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Unformatted text preview: LECTURE Third Edition SHAFTS: TORSION LOADING AND
DEFORMATION
• A. J. Clark School of Engineering •Department of Civil and Environmental Engineering 6
Chapter
3.1  3.5 by
Dr. Ibrahim A. Assakkaf
SPRING 2003
ENES 220 – Mechanics of Materials
Department of Civil and Environmental Engineering
University of Maryland, College Park LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading Slide No. 1
ENES 220 ©Assakkaf Introduction
– Members subjected to axial loads were
discussed previously.
– The procedure for deriving loaddeformation relationship for axially loaded
members was also illustrated.
– This chapter will present a similar
treatment of members subjected to torsion
by loads that to twist the members about
their longitudinal centroidal axes. 1 Slide No. 2 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading ENES 220 ©Assakkaf Torsional Loads on Circular Shafts
• Interested in stresses and strains of
circular shafts subjected to twisting
couples or torques
• Turbine exerts torque T on the shaft
• Shaft transmits the torque to the
generator
• Generator creates an equal and
opposite torque T’ Slide No. 3 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading ENES 220 ©Assakkaf Net Torque Due to Internal Stresses
• Net of the internal shearing stresses is an
internal torque, equal and opposite to the
applied torque,
T = ∫ ρ dF = ∫ ρ (τ dA)
• Although the net torque due to the shearing
stresses is known, the distribution of the
stresses is not
• Distribution of shearing stresses is statically
indeterminate – must consider shaft
deformations
• Unlike the normal stress due to axial loads,
the distribution of shearing stresses due to
torsional loads can not be assumed uniform. 2 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading Slide No. 4
ENES 220 ©Assakkaf Introduction
Cylindrical members Fig. 1 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading Slide No. 5
ENES 220 ©Assakkaf Introduction
Rectangular members Fig. 2 3 Slide No. 6 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading ENES 220 ©Assakkaf Introduction
– This chapter deals with members in the
form of concentric circular cylinders, solid
and hollow, subjected to torques about
their longitudinal geometric axes.
– Although this may seem like a somewhat a
special case, it is evident that many torquecarrying engineering members are
cylindrical in shape. Slide No. 7 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading ENES 220 ©Assakkaf Deformation of Circular Shaft
– Consider the following shaft
ab
cd Fig. 3 l T T l 4 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading Slide No. 8
ENES 220 ©Assakkaf Deformation of Circular Shaft
– In reference to the previous figure, the
following observations can be noted:
• The distance l between the outside
circumferential lines does not change
significantly as a result of the application of the
torque. However, the rectangles become
parallelograms whose sides have the same
length as those of the original rectangles. LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading Slide No. 9
ENES 220 ©Assakkaf • The circumferential lines do not become
zigzag; that is ; they remain in parallel planes.
• The original straight parallel longitudinal lines,
such as ab and cd, remain parallel to each
other but do not remain parallel to the
longitudinal axis of the member. These lines
become helices*.
*A helix is a path of a point that moves longitudinally
and circumferentially along a surface of a cylinder
at uniform rate 5 Slide No. 10 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading ENES 220 ©Assakkaf Deformation of Circular Shaft Subjected
to Torque T
Fig. 4
a T
T b Slide No. 11 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 3.5) Torsion Loading ENES 220 ©Assakkaf Deformation of a Bar of Square Cross
Section Subjected to Torque T
Fig. 5
a T
T b 6 Slide No. 12 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading ENES 220 ©Assakkaf Shaft Deformations
• From observation, the angle of twist of
the shaft is proportional to the applied
torque and to the shaft length.
φ ∝T φ∝L • When subjected to torsion, every crosssection of a circular shaft remains plane
and undistorted.
• Crosssections for hollow and solid circular
shafts remain plain and undistorted because a
circular shaft is axisymmetric.
• Crosssections of noncircular (nonaxisymmetric) shafts are distorted
when subjected to torsion. LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading Slide No. 13
ENES 220 ©Assakkaf An Important Property of Circular Shaft
– When a circular shaft is subjected to
torsion, every cross section remains plane
and disturbed
– In other words, while the various cross
sections along the shaft rotate through
different amounts, each cross section
rotates as a solid rigid slab. 7 Slide No. 14 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading ENES 220 ©Assakkaf An Important Property of Circular Shaft
– This illustrated in Fig.4b, which shows the
deformation in rubber model subjected to torsion.
– This property applies to circular shafts whether
solid or hollow.
– It does not apply to noncircular cross section.
When a bar of square cross section is subjected to
torsion, its various sections are warped and do not
remain plane (see Fig. 5.b) Slide No. 15 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading ENES 220 ©Assakkaf Stresses in Circular Shaft due to
Torsion
– Consider the following circular shaft that is
subjected to torsion T Fig. 6 B C A T T 8 Slide No. 16 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading ENES 220 ©Assakkaf Stresses in Circular Shaft due to
Torsion
– A section perpendicular to the axis of the
shaft can be passed at an arbitrary point C
as shown in Fig. 6.
– The Freebody diagram of the portion BC
of the shaft must include the elementary
shearing forces dF perpendicular to the
radius ρ of the shaft. Slide No. 17 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading ENES 220 ©Assakkaf Stresses in Circular Shaft due to
Torsion
B C T ρ T Fig. 7 dF = τρ dA 9 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading Slide No. 18
ENES 220 ©Assakkaf Stresses in Circular Shaft due to
Torsion
– But the conditions of equilibrium for BC
require that the system of these
elementary forces be equivalent to an
internal torque T.
– Denoting ρ the perpendicular distance from
the force dF to axis of the shaft, and
expressing that the sum of moments of LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsion Loading Slide No. 19
ENES 220 ©Assakkaf Stresses in Circular Shaft due to
Torsion
– of the shearing forces dF about the axis of
the shaft is equal in magnitude to the
torque T, we can write T = Tr = ∫ ρ dF = ∫ ρ τ dA area (1) area 10 Slide No. 20 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) ENES 220 ©Assakkaf Torsion Loading
Stresses in Circular Shaft due to
Torsion
B
C T T ρ T = Tr = ∫ ρ τ dA (2) area LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Shearing Strain Slide No. 21
ENES 220 ©Assakkaf If a plane transverse section before
twisting remains plane after twisting and
a diameter of the the section remains
straight, the distortion of the shaft of
Figure 7 will be as shown in the
following figures (Figs. 8 and 9): 11 Slide No. 22 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Shearing Strain ENES 220 ©Assakkaf Shearing Strain
• 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 is equal to angle of twist.
• It follows that
Lγ = ρφ or γ = ρφ
L • Shear strain is proportional to twist and radius
γ max = ρ
cφ
and γ = γ max
L
c Slide No. 23 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Shearing Strain ENES 220 ©Assakkaf Shearing Strain
c φ ρ L
Fig. 8 12 Slide No. 24 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) ENES 220 ©Assakkaf Torsional Shearing Strain
Shearing Strain γ
a′
a L ρ
φ
aa′
aa′
≈ sin
L
L
because γ is very small
tan γ = Fig. 9 γ a′
a L Slide No. 25 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Shearing Strain ENES 220 ©Assakkaf Shearing Strain
– From Fig. 9, the length aá can be
expressed as
– But aa′ = L tan γ = Lγ – Therefore, (3) aa′ = ρφ
Lγ = ρφ ⇒ γ = (4) ρφ
L (5) 13 Slide No. 26 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) ENES 220 ©Assakkaf Torsional Shearing Strain
Shearing Strain For radius ρ, the shearing strain for circular
shaft is
ρφ γρ = (6) L For radius c, the shearing strain for circular
shaft is γc = cφ
L (7) Slide No. 27 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) ENES 220 ©Assakkaf Torsional Shearing Strain
Shearing Strain
Combining Eqs. 6 and 7, gives γ ρ L γ cL
φ=
=
ρ
c
Therefore γρ = γc
c ρ (8) (9) 14 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Slide No. 28 Torsional Shearing Stress ENES 220 ©Assakkaf The Elastic Torsion Formula
If Hooke’s law applies, the shearing stress
τ is related to the shearing strain γ by the
equation τ = Gγ (10) where G = modulus of rigidity. Combining
Eqs. 9 and 10, results in τρ G = τc Gc ρ ⇒τρ = τc
c (11) ρ LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Slide No. 29 Torsional Shearing Stress ENES 220 ©Assakkaf The Elastic Torsion Formula
When Eq. 11 is substituted into Eq. 2, the
results will be as follows: ∫ ρ τ dA T = Tr = area τ τ = ∫ ρ c ρ dA = ∫ ρ c ρ dA
c
c 0
area c = τc
c ∫ρ 2 dA (12) area 15 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Slide No. 30 Torsional Shearing Stress ENES 220 ©Assakkaf Polar Moment of Inertia
The integral of equation 12 is called the
polar moment of inertia (polar second
moment of area).
It is given the symbol J. For a solid circular
shaft, the polar moment of inertia is given
by
c
4 J = ∫ ρ 2 dA = ∫ ρ 2 (2πρ dρ ) = πc 2 0 (13) LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Slide No. 31 Torsional Shearing Stress ENES 220 ©Assakkaf Polar Moment of Inertia
– For a circular annulus as shown, the polar
moment of inertia is given by
c
b c J = ∫ ρ dA = ∫ ρ 2 (2πρ dρ )
2 b • = πc
2 4 − πb
2 4 = π
2 (r 4
o − ri 4 ) (14) r0 = outer radius and ri = inner radius 16 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Slide No. 32 Torsional Shearing Stress ENES 220 ©Assakkaf Shearing Stress in Terms of Torque and
Polar Moment of Inertia
– In terms of the polar second moment J, Eq.
12 can be written as
T = Tr = τc
c ∫ρ 2 dA = area τcJ (15) c – Solving for shearing stress,
Tc
τc =
J (16) LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Slide No. 33 Torsional Shearing Stress ENES 220 ©Assakkaf Shearing Stress in Terms of Torque and
Polar Moment of Inertia τ max =
τρ = Tc
J Tρ
J (17a)
(18a) τ= shearing stress, T = applied torque
ρ = radius, and J = polar moment on inertia 17 Slide No. 34 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) ENES 220 ©Assakkaf Torsional Shearing Stress Distribution of Shearing Stress within
the Circular Cross Section
τ Tc
τc =
J τ τmax τmin τmax ro c ρ ρ ri Fig. 10 Slide No. 35 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Stresses in Elastic Range ENES 220 ©Assakkaf • Multiplying the previous equation by the
shear modulus,
Gγ = ρ
c Gγ max From Hooke’s Law, τ = Gγ , so
τ= ρ
c τ max The shearing stress varies linearly with the
radial position in the section. J = 1 π c4
2 • Recall that the sum of the moments from
the internal stress distribution is equal to
the torque on the shaft at the section,
τ
τ
T = ∫ ρτ dA = max ∫ ρ 2 dA = max J
c
c
J= ( 1 π c4
2
2 4
− c1 ) • The results are known as the elastic torsion
formulas,
τ max = Tc
Tρ
and τ =
J
J 18 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Shearing Stress Slide No. 36
ENES 220 ©Assakkaf Example 1
– A hollow cylindrical steel shaft is 1.5 m
long and has inner and outer diameters
equal to 40 mm and 60 mm. (a) What is
the largest torque which may be applied to
the shaft if the shearing stress is not to
exceed 120 MPa? (b) What is the
corresponding minimum value of the
shearing stress in the shaft? LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Shearing Stress Slide No. 37
ENES 220 ©Assakkaf Example 1 (cont’d)
T
60 mm
40 mm T 1.2 m Fig. 11 19 Slide No. 38 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Shearing Stress ENES 220 ©Assakkaf Example 1 (cont’d)
– (a) Largest Permissible Torque
Using Eq.17a τJ
Tc
⇒ Tmax = max
J
c τ max = (19) Using Eq.14 for claculating J ,
J= π (r
2 4
o ) − ri 4 = [(0.03) − (0.02) ]
2 π 4 4 = 1.021× 10 −6 m 4 Slide No. 39 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) ENES 220 ©Assakkaf Torsional Shearing Stress
Example 1 (cont’d) Substituting for J and τmax into Eq. 19, we have
T= ( )( ) Jτ max 1.021× 10 −6 120 × 106
=
= 4.05 kN ⋅ m
0.03
ro (b) Minimum Shearing Stress τρ = Tri 4.05 ×103 (0.02)
= 79.3 MPa
=
J
1.021×10 −6 20 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Displacements Slide No. 40
ENES 220 ©Assakkaf Angle of Twist in the Elastic Range
– Often, the amount of twist in a shaft is of
importance.
– Therefore, determination of angle of twist is
a common problem for the machine
designer.
– The fundamental equations that govern the
amount of twist were discussed previously LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Displacements Slide No. 41
ENES 220 ©Assakkaf Angle of Twist in the Elastic Range
– The basic equations that govern angle of
twist are
– Recall Eqs. 6, γρ = ρφ
L or γρ = ρ dθ
dL (20) 21 Slide No. 42 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Displacements ENES 220 ©Assakkaf Angle of Twist in the Elastic Range τc = Tc
J or τ ρ = Tρ
J (21) and
G= τ
γ (22) LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Displacements Slide No. 43
ENES 220 ©Assakkaf Angle of Twist in the Elastic Range
– Recall Eq. 17a and 7 τ max = Tc
J γ max = cθ
L – Combining these two equations, gives
θ= γ max L c
TL
=
GJ = τ max L Tc 1 L
= G c J G c 22 Slide No. 44 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) ENES 220 ©Assakkaf Torsional Displacements Angle of Twist in the Elastic Range
The angle of twist for a circular uniform
shaft subjected to external torque T is
given by TL
θ=
GJ (22) LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Displacements Slide No. 45
ENES 220 ©Assakkaf Angle of Twist in the Elastic Range
– Multiple Torques/Sizes
• The expression for the angle of twist of the
previous equation may be used only if the shaft
is homogeneous (constant G) and has a
uniform cross sectional area A, and is loaded at
its ends.
• If the shaft is loaded at other points, or if it
consists of several portions of various cross
sections, and materials, then 23 Slide No. 46 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) ENES 220 ©Assakkaf Torsional Displacements
Angle of Twist in the Elastic Range
– Multiple Torques/Sizes • It needs to be divided into components which
satisfy individually the required conditions for
application of the formula.
• Denoting respectively by Ti, Li, Ji, and Gi, the
internal torque, length, polar moment of area,
and modulus of rigidity corresponding to
component i,then n
n θ = ∑θ i = ∑
i =1 i =1 Ti Li
Gi J i (23) Slide No. 47 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) ENES 220 ©Assakkaf Torsional Displacements
n i =1 θ = ∑θ i = ∑ E1 E2 L2 Ti Li
Gi J i E3 L1
Fig. 12 n i =1 Multiple Torques/Sizes L3 Circular Shafts 24 Slide No. 48 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) ENES 220 ©Assakkaf Torsional Displacements Angle of Twist in the Elastic Range
The angle of twist of various parts of a
shaft of uniform member can be given by
n n θ = ∑θ i = ∑
i =1 i =1 Ti Li
Gi J i LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Angle of Twist in Elastic Range (24) Slide No. 49
ENES 220 ©Assakkaf • Recall that the angle of twist and maximum
shearing strain are related,
γ max = cφ
L • In the elastic range, the shearing strain and shear
are related by Hooke’s Law,
γ max = τ max
G = Tc
JG • Equating the expressions for shearing strain and
solving for the angle of twist,
φ= TL
JG • If the torsional loading or shaft crosssection
changes along the length, the angle of rotation is
found as the sum of segment rotations
φ =∑
i Ti Li
J i Gi 25 Slide No. 50 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) ENES 220 ©Assakkaf Torsional Displacements Angle of Twist in the Elastic Range
If the properties (T, G, or J) of the shaft are
functions of the length of the shaft, then
L T
dx
GJ
0 θ =∫ (25) Slide No. 51 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Displacements ENES 220 ©Assakkaf Angle of Twist in the Elastic Range
– Varying Properties
L T
dx
GJ
0 θ =∫ L
× x Fig. 13 26 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Displacements Slide No. 52
ENES 220 ©Assakkaf Example 2
– What torque should be applied to the end
of the shaft of Example 1 to produce a twist
of 20? Use the value G = 80 GPa for the
modulus of rigidity of steel. LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Displacements Slide No. 53
ENES 220 ©Assakkaf Example 2 (cont’d)
T
60 mm
40 mm T 1.2 m Fig. 14 27 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Slide No. 54
ENES 220 ©Assakkaf Torsional Displacements
Example 2 (cont’d)
Solving Eq. 22 for T, we get
JG
T=
θ
(26)
L
Substituting the given values
G = 80 × 109 Pa L = 1.5 m 2π rad −3 = 34.9 ×10 rad
3600 θ = 20 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Slide No. 55
ENES 220 ©Assakkaf Torsional Displacements
Example 2 (cont’d) From Example 1, J was computed to give a
value of 1.021×106 m4.
Therefore,using Eq. 26 ( )( )( 1.021×10 −6 80 ×109
JG
34.9 ×10 −3
θ=
T=
1.5
L
= 1.9 ×103 N ⋅ m = 1.9 kN ⋅ m ) 28 Slide No. 56 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Displacements ENES 220 ©Assakkaf Example 3
What angle of twist will create a shearing
stress of 70 MPa on the inner surface of
the hollow steel shaft of Examples 1 and
2?
T
60 mm
40 mm T 1.2 m Fig. 14 LECTURE 6. SHAFTS: TORSION LOADING AND DEFORMATION (3.1 – 3.5) Torsional Displacements Slide No. 57
ENES 220 ©Assakkaf Example 3 (cont’d)
τρ = Jτ ρ
Tρ
⇒T =
ρ
J (1.021×10 6 )(70 × 106 )
=
= 3.5735 kN ⋅ m
0.02
3.5735 × 103 (1.5)
TL
=
= 0.65625
φ=
GJ 80 × 109 (1.021× 10 −6 )
To obtain θ in degrees, we write θ = 0.65625 360
= 3.760
2π 29 ...
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