材料力学03_扭转

• From observation, the angle of twist of the shaft is proportional to the applied torque and to the shaft length.
T
L
ቤተ መጻሕፍቲ ባይዱ
• When subjected to torsion, every cross-section of a circular shaft remains plane and undistorted.
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Stress
Shear Stress • The internal torque is, as a matter of fact, the resultant of the shear stress acting on the cut plane. • The shear stress at any point is perpendicular to the radius of the point. • Distribution of shearing stresses is statically indeterminate – must consider shaft deformations. However, the resultant of the internal shearing stresses equals the internal torque T
• Cross sections of the bar do not change (still circular) in shape, but rotate around the longitudinal axis of the bar.
• For small deformation (twist), neither the length of the bar nor its radius will change. • The square grid on the surface become a rhombus grid, indicating a pure shear phenomenon. Shear strain equals the change of a right angle.
T
T
In the last thee figures, the right hand rule is applied when determine the direction of the twist moment, T.
T T
3D vector format
T B
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T
2D vector format
Stresses in Elastic Range
Shear Strain
• g

c
g max
Gg
• Multiplying the previous Eq. by the shear modulus,
Hooke’s Law

c
Gg max
• Use Hooke’s Law, t = Gg, in the above Eq. to give
g
A’
• Geometry. AOA : AA
Lg or
ABA : AA gL
g

L
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• Shear strain is proportional to twist and radius c g g max g max and c L
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t t T t dA t max dA max 2 dA max J c c c
J 2 dA = Polar moment of inertia of the cross section.
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Shear Stress
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• The results are known as the elastic torsion formulas, T Tc t t max J J
dF
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dF = t dA
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Shaft Deformations: Observations
• Under the torque, the right side end will rotate (w.r.t. the left side end) through an angle , the angle of twist.
Positive torque: A torque whose vector (right hand rule) directs outward from the shaft. Negative torque: A torque whose the vector (right hand rule) directs inward to the shaft. Note
Fx 0 :
s
45o
t max A0 s
s45 ( 2A0)
45
45o
( 2 A0 ) cos 45 0
t max
tmax A0
Element a is in pure shear. Element c is subjected to a tensile stress on two faces and compressive stress on the other two.
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• Please note that cross-sections of noncircular shafts are distorted when subjected to torsion.
Shearing Strain
• Under the torque, the right-side end will rotate (with respect to the left-side end) through the angle of twist . • Consider an interior section of the shaft. As a torsional load is applied, an element on the interior cylinder deforms into a rhombus. • Since the ends of the element remain planar, the shear strain equals the change of the right angle as shown. g B A
tmax A0
s45 = ±
Tc J
• Consider an element c at 45o to the shaft axis, static analysis shows that there will be only normal stress but no shear stress on the surfaces.
• Normal stresses, shearing stresses or a combination of both may be found for other orientations (e.g., element b as shown).
T
tmax s45
a c
T
Tc tmax = J
s45 ( 2A0)
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45
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tmax A0
tmax A0
Note that all stresses for elements a and c have the same magnitude
Torsional Failure Modes
Stress • When subject to torsion, maximum shear stress occurs on elements at the outer surface of the shaft, with faces parallel and perpendicular to the shaft axis. • Maximum normal stress occurs elements at 45 to the shaft axis. Failure Modes on
A
Applied Loads and Internal Forces
Applied Loads Internal Forces: Internal Torque • Typical applied loads are twist moments applied at different locations of a shaft. • If we pass a plane through point S to cut the shaft into two parts, keeping the left part as a free body, there must be an internal torque, TEC, acting on the cut plane to maintain equilibrium.
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TEC or a 2D FBD
TB
B
TE
E
S
TEC
• The calculation of internal torque is almost the same as the calculation of axial force, especially with the 2D free body diagram. Use SMx = 0 to calculate internal torque.
t t max c
The shearing stress varies linearly with the radial position in the section.