材料力学 Torsion

fraction of T that is resisted by the material contained within the outer region of
the shaft, which has an inner radius of c/2 and outer radius c.
The stress in the shaft varies linearly, such that τ=(ρ/c)τmax.
The shear strain varies linearly along
radial line:
c
max
The equation above also valid for circular tubes.
5.2 The Torsion Formula (扭转公式)
The shear stress varies linearly along radial line if the material is linear-elastic, and is a function of the radial position:
An element taken from the shaft
The angle of twist (扭转角) between the
front and rear faces of the element:
d dx
d
dx
The magnitude of the shear strain for any of these elements varies only with its radial distance from the axis of the shaft.
Chapter 5 Torsion
5.1 Torsional Deformation of a Circular Shaft
Torque is a moment that tends to twist a member about its longitudinal axis.
Deformation caused by torque:
shafts as an efficient means for transmitting torque, and thereby
saves materials.
Example 5.2
The shaft shown in Fig. 5-11a is supported by two bearings and is subjected to three torques. Determine the shear stress developed at points A and B, located at section a-a of the shaft, Fig. 5-11c.
4
c 02Tubula源自 shaftJJ2A
c2o4dA
ci4
A
2
(2
d
)
2
3d 2 1 4
A
4
co ci
Not only does the internal torque T develop a linear
distribution of shear stress along each radial line in the plane
The circle on any section remains circle; Each longitudinal grid line deforms into a
helix intersecting the circles at equal angle; The cross sections at the ends of the shaft remain flat; The length and the radius of the shaft remain unchanged.
c
m
ax
The torque acting on the section:
The polar moment
of inertia (极惯性矩)
T
A dA
A
c
m
dA
ax
max c
A 2dA
The shear stress at the intermediate distance ρ can be determined
of the cross-sectional area, but also an associated shear stress
distribution is developed along an axial plane.
Absolute maximum torsional stress is determined by drawing
with
Torsion Formula (扭转公式)
T
J
Only valid when the shaft is circular and the material behaves
linear-elastically
Solid Shaft
J
J
A
2cd4A
2 (2d) 2
A
3d
A
2 1 4
Therefore, the torque dT’ on the ring is
dT ' (dA) ( / c) max(2d)
For the ring area
T ' 2 max c 3d c c/2
T
'
15
32
m
axc3
This torque T’ can be expressed in terms of the applied torque T
a torque diagram to identify the maximum ratio of Tc/J when
the external torque or the radius is not constant.
Example 5.1
The solid shaft of radius c is subjected to a torque T, Fig. 5-10a. Determine the
max
Tc J
(
Tc / 2)c4
2T
c3
So
From the result, weTc'an11s65eeTmost of the torque is resisted by the
outer region of the shaft, which justifies the use of tubular