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

Combined Stresses at H
The normal and shear stresses acting at point H can be summarized on a stress element.
Stress transformation Results at H
The principal stresses and the maximum shear stress at H are shown in the accompanying
figure.
11.18 MPa
H
z
69.9 MPa
x
11.18 MPa
H
69.9 MPa
x
34.9 MPa
36.7 MPa
z
8.87°
71.6 MPa
1.745 MPa
Normal Stresses at K
The 320 N · m bending moment acting about the z axis creates normal stress that
varies over the depth of the shaft. Point K, however, is located on the z axis,
which is the neutral axis for this bending moment. Consequently, the bending
stress at K is zero.
K
H
Shear Stresses at K
The 102.4 N · m torque acting about the x axis creates shear stress at K. The magnitude
of this shear stress is the same as the stress magnitude at H: τ = 11.18 MPa.
The 640 N shear force acting vertically at the section of interest is also associated
with shear stress at point K. From Equation (9.8), the first moment of area for
a solid circular cross section is
K
H
Q d 3 3
(36 mm)
= = = 3,888 mm
12 12
3
H
The shear stress formula [Equation (9.2)] is used to calculate the shear stress:
3
VQ (640 N)(3,888 mm )
τ = = = 0.838 MPa
4
It (82,448 mm )(36 mm)
z
K
Combined Stresses at K
The normal and shear stresses acting at point K can be summarized on a stress element.
Note that, at point K, both shear stresses act downward on the +x face of the
stress element. After the proper shear stress direction has been established on one
face, the shear stress directions on the other three faces are known.
y
K
12.02 MPa
x
641