Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

102,400 N · mm = 102.4 N · m. Similarly, the moment about the z axis is the product of the
magnitude of the force and the distance in the x direction from points H and K to the gear
teeth: M z = (640 N) (500 mm) = 320,000 N · mm = 320 N · m. By inspection, these moments
act in the directions shown.
Alternative method: The geometry of this problem is relatively simple; therefore, the
equivalent moments can be determined readily by inspection. For situations that are more
complicated, it is sometimes easier to determine the equivalent moments from position
vectors and force vectors.
The position vector r from the section of interest to the point of application of the
load is r = 500 mm i + 160 mm k. The load acting on the gear teeth can be expressed as
the force vector F = −640 N j. The equivalent moment vector M can be determined from
the cross product M = r × F:
M = r × F =
i j k
500 0 160
0 −640 0
= 102,400 Nmm ⋅ i − 320,000 Nmm ⋅ k
For the coordinate axes used here, the axis of the shaft extends in the x direction; therefore,
the i component of the moment vector is recognized as a torque, while the k component
is simply a bending moment.
Section Properties
The shaft diameter is 36 mm. The polar moment of inertia will be required in order to
calculate the shear stress caused by the internal torque in the shaft:
J
π π
= d = (36 mm) = 164,896 mm
32 32
4 4 4
K
H
The moment of inertia of the shaft about the z centroidal axis is
I
z
π π
= d = (36 mm) = 82,448 mm
64 64
4 4 4
Normal Stresses at H
The 320 N · m bending moment acting about the z axis creates a normal stress that
varies over the depth of the shaft. At point H, the bending stress can be computed
from the flexure formula as
Mc (320,000 Nmm)(18mm)
⋅
σ x = =
= 69.9 MPa ( T)
4
I 82,448 mm
z
H
Shear Stress at H
The 102.4 N · m torque acting about the x axis creates shear stress at H. The magnitude
of this shear stress can be calculated from the elastic torsion formula:
K
Tc
τ = =
J
(102,400 Nmm)(36 ⋅ mm/2)
164,896 mm
4
= 11.18 MPa
The transverse shear stress associated with the 640 N shear force is zero at H.
640