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within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education

1/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP

CONTINUUM MECHANICS

(TORSION

as BOUNDARY VALUE PROBLEM - BVP)

Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union

within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education

2/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP

Body shape

:

straight prismatic bar with end surfaces perpendicular to

the bar axis, cross-section of arbitrary shape.

Loading:

distributed loading over end surfaces yielding torque as only

cross-sectional force, side surface free of loading, no volume forces.

Kinematics boundary conditions:

Bar fixed at one end (all

displacements and their derivatives vanish there).

M

S

Problem

formulation

In a further analysis we shall adopt the
assumption of replacing kinematics
conditions by statics ones (reaction
torque); the bar is considered as being
in the equilibrium but free to be
twisted (free torsion).

M

We will make also use of de Sain-Venant principle replacing distributed

loading with a torque

M

S

=

M

Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union

within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education

3/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP

x

3

x

1

x

2



























cos

sin

sin

cos

cos

cos

cos

1













r

r

r

u





x

1

x

2

A

A’

r

r’

Assume:

r’ =

r

























sin

cos

sin

cos

sin

sin

sin

2













r

r

r

u

Assume:





3

x







0

,

,

2

1



















sin

1

cos 









2

1

sin

x

r

u



















1

2

cos

x

r

u





Twist angle

per

unit length

(unit

angle)





1

,

0

,

0







3

2

1

x

x

u







3

1

2

x

x

u 









2

1

3

,x

x

u 

?

A

A’

1

1

Total

twist

angle

Distortion

function

Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union

within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education

4/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP



























i

j

j

i

ij

x

u

x

u

2

1





3

2

1

x

x

u







3

1

2

x

x

u 









2

1

3

,x

x

u 





















































2

1

2

3

3

2

23

2

1

2

1

x

x

x

u

x

u

0

2

2

22









x

u



0

3

3

33









x

u



0

1

1

11









x

u







0

2

1

2

1

3

3

1

2

2

1

12







































x

x

x

u

x

u























































1

2

1

3

3

1

13

2

1

2

1

x

x

x

u

x

u





























0

0

0

0

0



T

ij

kk

ij

ij

G











2

0



kk

































0

0

0

0

0

2G

T







13

2

1

23

'















x

G





31

1

2

13

'

















x

G



- distortion function

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5/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP

0























j

ij

i

x

P







31

1

2

13

'

















x

G

0

0

0

3

33

2

32

1

31

3

23

2

22

1

21

3

13

2

12

1

11























































x

x

x

x

x

x

x

x

x



















0

0

0

0

0

0

0

2

2

2

2

1

2













x

x





0









0

''

0

''

0

2

1

















G

G





32

2

1

23

'















x

G

The governing

equation of torsion

boundary value

problem

0

2







or:

Laplacian

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6/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP

Statics boundary conditions

On a bar surface





0

,

0

,

0

q





0

,

,

2

1





















0

'

'

2

2

1

1

1

2



























x

G

x

G

On bar ends:





1

,

0

,

0 









0

,

,

2

1





q

q

q





1

2

1

'















x

G

q





2

1

2

'













x

G

q

x

1

x

2

q

ν1

q

ν2





S

A

M

dA

x

q

x

q







2

1

1

2

















S

A

M

dA

x

x

x

x

G













2

1

2

1

2

1

'

'







+

M

S

S

S

GJ

M





J

S

Torsion inertia
moment

This is boundary value condition for distortion function differential
equation

By de Saint Venant

hypothesis

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7/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP





















2

1

23

2

x

x

GJ

M

S

S









GT

T

2















dA

x

x

x

x

J

A

S











2

1

2

1

2

1

'

'





S

S

GJ

M

























2

1

23

2

x

x





























1

2

13

2

x

x





























1

2

13

2

x

x

GJ

M

S

S

























2

1

23

x

x

J

M

S

S



























1

2

13

x

x

J

M

S

S







3

2

1

x

x

u







3

1

2

x

x

u 









2

1

3

,x

x

u 

0

2

2

2

2

1

2













x

x





0

2

1

1

1

2











































x

x

x

x













0

0

,

0

,

0





Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union

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8/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP





















2

1

,

x

f

x

f

n

0

2

2

2

2

1

2













x

x





2
R

x

1

x

2

n





0

,

2

2

2

2

1

2

1









R

x

x

x

x

f

2

1

2

1

4

4

x

x

n











2

1

2

,

2

x

x

n





R

x

R

x

n

n

2

1

,











0

2

2

1

1

1

2



































































R

x

x

x

R

x

x

x





0

2

2

1

1













R

x

x

R

x

x





0





Governing equation

and

boundary

condition are

homogeneous

Solid circular

shaft

Contour

equation:

No

distortion!

R

x

x

n

2

2

2

1

2

1









Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union

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9/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP













dA

x

x

x

x

J

A

S











2

1

2

1

2

1

'

'





S

S

GJ

M

























2

1

23

x

x

J

M

S

S



























1

2

13

x

x

J

M

S

S





0









0

2

2

2

2

1

J

dA

r

dA

x

x

J

A

A

S













2

0

13

x

J

M

S







1

0

23

x

J

M

S





r

J

M

x

x

J

M

S

S

0

2

1

2

1

0

2

23

2

13

















r

0

0

max

W

M

R

J

M

S

S







3

0

3

x

GJ

M

x

S









Twist angle

Unit twist
angle

Total twist angle for a shaft of
length

l

l

GJ

M

S

0

max





)

(r



Solid circular

shaft

1

x

2

x

Torsion section

modulus

Polar inertia moment

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10/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP

S

S

GJ

M





0





S

S

W

M



max



 



s

s

J

J 

 



s

s

W

W 

For

h



b

b

h/2



max

h

b

J

s

3





h

b

W

s

2





h/b

1

1,5

2

2,5

3

4

6

8





0,208

0,23

1

0,24

6

0,25

8

0,26

7

0,28

2

0.29

9

0,30

7

0,33

3



0,141

0,19

6

0,22

9

0,24

9

0,26

3

0,28

1

0.29

9

0,30

7

0,33

3

Rectangular bar

)

,

( b

h







h/2

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11/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP

Bars of open cross-
sections

Bars of closed cross-
sections

The behaviour of the above types of bars differs significantly when
subjected to the action of a torque. One can make a simple
experiment cutting a tube:

Thin-walled

bars

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within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education

12/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP





1



2





1

Assume: constant
distribution of shear
stress across of tube
thickness





2

Assume: prismatic tube
of varying wall thickness

From equilibrium condition:

const

















2

2

1

1

0

2

2

1

1

















min

max





const



Closed thin-walled

cross-sections

Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union

within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education

13/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP



d
s

dA

r(s)

s

ds

s

r

dA

)

(

2

1



 

 

dA

ds

s

r

M

s

s

s

2

)

(













const





S

dA

M

s

s





2

2







S

– area of the figure

embedded within central
curve

s

S

M

s





2



s

s

W

M



max



S

W

s

min

2







min

max

2





S

M

s



Closed thin-walled

cross-sections

Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union

within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education

14/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP

b

1

h

1

b

2

h

2

b

3

h

3







n

i

si

s

M

M

1







i

i

i

b

h 

Solutions for torsion of rectangular bars
obey:

s

si

GJ

M





si

si

i

W

M



max



i

i

i

si

h

b

J

3





i

i

i

si

h

b

W

2





A2











i

i

i

Si

i

h

b

G

M

3

i

i

i

si

h

b

G

M

3





A3







n

i

i

i

i

s

h

b

G

M

1

3



s

s

n

i

i

i

i

s

J

G

M

h

b

G

M







1

3





Assumptions:

Cross-section

partitioning:

A1

A2

A3

A1

Open thin-walled cross-

sections

Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union

within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education

15/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP

si

si

i

W

M



max



i

i

i

si

h

b

W

2





i

i

i

si

h

b

G

M

3





s

s

n

i

i

i

i

s

J

G

M

h

b

G

M







1

3





i

i

i

s

s

si

h

b

GJ

M

G

M

3





i

i

i

s

s

i

i

i

i

i

i

s

s

i

b

J

M

h

b

h

b

J

M















2

3

max

 

i

s

s

i

b

J

M

max

max

max



















i

i

i

s

s

i

b

J

M







max

max

max

For h

i

/b

i

>6



i

=



i

Open thin-walled cross-

sections

Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union

within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education

16/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP

 

i

s

s

b

J

M

max

max 





max

h/2

h/2

max



Open thin-walled cross-

sections

Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union

within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education

17/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP



d
s

dA

r(s)

s

S

– area of the figure

embedded within central
curve

s



min

max

2





S

M

s



Closed thin-walled

cross-sections

Project “The development of the didactic potential of Cracow University of Technology in the range of modern construction” is co-financed by the European Union

within the confines of the European Social Fund and realized under surveillance of Ministry of Science and Higher Education

18/17

M.Chrzanowski: Strength of Materials

SM1-12: Continuum Mechanics: Torsion as

BVP



stop