4.13. EXERCISES 237
L
0.5L
w
P
Figure 4.27: A simple support beam.
a width b D 1:00 ˙ 0:010
00
. e uniformly distributed load is w D 100 ˙ 10 (lb/in). e
concentrated force in the middle is P D 1500 ˙ 180 (lb). e yield strength S
y
of the
beam’s material follows a normal distribution with a mean
S
y
D 34,500 (psi) and a
standard deviation
S
y
D 3120 (psi). Calculate the reliability of the beam.
4.28. A cantilever beam as shown in Figure 4.28 is subjected to a concentrated force at the
free end P D 150 ˙ 80 (lb). e cross-section of the beam is a rectangular shape with
a height h D 2:00 ˙ 0:010
00
and a width b D 1:00 ˙ 0:010
00
. e length of the beam is
L D 20:0 ˙ 0:032
00
. e Young’s modulus of the beam material follows a normal distri-
bution with a mean
E
D 2:76 10
7
(psi) and a standard deviation
E
D 6:89 10
5
(psi). If the maximum allowable deflection of the beam is D 0:022
00
. Calculate the
reliability of this beam for a deformation issue.
L
P
Figure 4.28: A cantilever beam.
4.29. A shaft with a diameter d D 1:500 ˙ 0:005
00
is subjected to a torque T D 3000 ˙ 150
(lb/in) and a bending moment M D 9000 ˙ 600 (lb/in). e yield strength S
y
of the
beam’s material follows a normal distribution with a mean
S
y
D 34,500 (psi) and a
standard deviation
S
y
D 3120 (psi).
(a) Calculate the reliability of the shaft by using the MSS stress theory.
(b) Calculate the reliability of the shaft by using DE theory.
4.30. A thin-wall cylindrical vessel has a wall thickness t D 0:25 C 0:020
00
and an inner di-
ameter d D 40:0 ˙ 0:125
00
. e internal pressure of the fluid is p D 350 ˙50 (psi). e
238 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
yield strength S
y
of the beam’s material follows a normal distribution with a mean
S
y
D 34,500 (psi) and a standard deviation
S
y
D 3120 (psi).
(a) Calculate the reliability of the shaft by using the MSS stress theory.
(b) Calculate the reliability of the shaft by using DE theory.
4.31. A plane stress element of a component of a ductile material at the critical point is shown
in Figure 4.29. e normal compressive stress
x
(ksi), the normal tensile stress
y
, and
the shear stress
xy
follows normal distributions. eir distribution parameters are listed
in Table 4.50.
x
y
τ
xy
σ
y
σ
x
Figure 4.29: Schematic of a plane stress element at the critical point.
Table 4.50: e stresses in a plane stress element
σ
x
(ksi) σ
y
(ksi) τ
xy
(ksi)
μ
σ
x
σ
σ
x
μ
σ
y
σ
σ
y
μ
τ
xy
σ
τ
xy
16,200 1,550 6,800 540 13,000 1100
e column is made of a ductile material. e yield strength S
y
of the beam’s material
follows a normal distribution with a mean
S
y
D 34,500 (psi) and a standard deviation
S
y
D 3120 (psi). Calculate the reliability of the shaft by using the DE theory.
4.32. Schematic of a critical cross-section of a column is shown in Figure 4.30. It is subjected
to a compression force F
x
D 4000 ˙ 180 (lb) and a bending moment M
z
D 28;000 ˙
300 (lb/in). e geometric dimensions of the critical cross-section of the column are
h D 2:25 ˙ 0:010
00
and b D 1:00 ˙ 0:010
00
. Its ultimate tensile strength S
ut
follows a
normal distribution with a mean
S
ut
D 22:00 (ksi) and a standard deviation
S
ut
D 1:80
(ksi). e ultimate compression strength S
uc
follows a normal distribution with a mean
S
uc
D 82:00 (ksi) and a standard deviation
S
uc
D 10:50 (ksi). Calculate the reliability
of the column by using the MNS theory.
4.33. A plane stress element of a component of a brittle material at the critical point is shown
in Figure 4.31. e normal tensile stress
x
(ksi), the normal compression stress
y
and
4.13. EXERCISES 239
x
z
y
F
x
M
z
b
h
Figure 4.30: Schematic of compression force and a bending moment.
x
y
τ
xy
σ
y
σ
x
Figure 4.31: Schematic of a plane stress element.
the shear stress
xy
follows normal distributions. eir distribution parameters are listed
in Table 4.51.
Table 4.51: e stresses in a plane stress element
σ
x
(ksi) σ
y
(ksi) τ
xy
(ksi)
μ
σ
x
σ
σ
x
μ
σ
y
σ
σ
y
μ
τ
xy
σ
τ
xy
14.2 1.51 50.80 6.25 15.0 2.31
e column is made of brittle material. Its ultimate tensile strength S
ut
follows a normal
distribution with a mean
S
ut
D 22:00 (ksi) and standard deviation
S
ut
D 1:80 (ksi).
e ultimate compression strength S
uc
follows a normal distribution with a mean
S
uc
D
82:00 (ksi) and standard deviation
S
uc
D 10:50 (ksi). Calculate the reliability of this
component by using the BCM theory.
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