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

Stresses at H
The forces and moments acting at the section of interest will be evaluated sequentially
to determine the type, magnitude, and direction of any stresses created at H.
The 17 kN axial force creates compressive normal stress, which acts in the
y direction:
Fy
17,000 N
σ y = = = 8.351 MPa (C)
2
A 2,035.752 mm
The 3.2 kN force acting in the positive z direction creates transverse shear stress
(i.e., τ = VQ/It) throughout the cross section of the pipe. However, the magnitude of
the transverse shear stress is zero at point H.
The same 3.2 kN force acting in the positive z direction also creates a bending
moment at the section where H is located. The magnitude of the bending moment is
M x = (3.2 kN)(0.8 m) = 2.56 kN⋅m
By inspection, we observe that this bending moment about the x axis creates compressive
normal stress on the horizontal faces of the stress element at H:
49.011 MPa
Mc x
σ y =
Ix
(2.56 kN⋅m)(57 mm)(1,000 mm/m)(1, 000 N/kN)
=
4
2,977,287mm
= 49.011 MPa(C)
z
H
2.56 kN·m
x
The 3.75 kN ⋅ m torque acting about the y axis creates shear stress at H. The
magnitude of this shear stress can be calculated from the elastic torsion formula:
y
τ =
=
Tc
J
(3.75 kN⋅m)(57 mm)(1,000 mm/m)(1, 000 N/kN)
= 35.897 MPa
5,954,575 mm
4
35.897 MPa
H
3.75 kN·m
Combined Stresses at H
The normal and shear stresses acting at point H can be summarized on a stress element.
Note that, at point H, the torsion shear stress acts in the −x direction on the +y
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.
z
x
Multiple stresses acting at H. Summary of stresses acting at H.
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