Transcript Document
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Design of a Transmission Shaft
• If power is transferred to and from the
shaft by gears or sprocket wheels, the
shaft is subjected to transverse loading
as well as shear loading.
• Normal stresses due to transverse loads
may be large and should be included in
determination of maximum shearing
stress.
• Shearing stresses due to transverse
loads are usually small and
contribution to maximum shear stress
may be neglected.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
8-1
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Design of a Transmission Shaft
• At any section,
Mc
I
Tc
m
J
m
where M 2 M y2 M z2
• Maximum shearing stress,
2
2
Mc Tc
max m m 2
2
2I J
2
for a circular or annular cross - section, 2 I J
max
c
M2 T2
J
• Shaft section requirement,
J
c
min
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
M 2 T 2
max
all
8-2
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 8.3
SOLUTION:
• Determine the gear torques and
corresponding tangential forces.
• Find reactions at A and B.
• Identify critical shaft section from
torque and bending moment diagrams.
Solid shaft rotates at 480 rpm and
transmits 30 kW from the motor to
gears G and H; 20 kW is taken off at
gear G and 10 kW at gear H. Knowing
that all = 50 MPa, determine the
smallest permissible diameter for the
shaft.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
• Calculate minimum allowable shaft
diameter.
8-3
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 8.3
SOLUTION:
• Determine the gear torques and corresponding
tangential forces.
TE
P
30 kW
597 N m
2f 2 8 Hz
FE
TE 597 N m
3.73 kN
rE
0.16 m
TC
20 kW
398 N m
2 8 Hz
FC 6.63 kN
TD
10 kW
199 N m
2 8 Hz
FD 2.49 kN
• Find reactions at A and B.
Ay 0.932 kN
Az 6.22 kN
B y 2.80 kN
Bz 2.90 kN
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
8-4
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 8.3
• Identify critical shaft section from torque and
bending moment diagrams.
M
2
y
M z2 T 2
max
11602 3732 597 2
1357 N m
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
8-5
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 8.3
• Calculate minimum allowable shaft diameter.
M y2 M z2 T 2
J
c
all
1357 N m
27.14 10 6 m3
50 MPa
For a solid circular shaft,
J 3
c 27.14 10 6 m3
c 2
c 0.02585 m 25.85 m
d 2c 51.7 mm
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
8-6
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Stresses Under Combined Loadings
• Wish to determine stresses in slender
structural members subjected to
arbitrary loadings.
• Pass section through points of interest.
Determine force-couple system at
centroid of section required to maintain
equilibrium.
• System of internal forces consist of
three force components and three
couple vectors.
• Determine stress distribution by
applying the superposition principle.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
8-7
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Stresses Under Combined Loadings
• Axial force and in-plane couple vectors
contribute to normal stress distribution
in the section.
• Shear force components and twisting
couple contribute to shearing stress
distribution in the section.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
8-8
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Stresses Under Combined Loadings
• Normal and shearing stresses are used to
determine principal stresses, maximum
shearing stress and orientation of principal
planes.
• Analysis is valid only to extent that
conditions of applicability of superposition
principle and Saint-Venant’s principle are
met.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
8-9
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 8.5
SOLUTION:
• Determine internal forces in Section
EFG.
• Evaluate normal stress at H.
• Evaluate shearing stress at H.
Three forces are applied to a short
steel post as shown. Determine the
principle stresses, principal planes and
maximum shearing stress at point H.
• Calculate principal stresses and
maximum shearing stress.
Determine principal planes.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
8 - 10
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 8.5
SOLUTION:
• Determine internal forces in Section EFG.
Vx 30 kN P 50 kN Vz 75 kN
M x 50 kN 0.130 m 75 kN 0.200 m
8.5 kN m
M y 0 M z 30 kN 0.100 m 3 kN m
Note: Section properties,
A 0.040 m 0.140 m 5.6 103 m 2
1 0.040 m 0.140 m 3 9.15 10 6 m 4
I x 12
1 0.140 m 0.040 m 3 0.747 10 6 m 4
I z 12
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
8 - 11
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 8.5
• Evaluate normal stress at H.
y
P Mz a Mx b
A
Iz
Ix
3 kN m 0.020 m
5.6 10-3 m 2
0.747 106 m 4
50 kN
8.5 kN m 0.025 m
9.15 106 m 4
8.93 80.3 23.2 MPa 66.0 MPa
• Evaluate shearing stress at H.
Q A1 y1 0.040 m 0.045 m 0.0475 m
85.5 106 m3
Vz Q
75 kN 85.5 106 m3
yz
I xt
9.15 106 m 4 0.040 m
17.52 MPa
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
8 - 12
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 8.5
• Calculate principal stresses and maximum
shearing stress.
Determine principal planes.
max R 33.02 17.522 37.4 MPa
max OC R 33.0 37.4 70.4 MPa
min OC R 33.0 37.4 7.4 MPa
tan 2 p
CY 17.52
2 p 27.96
CD 33.0
p 13.98
max 37.4 MPa
max 70.4 MPa
min 7.4 MPa
p 13.98
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
8 - 13