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

m15.7 A rectangular post has cross-sectional dimensions
of 200 mm (height) by 80 mm (width). The post is subjected
to a concentrated force of 10 kN acting in the x–y plane at
an angle of 60° with the vertical direction. Determine the
stresses that act in the x and y directions at point B, which is
located on the front face of the post, 10 mm to the left of the
longitudinal centerline.
ExAmpLE 15.5
A 36 mm solid shaft supports a 640 N load as shown. Determine
the principal stresses and the maximum shear stress at
points H and K.
y
160 mm
Plan the Solution
The 640 N load applied to the gear will create a vertical
shear force, a torque, and a bending moment in the shaft at z
the section of interest. These internal forces will in turn create
normal and shear stresses at points H and K, but because
point H is located on the top of the shaft and point K is located
on the side of the shaft, the states of stress will differ
at the two points. We will begin the solution by determining
a system of forces and moments that acts at the section of interest and that is statically
equivalent to the 640 N load applied to the teeth of the gear. The normal and
shear stresses created by this equivalent force system will be computed and shown
in their proper directions on a stress element for both point H and point K. Stress
transformation calculations will be used to determine the principal stresses and
maximum shear stress for each stress element.
500 mm
z
K
H
K
640 N
x
y
H
d = 36 mm
SolutioN
Equivalent Force System
A system of forces and moments that is statically equivalent to the
640 N load can be readily determined for the section of interest.
The equivalent force at the section is equal to the 640 N load on
the gear. Since the line of action of the 640 N load does not pass
through the section that includes points H and K, the moments produced
by this load must be determined.
The moment about the x axis (i.e., a torque) is the product of
the magnitude of the force and the distance in the z direction from
the section of interest to the gear teeth: M x = (640 N)(160 mm) =
z
y
Shaft cross section.
320 N·m
H
K
102.4 N·m
640 N
x
639