Adam Paweł Zaborski
1. Torsion
Definitions
torsion angle – the angle between the final and initial radius vectors of a point, Fig. 1.1; please note, that
the angle is defined in the cross-section, its unit is [rad] or []; this is the rotation angle of one crosssection with respect to another

Fig. 1.1 Definition of the torsion angle
unit torsion angle – the torsion angle per meter
torsion moment, twisting couple, torque moment – the moment which vector is parallel to the bar axis; it
may be shown in different ways, Fig. 1.2
Fig. 1.2 Different symbols used to present a torque moment
The most common application of torsion is provided by transmission shafts, which can be either solid or
hollow.
Solution of BVP
de Saint-Venant’s assumptions
1. The projection of the cross-section onto its initial plane remains unchanged (as the rigid membrane),
Fig.1.3
2. The cross-section distortion consists in the movement “out-of-plane” only and is the same for all crosssections.
A'
w


A

v
Fig. 1.3 de S-V’s assumptions
The displacement components can be written:
  ' , sin   , cos   1,
v   sin , w   cos ,
,
v   xz, w   xy,
u  ( y, z ),
where:
 – torsion angle,
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.
Adam Paweł Zaborski
 – unit torsion angle,
 – the distortion function; it does not depend on x,
We calculate the strain matrix:
0

T  



1
2



y
z

1
2


G


z



 y 
,
0


0

0
and the stress matrix:
0

T  



G


y
z
 y 

0


0


z
0
From the first Navier’s equation we get:
 2  0
(1)
remaining equations are identically fulfilled. We see that the distortion function is harmonic.
The static boundary conditions are:
 on the side surface:


y
 

z
 z m

 y n  0,
the remaining conditions are identically fulfilled,
 on the bottoms:

q vy  G

y

(2)

 z  1,
q vz  G

z

 y  1 ,
(3)
(1st equation is identically fulfilled).
From the kinematic conditions of fixing at the point O, we have:
0,0,0  0,

y

z
 0,
 0.
(4)
The Laplace equation (1), with zero boundary conditions with the partial derivatives (2), makes so-called
Neumann’s problem, which has a solution accurate up to a constant. For the cross-section with one
symmetry axis is:


 z  0 .
y
Because we can do nothing with the remaining conditions on the bottoms (3), we define the loading in the
same form as for the shear stresses:


(5)
 vy  G y  z ,  vz  G z  y .

Calculating the torque, we have:


 

M s   qvz y  qvy z dF   G
F

z



 y y  G

y


 z z dF  GJ s
(6)
F
where the torsion inertia moment is defined as:
J s  


z
y

y

z  y 2  z 2 dF
(7)
F
In the general case the torsion inertia depends on the distortion function and indirectly on the BVP’s
solution.
We get eventually the relationship between the unit torsion angle, torsion moment and the torsion rigidity
GIs:

Ms
.
GJ s
The unit of the unit torsion angle is [1/m = rad/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.
Adam Paweł Zaborski
Torsion of bars with axisymmetric cross-section
Basic formulae
In the particular case of axial symmetry (a circle, a ring), the distortion function is identically equal to zero
and the torsion inertia moment reduces to the polar inertia moment:
( y, ) z  0  J s  J 0
It means that the plane cross-section remains plane after the deformation, it doesn’t deplane.
The stresses, (5), become:
 vy  Gz,
 vz  Gy.
z
 xz
y
 xy
Fig. 1.4. Shear stress distribution
The resultant shear stress is:
   2xy   2xz  G z 2  y 2  Gr 
Ms
r , [MPa].
J0
The shearing stress varies linearly with the distance from the axis of the shaft, Fig. 1.4.
The torsion angle is calculated from the unit torsion angle provided that torsion moment is constant:
Ms
d

dx
GJ 0
    dx 
l
M sls
[1 = rad]
GJ 0
Note, that the torsion angle is dimensionless, so the angle unit in the above formula is a radian, not a
degree.
Design conditions
There are two main requirements:
 the requirement of the strength and
 the requirement of usability.
The first requirement means that every structure has to sustain applied load. The second one consists on
several demands of durability, rigidity, resistance to severe weather conditions and so on. From the point of
view of the strength of materials course two principal design conditions should be listed:
 the ultimate limit state
 the serviceability limit state, usually the stiffness of the structure.
In the particular case of the torsion, the ultimate limit state is checked by:
max( )  Rt
where Rt is the shear strength of a given material. The suitable value can be found in the standards.
Because the maximal value of the stress is attained at the side surface for r  R , we introduce so-called the
cross-section twist factor W0 :
def
W0 
J0
R
,
so:
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.
Adam Paweł Zaborski
max( ) 
M0
M
R 0
J0
W0
The stiffness condition means that the unit torsion angle or the torsion angle doesn’t exceed acceptable
(permissible) values:
   acceptable , or    acceptable
Examples
Example 1.1
The shaft with the diameter of 4 cm and length 2 m is fixed at one end and loaded at another end by such
torque that the point A on the side surface displaced to the point A’. The arc AA’ length is 1 mm.
Determine the torsion angle, the unit torsion angle, the torque, the maximal shear stress and the angular
strain on the side surface. G = 80 GPa.
Solution
From the definition of the twist angle we calculate:

0.1
0.2
2 
 0.05 [rad].
d
4
For constant torsion moment we have:

 0.05

 0.025
l
2
[rad/m].
Using the formula of the unit torsion angle we get:
M s  G J s   80 10 9 
0.04 4
 0.025  500 [Nm]
32
(quite important, the average value for the VW golf is about 150 Nm).
The maximal stress value is:
 max 
M 0 GJ 0 
d

R  G  80 10 9  0.025  0.02  40 10 6 [Pa]  40 [MPa].
W0
J0
2
From the Hook equation we determine
  2 
2 max
d
   0.025  0.02  0.0005
2G
2
.
Example 1.2
Determine the twist angle between section A and section E of the shaft in Fig. 1.5, d1  4 cm, d 2  3 cm, G =
80 GPa.
70
40
A
30 Nm
C
D
B
0.4
0.4
d1
0.3
E
0.5 m
d2
Fig. 1.5 Twisted shaft
Solution
The formula of the twist angle:

M 0l
GJ 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 within the confines of the European Social Fund
and realized under surveillance of Ministry of Science and Higher Education.
Adam Paweł Zaborski
is valid only in the case of all parameters constant, so we have to cut the shaft into the interval with
constant cross-section inertia moment and constant torque: AB, BC, CD and DE.
We have:
AB: l AB  0.4 m, M AB  30  40  70  60 Nm, J AB 
d 14
60  0.4
 2.51 10 7 m4,  AB 
 0.00120 rad
32
80 10 9  2.51 10 7
d 14
10  0.4
 2.51 10 7 m4,  BC 
 0.000199 rad
32
80 10 9  2.51 10 7
d 4
10  0.3
 2  7.95 10 8 m4,  CD 
 0.000472 rad
32
80 10 9  7.95 10 8
BC: l BC  0.4 m, M BC  30  40  10 Nm, J BC 
CD: l CD  0.3 m, M CD  30  40  10 Nm, J CD
DE: l DE  0.5 m, M DE  30 Nm, J DE 
d 24
30  0.5
 7.95 10 8 m4,  DE 
 0.00236 rad
32
80 10 9  7.95 10 8
and the total angle of twist is:
 AE   AB   BC   CD   DE  0.00120  0.000199  0.000472  0.00236  0.00289 rad.
Fig. 1.6 presents the diagram of the twist angle.
0,003
0,0025
0,002
0,0015
0,001
0,0005
0
0
0,5
1
1,5
Fig. 1.6 Diagram of the twist angle
Addendum
Glossary
torsion angle, angle of twist – kąt skręcenia
unit torsion angle – jednostkowy kąt skręcenia
torsion moment – moment skręcający
torsion inertia moment – moment bezwładności na skręcanie
distortion – spaczenie, deplanacja
torsion rigidity – sztywność skręcania
shaft – wał, wałek
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.