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

z 2
z 1
dimensions of the assembly are x 1 = 1.35 m, y 1 = 1.0 m, y 2 = 2.3 m,
and z 1 = 1.6 m. Determine the normal and shear stresses on the
outer surface of the pipe at (a) point A and (b) point C. Show these
stresses on an appropriate sketch.
P z
P y
P x
y 2
y
z 1
A B
C
P z
P y
P x
y 2
y 1
z
FIGURE p15.46/47/48/49
D
x
z
y
C
D
A
B
y 1
p15.49 The piping assembly shown in Figure P15.46/47/48/49
consists of stainless steel pipe that has an outside diameter of 200 mm
and a wall thickness of 8 mm. The assembly is subjected to concentrated
loads P x = 2,400 N, P y = 0, and P z = 1,100 N, as well as an
internal fluid pressure of 900 kPa that acts in all of the pipes. The
dimensions of the assembly are y 1 = 0.7 m, y 2 = 1.8 m, z 1 = 1.3 m,
and z 2 = 1.1 m. Determine the normal and shear stresses on the
outer surface of the pipe at (a) point A and (b) point D. Show these
stresses on an appropriate sketch.
p15.50 The piping assembly shown in Figure P15.50/51 consists
of stainless steel pipe that has an outside diameter of 275 mm
and a wall thickness of 9 mm. The assembly is subjected to concentrated
loads P x = 3.2 kN, P y = 5.4 kN, and P z = 1.3 kN, as well as an
internal fluid pressure of 1,400 kPa that acts in all of the pipes. The
FIGURE p15.50/51
p15.51 The piping assembly shown in Figure P15.50/51 consists
of stainless steel pipe that has an outside diameter of 275 mm
and a wall thickness of 9 mm. The assembly is subjected to concentrated
loads P x = 3.2 kN, P y = 5.4 kN, and P z = 1.3 kN, as well as an
internal fluid pressure of 1,400 kPa that acts in all of the pipes. The
dimensions of the assembly are x 1 = 1.35 m, y 1 = 1.0 m, y 2 = 2.3 m,
and z 1 = 1.6 m. Determine the normal and shear stresses on the
outer surface of the pipe at (a) point B and (b) point D. Show these
stresses on an appropriate sketch.
x 1
x
15.5 Theories of Failure
A tension test of an axially loaded member is easy to conduct, and the results, for many
types of materials, are well known. When such a member fails, the failure occurs at a
specific principal stress (i.e., the axial stress), a definite axial strain, a maximum shear
stress equal to one-half of the axial stress, and a specific amount of strain energy per unit
volume of stressed material. Since all of these limits are reached simultaneously for an
axial load, it makes no difference which criterion (stress, strain, or energy) is used for
predicting failure in another axially loaded member of the same material.
For an element subjected to biaxial or triaxial loading, however, the situation is more
complicated because the limits of normal stress, normal strain, shear stress, and strain
energy existing at failure are not reached simultaneously. In other words, the precise cause
of failure, in general, is unknown. In such cases, it becomes important to determine the best
criterion for predicting failure, because test results are difficult to obtain and the possible
combinations of loads are endless. Several theories have been proposed for predicting the
failure of various types of material subjected to many combinations of loads. Unfortunately,
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