Chapter 9 Torsion of Thin-Walled Tubes

Summary of Saint-Venant Torsion Theory

Warping function, Ψ

shear stress ( )xy G zy

∂Ψτ = β −

∂, ( )xz G y

z∂Ψ

τ = β +∂

compatibility relationship automatic

equilibrium equations 2 0∇ Ψ =

boundary conditions 0xy xzdz dyds ds

τ − τ =

2 21 ( )2

dz dy d y zy ds z ds ds

∂Ψ ∂Ψ− = +

∂ ∂

torque ( )xz xy

AT y z dA= τ − τ∫∫

( )A

T G y z dA G Jz y

∂Ψ ∂Ψ= β − + β∫∫

∂ ∂

Prandtl stress function, φ

shear stress xy z∂φ

τ =∂

, xz y∂φ

τ = −∂

compatibility relationship 2 2G∇ φ = − β

equilibrium equations automatic

boundary conditions 0dφ =

torque 2

AT dA= φ∫∫

Membrane Analogy

y

z

S dy

S dz

yx

S dzS dz

p dy dzy∂ζ∂

dyy y y∂ζ ∂ ∂ζ

+∂ ∂ ∂

ζ(y,z)

y

z

S dy

S dz

yx

S dzS dz

p dy dzy∂ζ∂

dyy y y∂ζ ∂ ∂ζ

+∂ ∂ ∂

ζ(y,z)

sin sin

sin sin 0

xdF S dz S dz dyy y y y

S dy S dy dz p dy dzz z z z

⎛ ⎞ ⎛ ⎞∂ζ ∂ζ ∂ ∂ζ= − + +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

∂ζ ∂ζ ∂ ∂ζ⎛ ⎞ ⎛ ⎞− + + + =⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

2 2

2 2 0S dz dy S dy dz p dy dzy z∂ ζ ∂ ζ

+ + =∂ ∂

2 2

22 2

pSy z

∂ ζ ∂ ζ∇ ζ = + = −

∂ ∂

compatibility relationship:

2 2G∇ φ = − β

boundary condition: 0dφ =

Examples of Solid Cross Sections

elliptical cross section rectangular cross section

triangular cross section I-beam cross section

Discontinuous Boundary

y

z

yx ζ(y,z)

ζ = 0

ζ = ζ1p

y

z

yx ζ(y,z)

ζ = 0

ζ = ζ1p

elliptical cross section with hole rectangular cross section with hole

Narrow Rectangular Cross Section

b >> t

b

t y

z

b

t y

z

equilibrium in x-direction

2

2 0xdF S dy dz p dy dzz∂ ζ

= + =∂

2

2pSz

∂ ζ= −

∂

0p zz S∂ζ

= − +∂

2

2p z CS

ζ = − +

2( ) 0

2 8t pz t C

Sζ =± = − + =

2

8pC tS

=

2

22 4p t zS⎛ ⎞

ζ = −⎜ ⎟⎜ ⎟⎝ ⎠

22

4tG z

⎛ ⎞φ = β −⎜ ⎟⎜ ⎟

⎝ ⎠

torque

2/ 2 2

/ 22 2

4

t

A z t

tT dA G b z dz=−

⎛ ⎞= φ = β −⎜ ⎟∫∫ ∫ ⎜ ⎟

⎝ ⎠

/ 22 3

/ 2

24 3

t

t

t z zT G b−

⎡ ⎤= β −⎢ ⎥

⎢ ⎥⎣ ⎦

313

T G bt= β

shear stress

2xy G zz∂φ

τ = = − β∂

, 0xz y∂φ

τ = − ≈∂

max 23TG tbt

τ = β =

torsional stiffness

( )tTK JG

= =β

31

3K bt=

torque contributions

bulk end( )xy xzA

T z y dA T T= − τ + τ = +∫∫

/ 2 2

bulk/ 2

2t

xyA z t

T z dA G b z dz=−

= − τ = β∫∫ ∫

3bulk

1 16 2

T G bt T= β =

End Correction

h

y = −b/2 y = +b/2

hh

y = −b/2 y = +b/2

h

1D approximation:

22 if and 0 else

4 2 2t b bG z y

⎛ ⎞φ = β − − ≤ ≤ φ =⎜ ⎟⎜ ⎟

⎝ ⎠

0 if and if2 2 2xzb b by y

y y∂φ ∂φ

τ = − = − < < = ±∞ =∂ ∂

∓

2D “tapered” approximation:

2 22 2 2if and else

4 2 2 4

b yt b b tG z h y h G zh

−⎛ ⎞ ⎛ ⎞φ = β − − + ≤ ≤ − φ = β −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

end xzdT y dA= τ

2/ 2 / 2

2end

/ 2 / 24 2

t bxz

A z t b h

t b bT y dA G z dz y dyh=− −

⎡ ⎤⎡ ⎤⎛ ⎞ ⎛ ⎞= τ ≈ β − −⎢ ⎥⎢ ⎥⎜ ⎟∫∫ ∫ ∫ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎢ ⎥⎢ ⎥⎝ ⎠⎣ ⎦ ⎣ ⎦

3end

1 16 2

T G bt T= β =

General Thin-Walled Shapes

TKG

=β

R

tτ

R

tττ

3 31 23 3

K bt R tπ= ≤

b2

b 1

t1

t 2

bt = t(s)

s

b2

b 1

t1

t 2

b2

b 1

t1

t 2

bt = t(s)

s

bt = t(s)

s

31

13

ni i

iK b t

== ∑ 3

0

1 ( )3

bK t s ds= ∫

Single-Cell Thin-Walled Tube

y

L

zx

T

A

D

B

C

t

τ1 τ2A

D

B

C

dx

t1

t2t

P

dΓ

ds

q ds

r

y

L

zx

T

A

D

B

C

ty

L

zx

T

A

D

B

C

t

τ1 τ2A

D

B

C

dx

t1

t2

τ1 τ2A

D

B

C

dx

t1t1

t2t2t

P

dΓ

ds

q ds

rt

P

dΓ

ds

q ds

r

1 1 2 2 0xdF t dx t dx= − τ + τ =

constant shear flow q

1 1 2 2q t t= τ = τ

twisting moment T

dT dF r t ds r q ds r= = τ = 12

d ds rΓ =

2dT q d= Γ

2T q= Γ

2Tq =Γ

surf( )2T

tτ = τ =

Γ

Rate of Twist, Torsional Stiffness

total strain energy over length L

20

12 2V

W T L U U dV Lt dsGτ

= β = = =∫∫∫ ∫

2

212 8

T L dsT LtG

β = ∫Γ

Bredt’s formula

24T ds

tGβ = ∫

Γ

2q dsG t

β = ∫Γ

Example: closed tube of circular cross section:

R

t

R

t

closed24

TG t

β =Γ

, 2 R= π , 2RΓ = π

closed32

TG R t

β =π

closed 32K R t= π

Warping Displacement

z

y

x, uT

T

Q

axis of twist

z

y

x, uT

T

Qaxis of twist

closed cell open cell

z

y

x, uT

T

Q

axis of twist

z

y

x, uT

T

Qaxis of twist

closed cell open cell

closed 32K R t= π

open 3 closed23

K Rt Kπ= <<

closed 2

open 23K RK t

=

given torque T:

closed

22 2T T

t R tτ = =

Γ π

openopensurf 2 2

3 3( )2

T Tbt R t

τ = τ = =π

open

closed 3 Rt

τ=

τ

Multicell Tubes

1 2 0axial wF q L q L q L= − + + = 1 2wq q q= −

1 2 w

CBA ADC ACT q r ds q r ds q r ds= + +∫ ∫ ∫

1 1 2 22( ) 2( ) 2w w w wT q q q= Γ + Γ + Γ − Γ − Γ

1 1 2 22 2T q q= Γ + Γ

For general n-cell cross section:

12

ni i

iT q

== Γ∑

shear flow qi in the ith loop:

12 i

ii i

qG dst

β = ∫Γ

Restraint of Warping

Example: I Sections

symmetric loading u(x = 0, y, z) = 0

top view of the upper flange with differential element

axial view

( )xβ = β

bending of the upper flange: 2

2f fvM E I

x∂

= −∂

3

12f

ft b

I =

2hv = − θ

2

21 12 2f f fM E I h E I h

xx∂ θ ∂β

= =∂∂

2

212

ff f

MV E I h

x x

∂ ∂ β= − = −

∂ ∂

torque due to shear force in the flanges:

f fT V h=

total torque:

2

22

12f fT V h K G E I h K G

x∂ β

= + β = − + β∂

3 22

2 24f

ft b hhJ Iω = =

(called the sectorial area of inertia)

2

2T E J K Gxω∂ β

= − + β∂

2

2K G TE J E Jx ω ω

∂ β− β = −

∂

2 2

22

k TkG Kx

∂ β− β = −

∂ where 2

2(1 )K G KkE J Jω ω

= =+ ν

1 2( ) sinh cosh Tx C kx C kx G Kβ = + +

boundary conditions:

( 0) 0xβ = = (no warping at the center)

0x Lx =

∂β =∂

(no bending moment at the end)

2TC G K= − and 1 tanhTC kLG K=

( )( ) tanh sinh cosh 1Tx kL kx kxG Kβ = − +

surf( ) G tτ =τ = β

0

tanh( ) ( ) 1 ( )L T L kL T LL x dx kLG K kL G K

⎛ ⎞θ = β = − = κ∫ ⎜ ⎟⎝ ⎠

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50Normalized Length, kL

Nor

mal

ized

Tw

ist, κ(

kL)

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 500 10 20 30 40 50Normalized Length, kL

Nor

mal

ized

Tw

ist, κ(

kL)

( )( ) tanh cosh sinh2f

fE I hT k

M x kL kx kxG K= −

( )2

( ) tanh sinh cosh2f

fE I hT k

V x kL kx kxG K= − −