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MECHANICAL DESIG AND MACHINE ELEMENTS
ME 322A
Jing Zheng
Problem 3-3 (7.7%)
Statements: Draw a free-body diagram of the pedal-arm assembly from a bicycle with
the pedal arms in the horizontal position and dimensions as shown in Figure P3-1.
(Consider the two arms, pedals and pivot as one piece). Assuming a rider-applied force of
1500N at the pedal, determine the torque applied to the chain sprocket and the maximum
bending moment and torque in the pedal arm.
Hints:
(a) (40%) There are no applied torques, only applied forces. It will make the
following calculation simple if put one of the pedal or arm with a section through
the origin.
(b) (20%) Page 78, Static Load Analysis. Moment (or Torque) = Force Distance.
(c) (40%) Again, Page 78, Static Load Analysis. Use the appropriate equilibrium
condition from the (3.3a) to solve the unknowns.
MECHANICAL DESIG AND MACHINE ELEMENTS
ME 322A
Jing Zheng
Problem 4-3 (15.3%)
Statements: For the bicycle pedal-arm assembly in Figure 4-1 with rider-applied force of
1500 M at the pedal, determine the maximum principal stress in the pedal arm if its crosssection is 15 mm in diameter. The pedal attaches to the pedal arm with a 12 mm screw
thread. What is the stress in the pedal screw?
Hints:
(a) (20%) Page 180, Example 4-9 shows a similar situation. Find the correspondent A
and B point as shown in Figure 4-30 (page 180).
(b) (20%) Page 181, step 5 of the Example 4-9 shows how to calculate the normal
bending stress and torsional shear stress at point A. The cross section properties
can be found on Appendix A (page 939-940).
(c) (10%) Section 4.3, page 141~143 shows the general steps to calculate the
principle stress. Page 181, step 6 of the Example 4-9 shows how the principle
stress is calculated in similar problem. Do not copy the result blindly since the
coordinates you used may not be exactly the same as the example used.
(d) (20%) Page 169, equation 4.15c shows how the maximum transverse shear stress
at the neutral axis of a round road is calculated. Point B is in pure shear. Example
4-9, step 7 shows how a similar case is analyzed.
(e) (10%) see hint C to find the principal stress value at the point B
(f) (20%) Determine the stress in the pedal screw. First, find the bending moment in
the screw from the rider-applied force. Calculate the maximum normal stress due
to bending in the screw. Is the maximum principal stress in the screw the
maximum normal stress? Why it is or why it is not?
MECHANICAL DESIG AND MACHINE ELEMENTS
ME 322A
Jing Zheng
Problem 5-3 (7.7%)
Statements: For the bicycle pedal-arm assembly in Figure 5-1 with rider-applied force of
1500 M at the pedal, determine the von Mises stress in the 15-mm-dia pedal arm. The
pedal attaches to the arm with a 12-mm thread. Find the von Mises stress in the screw.
Find the safety factor against static failure if the material has Sy = 350 Mpa.
Hints:
(a) (50%) The maximum principal stresses in the pedal arm are given by problem 4.3
solutions. The von Mises stress can be calculated by using the equation 5.7c, page
245. For the safety factor, look at the Safety Factor section on the next page.
(b) (50%) The maximum principal stresses in the screw are also given by problem 4.3
solutions. For the von Mises stress and safety factor, look at Hint A.
MECHANICAL DESIG AND MACHINE ELEMENTS
ME 322A
Jing Zheng
Problem 3-4 (11.5%)
Statements: The trailer hitch from Figure 1-1 (p.11) has loads applied as shown in Figure
P3-2. The tongue weight of 100 kg acts downward and the pull force of 4905 N acts
horizontally. Using the dimensions of the ball bracket in Figure 1-5 (p.14), draw a freebody diagram of the ball bracket and find the tensile and shear loads applied to the two
bolts that attach the bracket to the channel in Figure 1-1.
Assumptions:
1) Nuts are just snug tight (no pre-load), which is worst case.
2) All reactions will be concentrated loads, not distributed loads/pressures.
Hints:
(a) (10%) Find the weight on the tongue.
(b) (40%) There are two external forces Fpull and Wtongue acting on the ball. To keep the
ball bracket from rotating, there should be counter forces acting on the bracket. It
is very important to include all the forces acting on the ball and the bracket. There
are two vertical forces and three horizontal forces. (Apparently there is a force
provided by the bolt, which connect the bracket to the C-beam. However, is this
force enough to keep the bracket in steady state?)
(c) (50%) Find all the forces that can satisfy equation 3.3a or 3.3b (Page 78).
MECHANICAL DESIG AND MACHINE ELEMENTS
ME 322A
Jing Zheng
Problem 3-5 (3.8%)
Statements: For the trailer hitch of Problem 3-4, determine the horizontal force that will
result on the ball from accelerating a 2000-kg trailer to 60 m/s in 20 sec.
Assumptions:
1) Constant acceleration
2) Rolling resistance of tires and bearing is negligible
Hints:
(a) (50%) Calculate the acceleration
(b) (50%) F = m a
MECHANICAL DESIG AND MACHINE ELEMENTS
ME 322A
Jing Zheng
Problem 4-5 (23.1%)
Statements: Repeat Problem 4-4 for the loading conditions of Problem 3-5, i.e.,
determine the stresses due to a horizontal force that will result on the ball from
accelerating a 2000 kg trailer to 60 m/sec in 20 sec. Assume a constant acceleration.
From Problem 3-5, the pull force is 6000 N. Determine:
(a) The principal stresses in the shank of the ball where it joins the ball bracket.
(b) The bearing stress in the ball bracket hole.
(c) The tearout stress in the ball bracket.
(d) The normal and shear stresses in the 19-mm diameter attachment bolts.
(e) The principal stresses in the ball bracket as a cantilever.
Assumptions: The bolts are snug tight. All reactions are concentrated loads, not
distributed loads.
Hints:
(a) (50%) Draw separated free body diagrams of the ball and the bracket. It is very
important to include all the forces acting on the ball and the bracket. For the ball,
there are two vertical forces and three horizontal forces; for the bracket, there are
two vertical forces and four horizontal forces. Find all the forces that can satisfy
equation 3.3a or 3.3b (Page 78) and solve for the unknown forces. Then, find the
principal stresses.
(b) (10%) The bearing stress is the pulling force over the bearing area. See Direct
Bearing, Page 152, and equation 4.10a, we can assume that there is no clearance
between the pin and hole.
(c) (10%) The tearout stress is the pulling force over the tearout area. See Tearout
Failure, Page 151, and equation 4.9.
(d) (10%) Remember there are two attachment bolts.
(e) (20%)
Ball bracket as
a cantilever
MECHANICAL DESIG AND MACHINE ELEMENTS
ME 322A
Jing Zheng
Problem 5-5 (15.3%)
Statements: Repeat Problem 5-4 for the loading conditions of Problem 3-5, i.e.,
determine the horizontal force that will result on the ball from accelerating a 2000-kg
trailer to 60 m/sec in 20 sec. Assume constant acceleration. From Problem 3-5, the pull
force is 6000 N. Determine static safety factors for:
(a) The shank of the ball where it joins the ball bracket.
(b) Bearing failure in the ball bracket hole.
(c) Tearout failure in the ball bracket.
(d) Tensile failure in the 19-mm diameter attachment holes.
(e) Bending failure in the ball bracket as a cantilever.
Hints:
(a) (20%) Found the von Mises stress, then the safety factor. Page 245, 246.
(b) (20%) Found the von Mises stress, then the safety factor. Page 245, 246.
(c) (20%) Found the von Mises stress, then the safety factor. Page 245, 246.
(d) (20%) Found the von Mises stress, then the safety factor. Page 245, 246.
(e) (20%) Found the von Mises stress, then the safety factor. Page 245, 246.
MECHANICAL DESIG AND MACHINE ELEMENTS
ME 322A
Jing Zheng
Problem 5-14 (15.6%)
Statements: Figure P5-5 shows a child’s toy called a pogo stick. The child stands on the
pads, applying half her weight on each side. She jumps off the ground, holding the pads
up against her feet, and bounces along with the spring cushioning the impact and storing
energy to help each rebound. Assume a 60-lb child and a spring constant of 100lb/in. the
pogo stick weighs 5 lb. Design the aluminum cantilever beam sections on which she
stands to survive jumping 2 in off the ground with a safety factor of 2. Use 1100 series
aluminum. Define and size the beam shape.
Hints:
(a) (5%) The property of the 1100 series aluminum on Appendix C. There are two
condition of the alloy. Choose one that you would like to use.
(b) (5%) From Problem 3-14, find the total dynamic force on both foot and the load
on each support.
(c) (15%) Choose a cross-section and length for each aluminum support. You can
choose any shape and configuration as long as the assumption is reasonable and
you can calculate its stress loading.
(d) (5%) Calculate the maximum bending moment.
(e) (70%) At this stage, you should familiar on how to calculate principal stress, the
von Mises stress and safety factor. With the bending stress and safety factor
known, it is straight forward to find the cross-section dimension requirements.
Make sure the calculation is consistent with the assumption you made and the
cross-section configuration you chose.
ME 322A
Jing Zheng
Problem 3-3 (7.7%)
Statements: Draw a free-body diagram of the pedal-arm assembly from a bicycle with
the pedal arms in the horizontal position and dimensions as shown in Figure P3-1.
(Consider the two arms, pedals and pivot as one piece). Assuming a rider-applied force of
1500N at the pedal, determine the torque applied to the chain sprocket and the maximum
bending moment and torque in the pedal arm.
Hints:
(a) (40%) There are no applied torques, only applied forces. It will make the
following calculation simple if put one of the pedal or arm with a section through
the origin.
(b) (20%) Page 78, Static Load Analysis. Moment (or Torque) = Force Distance.
(c) (40%) Again, Page 78, Static Load Analysis. Use the appropriate equilibrium
condition from the (3.3a) to solve the unknowns.
MECHANICAL DESIG AND MACHINE ELEMENTS
ME 322A
Jing Zheng
Problem 4-3 (15.3%)
Statements: For the bicycle pedal-arm assembly in Figure 4-1 with rider-applied force of
1500 M at the pedal, determine the maximum principal stress in the pedal arm if its crosssection is 15 mm in diameter. The pedal attaches to the pedal arm with a 12 mm screw
thread. What is the stress in the pedal screw?
Hints:
(a) (20%) Page 180, Example 4-9 shows a similar situation. Find the correspondent A
and B point as shown in Figure 4-30 (page 180).
(b) (20%) Page 181, step 5 of the Example 4-9 shows how to calculate the normal
bending stress and torsional shear stress at point A. The cross section properties
can be found on Appendix A (page 939-940).
(c) (10%) Section 4.3, page 141~143 shows the general steps to calculate the
principle stress. Page 181, step 6 of the Example 4-9 shows how the principle
stress is calculated in similar problem. Do not copy the result blindly since the
coordinates you used may not be exactly the same as the example used.
(d) (20%) Page 169, equation 4.15c shows how the maximum transverse shear stress
at the neutral axis of a round road is calculated. Point B is in pure shear. Example
4-9, step 7 shows how a similar case is analyzed.
(e) (10%) see hint C to find the principal stress value at the point B
(f) (20%) Determine the stress in the pedal screw. First, find the bending moment in
the screw from the rider-applied force. Calculate the maximum normal stress due
to bending in the screw. Is the maximum principal stress in the screw the
maximum normal stress? Why it is or why it is not?
MECHANICAL DESIG AND MACHINE ELEMENTS
ME 322A
Jing Zheng
Problem 5-3 (7.7%)
Statements: For the bicycle pedal-arm assembly in Figure 5-1 with rider-applied force of
1500 M at the pedal, determine the von Mises stress in the 15-mm-dia pedal arm. The
pedal attaches to the arm with a 12-mm thread. Find the von Mises stress in the screw.
Find the safety factor against static failure if the material has Sy = 350 Mpa.
Hints:
(a) (50%) The maximum principal stresses in the pedal arm are given by problem 4.3
solutions. The von Mises stress can be calculated by using the equation 5.7c, page
245. For the safety factor, look at the Safety Factor section on the next page.
(b) (50%) The maximum principal stresses in the screw are also given by problem 4.3
solutions. For the von Mises stress and safety factor, look at Hint A.
MECHANICAL DESIG AND MACHINE ELEMENTS
ME 322A
Jing Zheng
Problem 3-4 (11.5%)
Statements: The trailer hitch from Figure 1-1 (p.11) has loads applied as shown in Figure
P3-2. The tongue weight of 100 kg acts downward and the pull force of 4905 N acts
horizontally. Using the dimensions of the ball bracket in Figure 1-5 (p.14), draw a freebody diagram of the ball bracket and find the tensile and shear loads applied to the two
bolts that attach the bracket to the channel in Figure 1-1.
Assumptions:
1) Nuts are just snug tight (no pre-load), which is worst case.
2) All reactions will be concentrated loads, not distributed loads/pressures.
Hints:
(a) (10%) Find the weight on the tongue.
(b) (40%) There are two external forces Fpull and Wtongue acting on the ball. To keep the
ball bracket from rotating, there should be counter forces acting on the bracket. It
is very important to include all the forces acting on the ball and the bracket. There
are two vertical forces and three horizontal forces. (Apparently there is a force
provided by the bolt, which connect the bracket to the C-beam. However, is this
force enough to keep the bracket in steady state?)
(c) (50%) Find all the forces that can satisfy equation 3.3a or 3.3b (Page 78).
MECHANICAL DESIG AND MACHINE ELEMENTS
ME 322A
Jing Zheng
Problem 3-5 (3.8%)
Statements: For the trailer hitch of Problem 3-4, determine the horizontal force that will
result on the ball from accelerating a 2000-kg trailer to 60 m/s in 20 sec.
Assumptions:
1) Constant acceleration
2) Rolling resistance of tires and bearing is negligible
Hints:
(a) (50%) Calculate the acceleration
(b) (50%) F = m a
MECHANICAL DESIG AND MACHINE ELEMENTS
ME 322A
Jing Zheng
Problem 4-5 (23.1%)
Statements: Repeat Problem 4-4 for the loading conditions of Problem 3-5, i.e.,
determine the stresses due to a horizontal force that will result on the ball from
accelerating a 2000 kg trailer to 60 m/sec in 20 sec. Assume a constant acceleration.
From Problem 3-5, the pull force is 6000 N. Determine:
(a) The principal stresses in the shank of the ball where it joins the ball bracket.
(b) The bearing stress in the ball bracket hole.
(c) The tearout stress in the ball bracket.
(d) The normal and shear stresses in the 19-mm diameter attachment bolts.
(e) The principal stresses in the ball bracket as a cantilever.
Assumptions: The bolts are snug tight. All reactions are concentrated loads, not
distributed loads.
Hints:
(a) (50%) Draw separated free body diagrams of the ball and the bracket. It is very
important to include all the forces acting on the ball and the bracket. For the ball,
there are two vertical forces and three horizontal forces; for the bracket, there are
two vertical forces and four horizontal forces. Find all the forces that can satisfy
equation 3.3a or 3.3b (Page 78) and solve for the unknown forces. Then, find the
principal stresses.
(b) (10%) The bearing stress is the pulling force over the bearing area. See Direct
Bearing, Page 152, and equation 4.10a, we can assume that there is no clearance
between the pin and hole.
(c) (10%) The tearout stress is the pulling force over the tearout area. See Tearout
Failure, Page 151, and equation 4.9.
(d) (10%) Remember there are two attachment bolts.
(e) (20%)
Ball bracket as
a cantilever
MECHANICAL DESIG AND MACHINE ELEMENTS
ME 322A
Jing Zheng
Problem 5-5 (15.3%)
Statements: Repeat Problem 5-4 for the loading conditions of Problem 3-5, i.e.,
determine the horizontal force that will result on the ball from accelerating a 2000-kg
trailer to 60 m/sec in 20 sec. Assume constant acceleration. From Problem 3-5, the pull
force is 6000 N. Determine static safety factors for:
(a) The shank of the ball where it joins the ball bracket.
(b) Bearing failure in the ball bracket hole.
(c) Tearout failure in the ball bracket.
(d) Tensile failure in the 19-mm diameter attachment holes.
(e) Bending failure in the ball bracket as a cantilever.
Hints:
(a) (20%) Found the von Mises stress, then the safety factor. Page 245, 246.
(b) (20%) Found the von Mises stress, then the safety factor. Page 245, 246.
(c) (20%) Found the von Mises stress, then the safety factor. Page 245, 246.
(d) (20%) Found the von Mises stress, then the safety factor. Page 245, 246.
(e) (20%) Found the von Mises stress, then the safety factor. Page 245, 246.
MECHANICAL DESIG AND MACHINE ELEMENTS
ME 322A
Jing Zheng
Problem 5-14 (15.6%)
Statements: Figure P5-5 shows a child’s toy called a pogo stick. The child stands on the
pads, applying half her weight on each side. She jumps off the ground, holding the pads
up against her feet, and bounces along with the spring cushioning the impact and storing
energy to help each rebound. Assume a 60-lb child and a spring constant of 100lb/in. the
pogo stick weighs 5 lb. Design the aluminum cantilever beam sections on which she
stands to survive jumping 2 in off the ground with a safety factor of 2. Use 1100 series
aluminum. Define and size the beam shape.
Hints:
(a) (5%) The property of the 1100 series aluminum on Appendix C. There are two
condition of the alloy. Choose one that you would like to use.
(b) (5%) From Problem 3-14, find the total dynamic force on both foot and the load
on each support.
(c) (15%) Choose a cross-section and length for each aluminum support. You can
choose any shape and configuration as long as the assumption is reasonable and
you can calculate its stress loading.
(d) (5%) Calculate the maximum bending moment.
(e) (70%) At this stage, you should familiar on how to calculate principal stress, the
von Mises stress and safety factor. With the bending stress and safety factor
known, it is straight forward to find the cross-section dimension requirements.
Make sure the calculation is consistent with the assumption you made and the
cross-section configuration you chose.
