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
2f 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  103 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  106 m 4
50 kN

8.5 kN  m 0.025 m 
9.15  106 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  106 m3



Vz Q
75 kN  85.5  106 m3
 yz 

I xt
9.15  106 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