工程力学英文版课件10 Shear Stresses and Strains,Torsion

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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
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.
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Βιβλιοθήκη Baidu
Shear Strain
The change in angle that occurs between two line segments that were originally perpendicular to one another is referred to as shear strain.
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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
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§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
y
zy xy zy xy 0
zy zy
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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).
a

a’
d
d’

b


c

This angle is denoted by (gamma) and is measured in radians (rad).
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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
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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.
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O1
dx
d
O2

E

C
G G H H
F
A
From the figure, we have
C B
D D
GG d GG d EG dx
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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.
( x)
deformed plane y x x
T
undeformed plane
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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.
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.
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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.
E G 2(1 )
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§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.
avg
A
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F
F
V
V
avg : average shear stress at the section, which is
assumed to be the same at each point located on the section. V : internal resultant shear force at the section determined from the equations of equilibrium. A: area at the section.
G
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Here G is called the shear modulus of elasticity or the modulus of rigidity. Notice that the units of measurement for G will be the same as stress, since is measured in radians, a dimensionless quantity. It will be shown later that the three material constants, modulus of elasticity, E, modulus of rigidity, G, and Poisson’s ratio, , are actually related by the equation
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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