Prandtl Stress Function Summary

( )( )

( )

,x x

y

z

u u y z

u z x

u y x

φ

φ

=

= −

=

xxz

xxy

uG yzuG zy

σ φ

σ φ

∂⎛ ⎞′= +⎜ ⎟∂⎝ ⎠⎛ ⎞∂ ′= −⎜ ⎟∂⎝ ⎠

satisfy equilibrium equation

by taking

0xy xz

y zσ σ∂ ∂

+ =∂ ∂

xy

xz

Gz

Gy

σ φ

σ φ

∂Φ′=∂∂Φ′= −∂

Φ … Prandtl stresss function ( l2 )

(1)

From (1) 2 2 2

2 2xuG G G G G

y z y zφ φ φ φ∂ ∂ Φ ∂ Φ′ ′ ′ ′= − − = +

∂ ∂ ∂ ∂2 2

2 2 2y z

∂ Φ ∂ Φ+ = −

∂ ∂Poisson’s equation

"compatibility"

2 2

2 2 2y z

∂ Φ ∂ Φ+ = −

∂ ∂in A

A

Cn

ts

0xn xy y xz zdn n Gds

σ σ σ φ Φ′= + = =

constant 0Φ = = on C

Torque-twist

effT G Jφ′= 2effJ dA= Φ∫∫

( )nt on Ceff on C

TJ n

σ ⎛ ∂Φ ⎞= −⎜ ⎟∂⎝ ⎠

( )maxmax

on Ceff on C

TJ n

τ ⎛ ∂Φ ⎞= −⎜ ⎟∂⎝ ⎠

bar cross section

T

For the warping displacement

x

x

u yz yu zy z

φ φ

φ φ

∂ ∂Φ′ ′= − −∂ ∂

∂ ∂Φ′ ′= +∂ ∂

xu yz dzy

yz dyz

φ

φ

⎛ ⎞∂Φ′= − +⎜ ⎟∂⎝ ⎠⎛ ∂Φ ⎞′= +⎜ ⎟∂⎝ ⎠

∫

∫

Thin rectangular cross section (neglect ends)

y

z

t

22

4t y

⎛ ⎞Φ = −⎜ ⎟

⎝ ⎠

/ 23

/ 2

123

y t

effy t

J bdy bt=+

=−

= Φ =∫

max 2/ 2

3

eff y t

T TJ y bt

τ=±

⎛ ⎞∂Φ= =⎜ ⎟∂⎝ ⎠

∓

xu yzφ′=

Membrane Analogy

p w

s s

2 2

2 2

w w py z s

∂ ∂+ = −

∂ ∂ 0w = on the boundary

2s wp

Φ =

22

4t y

⎛ ⎞Φ = −⎜ ⎟

⎝ ⎠

For cross sections with holes we also need to satisfy

0Φ =

KΦ =

0xhole

du =∫

2 holehole

ds G An

φ⎛ ∂Φ ⎞ ′− =⎜ ⎟∂⎝ ⎠∫

or 2 holehole

ds G Aτ φ′=∫

If one has multiple holes, this additional condition is applied at each hole to solve for the multiple unknown constants

additionalunknown

supplementary condition to determine K

Torsion of Thin, Closed Sections

K 1 K 2

τ τa b

c

c c

τ

a

ab b

a a

ta

Φ = 0

Φ = K1

b b

t b

Φ = KΦ = K

12 Φ = K

t c

2

c c

τa =K1

ta

, τb =K1 − K2

tb, τc =

K2

tc

K 1

K 2

q 1 =

=q 2

q 1 q2

1 1 1 2 1 2 2 2, ,a a b b c cq t K q q q t K K q t Kτ τ τ= = = − = = − = =

shear flows

q1

-q2

q - q2 1

shear flows into or out of a junction are conserved

0outq =∑

K 1

K 2

q 1 =

=q 2

q 1 q2

1Ω2Ω

iΩ … area enclosed by centerline of ith "cell'

Torque-shear flow 2 i ii

T q= Ω∑

for each cell 12 i ithcell

qdsG t

φ′=Ω ∫

maxmax

qKt

τ ⎛ ⎞= ⎜ ⎟⎝ ⎠

warping is generally small for closed sections

2 holehole

ds G Aτ φ′=∫

Torsion of a Thin Closed Section(multiple cells)

1q 2q

cell 1 cell 2

1

2

1 1 2 2

1

2

2 21

2

12

C

C

T q qq ds

G t

q dsG t

φ

φ

= Ω + Ω

′ =Ω

′ =Ω

∫

∫

1. If the torque T is known, then q1 and q2 are first found in terms of the unknown φ' from Eqs. (2) and (3). These qm 's are then placed into Eq.(1) which is solved for the unknown φ' . Once φ' is known in this manner, the qm 's are completely determined.

2. If φ' is known, Eqs.(2) and ( 3) can be solved directly for the qm 's and then Eq.(1) can be used to find the torque, T

(1)

(2)

(3)

( the q in Eqs.(2) and (3) is the total q flowing ina given cross section, i.e it is q1 – q2 flowingin the vertical section)

Pr⊥ Tq

Ω = area contained within the centerline of the cross section

For a single cell, we can write these more explicitly

2C

C

T qr ds

q r ds q

⊥

⊥

=

= = Ω

∫

∫ 2Tq =Ω

2

12

4

C

C

q dsG tT ds

G t

φ′ =Ω

=Ω

∫

∫effT GJ φ′=

where24

eff

C

J dst

Ω=

∫

maxmax min2 2

T Tt t

τ ⎛ ⎞= =⎜ ⎟Ω Ω⎝ ⎠

(no stress concentrations)

Torsion of a Thin Closed Section(single cell)

Pr⊥

2Tq =Ω

T

effT GJ φ′=

where24

eff

C

J dst

Ω=

∫

q

Ω = area contained within the centerline of the cross section

1. If T is known, q follows directly from Eq. (1),φ' is found from Eq.(2)

(1)

(2)

2. If φ' is known, T follows from Eq.(2),and q is then found from Eq. (1)

Torsion of a Thin Closed Section(single cell)

The shear stress is not quite uniform across the thickness for thin closed sections

t

yields uniform stress

The difference looks much like that for an open section

t

so as a small correction factor: ( )2

34 13eff

C

C

J t s dsdst

Ω= + ∫

∫

Torsion of closed sections with fins

23 34 1 1

3 3c fT T T G t ds t dsdst

φ

⎡ ⎤⎢ ⎥Ω′= + = + +⎢ ⎥⎢ ⎥⎣ ⎦

∑ ∑∫ ∫∫

In the closed sectionmax

min2cTt

τ =Ω

Jeff closed Jeff for a fin (allows varaiablethickness)

( )c eff closedT G Jφ′=

In a fin

( )max

max

f

eff fin

T tJ

τ⎛ ⎞⎜ ⎟=⎜ ⎟⎝ ⎠

( )f eff finT G Jφ′=

We can write this also in terms of the values since

total f others

total f others

J J J

T T T

= +

= +

f f

others others

total total

T G J

T G JT G J

φ

φφ

′=

′=′=

so

( ) ( )f total

eff efffin total

T T GJ J

φ′= =( )max

max

total

eff total

T tJ

τ⎛ ⎞⎜ ⎟=⎜ ⎟⎝ ⎠