φ@y, zD so that
τ = −∇ˆ
φ with ∇ˆ
⋅ x̂ = 2 and ∇ˆ
⋅ x = 0 ⇒
∇ ⋅ τ = −∇ ⋅ ∇ˆ
φ =. .. = 0 ⇒ ∇ ⋅ τ := 0 with qx = 0
Consider also
∇ˆ
⋅ τ =
−∇ˆ
⋅ ∇ˆ
φ = −∆φ = G ∇ˆ
⋅ γ = G ∇ˆ
⋅ H∇ ux + θ x̂L = G H∇ˆ ⋅ ∇ ux + θ ∇ˆ
⋅ x̂L = 2 G θ ⇒
∴ ∆φ + 2 G θ = 0
fl Given θ , we may compute φ w.r.t boundary conditons!!! fl τ = −∇ˆ
φ
28
Material and Computational Mechanics Group 6
How do we determine the boundary conditions?
fl no shear stress is allowed on the boundaries of the cross section, cf. fig. x fl considertraction vector w.r.t outward normal
t =ikjjjj 0
τxy
τxz
τxy
00
τxz
00
y{zzzz ikjjjj nx = 0
nynz
y{zzzz =ikjjjj τxy ny + τxz nz
00
y{zzzz ⇒
τ̄ = ny τxy + nz τxz = n ⋅ τ = 0 ∀ x ∈ Γ
Interpretation:
n ⋅ τ = −n ⋅ ∇ˆ
φ =. .. = n̂ ⋅ ∇φ = dφds
= 0 ⇒ φ = C ∀ x ∈ Γ
Note!
n ⋅ ∇ˆ
φ = −ny ∂ φ∂z
+ nz ∂ φ∂y
= −J−nz ∂ φ∂y
+ ny ∂ φ∂z
N = −n̂ ⋅ ∇φ
Note! C = 0 ∀ x ∈ Γ normally chosen for outer boundary of the cross section.