4.9. RELIABILITY OF A BEAM UNDER BENDING MOMENT 205
Table 4.32: e distribution parameters of random variables in Equation (a)
G (psi)
Normal distribution
d
1
Normal distribution
d
2
(in)
Normal distribution
L
1
(in)
Normal distribution
μ
G
σ
G
μ
d
1
σ
d
1
μ
d
2
σ
d
2
μ
L
1
σ
L
1
1.117×10
7
2.793×10
5
0.75 0.00125 1.000 0.00125 10.000 0.0025
L
2
(in)
Normal distribution
L
3
(in)
Normal distribution
T
A
(lb/in)
Normal distribution
T
C
(lb/in)
Normal distribution
μ
L
2
σ
L
2
μ
L
3
σ
L
3
μ
T
A
σ
T
A
μ
T
C
σ
T
C
5.00 0.0025 5.000 0.0025 560 20 1380 30
3. e reliability of the shaft under torsion for a deformation issue.
e Monte Carlo method has been discussed in Section 3.8. We can follow the Monte
Carlo method and the program flowchart in Figure 3.8 to create a MATLAB program.
Since the simulation problem is not complicated, we can use the trial number N D
1,598,400 for a key component from Table 3.2 in Section 3.8. e estimated reliability
of this component R is:
R D 0:9884:
4.9 RELIABILITY OF A BEAM UNDER BENDING
MOMENT
4.9.1 RELIABILITY OF A BEAM UNDER BENDING FOR A STRENGTH
ISSUE
A beam is a long structural element which has a ratio of span to the largest dimension of the
cross-section more than ten. A beam primarily resists loads applied laterally to the beam axis. e
following equation can calculate the maximum normal stress induced by the bending moment
on the critical cross-section:
max
D K
t
Mc
I
D K
t
M
I=c
D K
t
M
Z
; (4.34)
where
max
is the maximum bending stress on the critical cross-section of a beam, which could
be maximum tensile normal stress or maximum compression normal stress. K
t
is stress concen-
tration factor. M is the resultant internal bending moment on the critical cross-section. I is the
moment of inertia of the critical cross-section and c is the largest distance between a point on
206 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
the outer layer of the cross-section and the neutral axis. Z is section modulus of the cross-section
and is equal to I =c.
For a beam, the normal stress by bending moment typically is significantly larger than
the shear stress induced by the shearing force. erefore, the shear stress is negligible. When
maximum normal stress due to a bending moment in a beam on critical cross-section exceeds
material strengths such as the yield strength for ductile material and ultimate tensile strength
for brittle material, the beam will be treated as a failure.
For a beam made of brittle material, the limit state function of a beam will be:
g
.
S
u
; K
t
; Z; M
/
D S
u
K
t
M
Z
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure:
(4.35)
For a beam made of ductile material, the limit state function of a beam will be:
g
S
y
; K
t
; Z; M
D S
y
K
t
M
Z
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure;
(4.36)
where S
u
and S
y
are the material’s ultimate strength and yield strength. e rest of the param-
eters has the same meaning as those in Equation (4.34).
When a PDF can describe loadings, we can use Equations (4.35) and (4.36) to calculate
the reliability of a component by using the H-L method, R-F method, or Monte Carlo method,
which has been discussed in Chapter 3.
When loadings on a beam are described by a PMF, we can use the total probability theo-
rem Equation (2.24) in Chapter 2 to calculate the reliability of a beam. e reliability of a beam
under such loadings will be:
R D
n
X
iD1
.
p
i
R
i
/
; (4.37)
where p
i
is the PMF when the bending moment M is equal to M
i
. R
i
is the reliability of the
beam under the bending moment M
i
, which can be calculated by using the limit state func-
tions (4.35) or (4.36). R is the reliability of the beam under such bending moments M .
We will show two examples to demonstrate how to use the limit state functions (4.35)
and (4.36) to calculate the reliability of a beam under a bending moment.
Example 4.17
A schematic of a beam with a round cross-section is subjected to two concentrated forces P
B
and P
D
as shown in Figure 4.14. e beam is supported at the sections A and C . e beam
is made of a ductile material. e yield strength S
y
of the beam’s material follows a normal
4.9. RELIABILITY OF A BEAM UNDER BENDING MOMENT 207
P
B
P
D
P
B
P
P
D
P
A
B
C
D
L
1
L
2
L
3
Figure 4.14: A schematic of a beam under two concentrated loads.
distribution with a mean
S
y
D 34,500 (psi) and a standard deviation
S
y
D 3120 (psi). e
beam is a round bar with a diameter d D 1:375 ˙ 0:005
00
. e concentrated loads at sections B
and D are: P
B
D 900 ˙ 100 (lb) and P
D
D 600 ˙ 60 (lb). e geometric dimensions L
1
, L
2
,
and L
3
of the beam are: L
1
D 10 ˙ 0:010
00
, L
2
D 10 ˙ 0:010
00
, and L
3
D 5 ˙ 0:010
00
. Use the
H-L method to calculate the reliability of the beam.
1. e maximum bending moment and maximum bending stress.
According to the shear force and bending moment diagrams of this problem, the maximum
bending moment will be in section B and is equal to:
M
B
D
P
B
L
2
C P
D
L
3
L
1
C L
2
L
1
: (a)
Per Equation (4.33), the maximum bending stress at the cross-section B with a maximum
bending moment of the beam will be:
max
D
Mc
I
D
P
B
L
2
CP
D
L
3
L
1
CL
2
L
1
.d=2/
d
4
=64
D
32.P
B
L
2
C P
D
L
3
/L
1
d
3
.L
1
C L
2
/
: (b)
2. e limit state function of this example.
208 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
Since the beam materials is a ductile material, the limit state function of this beam per
Equation (4.35) will be:
g
d; L
1
; L
2
; L
3
; P
B
; P
D
; S
y
D S
y
32.P
B
L
2
C P
D
L
3
/L
1
d
3
.L
1
C L
2
/
D
8
ˆ
<
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(c)
ere are seven random variables in the limit state function (c). All dimensional random
variables d; L
1
, L
2
, and L
2
can be treated as normal distributions. We can use Equa-
tion (
4.1) to calculate their means and standard deviations. e concentrated loading
P
B
and P
D
can also be treated as normal distributions. We can use Equation (4.2) to calculate
their means and standard deviations. All distribution parameters of random variables in
the limit state function (c) are displayed in Table 4.33.
Table 4.33: e distribution parameters of random variables in Equation (c)
S
sy
(psi)
d (in)
L
1
(in) L
2
(in)
μ
S
y
σ
S
y
μ
d
σ
d
μ
L
1
σ
L
1
μ
L
2
σ
L
2
34500 3120 1.375 0.00125 10 0.0025 10 0.0025
L
3
(in) P
B
(lb) P
D
(lb)
μ
L
3
σ
L
3
μ
P
B
σ
P
B
μ
P
D
σ
P
D
5 0.0025 900 25 600 15
3. Reliability R of the beam under bending.
e limit state function (c) contains seven normally distributed random variables and is
a nonlinear function. We will follow the H-L method discussed in Section 3.6 and the
program flowchart in Figure 3.6 to create a MATLAB program. e iterative results are
listed in Table 4.34. From the iterative results, the reliability index ˇ and corresponding
reliability R of the beam in this example are:
ˇ D 2:7810 R D ˆ
.
2:7810
/
D 0:9973:
4.9. RELIABILITY OF A BEAM UNDER BENDING MOMENT 209
Table 4.34: e iterative results of Example 4.17 by the H-L method
#
d
*
L
1
*
L
2
*
L
3
*
P
B
*
P
D
*
S
y
*
β
*
|∆β
*
|
1
1.4375 10 10 5 900 600 25718.04 2.785902
2
1.437448 10.00001 10 5.000006 909.4704 601.7047 25912.47 2.780998 0.004903
3
1.437447 10.00001 10 5.000006 909.4547 601.7018 25912.17 2.781004 5.89E-06
Example 4.18
A schematic of a segment of a beam is shown in Figure 4.15. A PMF can describe the internal
bending moments on the stepped section.
P
.
M D M
i
/
D
(
p
1
D 0:72 M
i
D 1600 (lb/in)
p
2
D 0:28 M
i
D 2000 (lb/in):
(a)
e dimensions of the stepped cross-section are the heights h
1
D 1 ˙ 0:005
00
; h
2
D 1:5 ˙
0:005
00
and the thickness b D 0:5 ˙ 0:005
00
. e radius of the fillet is r D 1=8
00
. e beam ma-
terial is brittle. e ultimate material strength S
u
(ksi) follows a lognormal distribution with a
log mean
ln S
u
D 4:10 and a standard deviation
ln S
u
D 0:196. Calculate the reliability of the
stepped beam.
h
1
= 1 ± 0.005˝
r = 1/8˝
b = 0.5 ± 0.005˝
h
2
= 1.5 ± 0.005˝
Figure 4.15: A schematic of a segment beam.
Solution:
1. e maximum bending stress of the beam.
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