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

Stress transformation Results at K
The principal stresses and the maximum shear stress at K are shown in the accompanying
figure.
5,940 psi
2,360 psi
z
y
11,887 psi
K
5,804 psi
22.2°
8,310 psi
14,250 psi
ExAmpLE 15.7
A piping system transports a fluid that has an internal pressure
of 1,500 kPa. In addition to being subject to the fluid pressure,
the piping supports a vertical load of 9 kN and a horizontal
load of 13 kN (acting in the +x direction) at flange A. The pipe
has an outside diameter D = 200 mm and an inside diameter
d = 176 mm. Determine the principal stresses, the maximum
shear stress, and the absolute maximum shear stress at points
H and K.
Plan the Solution
The analysis begins by determining the statically equivalent
system of forces and moments acting internally at the section
that contains points H and K. 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. The internal pressure of the fluid also
creates normal stresses, which act longitudinally and circumferentially
in the pipe wall. These stresses will be computed
and included on the stress elements for H and K. Stress transformation
calculations will be used to determine the principal
stresses, the maximum shear stress, and the absolute maximum
shear stress for each stress element.
SolutioN
Equivalent Force System
A system of forces and moments that is statically equivalent to the loads
applied at flange A can be determined for the section of interest.
The equivalent forces are simply equal to the applied loads. A 13 kN
force acts in the +x direction, a 9 kN force acts in the −y direction, and
there is no force acting in the z direction.
The equivalent moments acting at the section of interest can be determined
by considering each load in turn. The 9 kN load acting at A creates
a moment of (9 kN)(1.2 m) = 10.8 kN · m, which acts about the +x axis.
The 13 kN load acting horizontally at A creates two moment components:
z
0.65 m
9 kN
z
A
B
13 kN
13 kN
1.2 m
9 kN
H
K
Equivalent forces at the section that contains
points H and K.
H
y
K
H
y
y
d = 176 mm
D = 200 mm
Cross section.
C
K
x
x
x
647