13
14
O1
dx
d
O2

E

C
G G H H
F
A
C B
D D
The angle , is indicated on the element. It can be related to the length dx of the element and the difference in the angle of rotation, d , between the shaded faces.
a

a’
d
d’

b


c

This angle is denoted by (gamma) and is measured in radians (rad).
7
Hooke’s Law for Shear The behaviour of a material subjected to pure shear can be studied in a laboratory by using specimens in the shape of thin circular tubes and subjecting them to a torsion loading. Experiments show that as for normal stress and strain, shear stress is proportional to shear strain as long as stress does not exceed the proportional limit. Hooke’s law for shear can be written as
E G 2(1 )
9
§12-2 Torsion of a Circular Shaft
Torque is a moment that tends to twist a member about its longitudinal axis. Its effect is of primary concern in the design of axles or drive shafts used in vehicles and machinery.
( x)
deformed plane y x x
T
undeformed plane
12
The angle ( x) , so defined, is called the angle of twist. It depends on the position x and will vary along the shaft as shown. In order to understand how this distortion strains the material, a small piece located at a distance x from the fixed end is now isolated from the shaft. The back face will rotate by ( x), and the front face by ( x) d . As a result, the difference in these rotations, d , causes the element to be subjected to a shear strain.
11
If the shaft is fixed at one end and a torque is applied to its other end, the shaded plane in figure will distort into a skewed form as shown. Here a radial line located on the cross section at a distance x from the fixed end of the shaft will rotate through an angle ( x). z
1
Shear Stresses and Strains; Torsion
§12–1 Shear Stresses and Strains;
Hooke’s Law for Shear
§12–2 Torsion of a Circular Shaft
§12–3 Problems Involving Deformation
zy xy z yz xz y 0
zy
zy
zy yz
zy zy yz yz
yz
x
∆y
yz
∆z
∆x
y
All four shear stresses must have equal magnitude and be directed either toward or away from each other at opposite edges of the element.
4
Complementary property of shear
z Consider a volume element of material taken at a zy point located on the surface yz ∆z yz of any sectioned area on ∆x zy which the average shear ∆y stress acts. Consider force x equilibrium in the y direction, then,
and Stress in a Circular Shaft
2
§12-1 Shear Stresses and Strains; Hooke’s Law for Shear
Shear Stress The intensity of force, or force per unit area, acting tangent to ∆A is called the shear stress. The average shear stress distributed over each sectioned area that develops this shear force is defined by V
By inspection, twisting causes the circles to remain circles, and each longitudinal grid line deforms into a helix that intersects the circles at equal angles. Also, the cross sections at the ends of the shaft remain flat ― that is, they do not warp or bulge in or out ― and radial lines on these ends remain straight during the deformation. From these observations we can assume that if the angle of rotation is small, the length of the shaft and its radius will remain unchanged.
y
zy xy zy xy 0
zy zy
5
And in a similar manner, force equilibrium in the z direction yields yz yz . Finally, taking moments about the x axis, z
When the torque is applied, the circles and longitudinal grid lines originally marked on the shaft tend to distort into the pattern shown in Fig. 10 (b).
15O1dx NhomakorabeadO2

E

C
G G H H
F
A
From the figure, we have
C B
D D
GG d GG d EG dx
16

d
dx
Since dx and d are the same for all elements located at points on the cross section at x, then d / dx is constant, and the equation states that the magnitude of the shear strain for any of these elements varies only with its radial distance from the axis of the shaft. In other words, the shear strain within the shaft varies linearly along any radial line, from zero at the axis of the shaft to a maximum max at its outer boundary.
17
If the material is linear-elastic, then Hooke’s law applies, G , and consequently a linear variation in shear strain, as noted in the previous section, leads to a corresponding linear variation in shear stress along any radial line on the cross section. Hence, like the shear strain variation, for a solid shaft, will vary from zero at the shaft’s longitudinal axis to a maximum value, max , at its outer surface.