212 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
Table 4.37: e iterative results of Example 4.18 for the limit state function (d)
Iterative #
S
u
*
K
t
*
b
*
h
1
*
β
*
|∆β
*
|
1 61.51051 1.72 0.5 0.81921 2.086685
2 35.95712 1.76435 0.499968 1.085224 1.909968 0.176717
3 41.5102 1.759636 0.49997 1.008678 1.87811 0.031858
4 42.21348 1.759073 0.499971 1.00008 1.877739 0.000371
5 42.22275 1.759077 0.499971 0.999971 1.877739 5.84E-08
4.9.2 RELIABILITY OF A BEAM UNDER BENDING FOR A
DEFLECTION ISSUE
When the maximum deflection of a beam exceeds the allowable deflection, the beam is treated
as a failure. e general limit state function of a beam under bending for a deflection is:
g
.
y
max
/
D y
max
D
8
ˆ
<
ˆ
:
> 0 Safe
D 0 Limit state
< 0 Failure;
(4.38)
where is the allowable deflection and is typically treated as a deterministic constant. y
max
is
the maximum deflection of a beam and is a function of other random variables.
ere is no general formula for the maximum deflection of a beam. It will be determined
per case. For a simple support beam under a concentrated load at the middle of the beam, as
shown in Figure 4.16a, the limit state function for a deflection issue is:
g
.
E; I; L; P
/
D y
max
D
PL
3
48EI
D
8
ˆ
<
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure;
(4.39)
where E is the beam’s material Young’s modulus, I is the moment of inertia of the cross-section
with a maximum deflection, P is the external concentrate forces on the middle of the beam, and
L is the beam length. Typically, E; I; L, and P are random variables.
For a cantilever beam under a concentrated load at the free end of the beam, as shown in
Figure 4.16b, the limit state function for a deflection issue is:
g
.
E; I; L; P
/
D y
max
D
PL
3
3EI
D
8
ˆ
<
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure;
(4.40)
where P is a concentrated force at the free end of a cantilever beam. e rests of the symbols
have the same meaning as those in Equation (4.39).
4.9. RELIABILITY OF A BEAM UNDER BENDING MOMENT 213
L L
0.5L
P
P
(a) Simply Supported Beam (b) Cantilever Beam
Figure 4.16: Two typical beams.
We will use one example to demonstrate how to calculate the reliability of a beam for a
deflection issue.
Example 4.19
A simple support beam as shown in Figure 4.17 is subjected to a uniform distributed load
w D 10 ˙ 1:5 (lb/in). e span of the beam is L D 20 ˙ 0:065
00
. e beam has a constant rect-
angular cross-section with the height h D 1:25 ˙ 0:005
00
and the thickness b D 0:5 ˙ 0:005
00
.
e Young’s modulus of the beam material follows a normal distribution 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 de-
flection of the beam is D 0:010
00
. Calculate the reliability of this beam for a deformation issue.
L
w
Figure 4.17: A simple support beam under a uniform distributed load w.
Solution:
1. e maximum deflection of the beam under a uniform distributed load.
For a simple support beam under a uniform distributed load, the maximum deflection of
the beam will happen in the middle of the beam and can be calculated by the following
equation:
y
max
D
5wL
4
384EI
D
5wL
4
384E.bh
3
=12/
D
5wL
4
32Ebh
3
; (a)
where E is material Young’s modulus, w is the uniform distributed load, L is the span of
the beam, and h and b are the height and thickness of the beam.
214 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
2. e limit state function of the beam for a deformation issue.
e limit state function of this beam per Equation (4.38) will be:
g
.
E; w; b; h; L
/
D y
max
D 0:010
5wL
4
32Ebh
3
D
8
ˆ
<
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(b)
e uniform distributed load can be treated as a normal distribution per Equation (4.2).
e geometric dimensions can be treated as normal distributions per Equation (4.1). ere
are five normal distributed random variables in the limit state function (b). eir distribu-
tion parameters are listed in Table 4.38.
Table 4.38: e distribution parameters of random variables in Equation (b)
E (psi)
w (lb/in) b (in) h (in) L (in)
μ
E
σ
E
μ
w
σ
w
μ
b
σ
b
μ
h
σ
h
μ
L
σ
L
2.76×10
7
6.89×10
5
10 0.375 0.5 0.00125 1.25 0.00125 20 0.01625
3. e reliability of the beam for a deformation issue.
e limit state function (b) is a nonlinear equation with five normal distributed ran-
dom variable. We can 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 4.39. From the iterative results, the reliability index ˇ and corresponding reliability
R of this beam are:
ˇ D 1:68155 R D ˆ
.
1:68155
/
D 0:9537:
Table 4.39: e iterative results of Example 4.19 by the H-L method
#
E
*
w
*
b
*
h
*
L
*
β
*
|∆β
*
|
1 27,600,000 10 0.5 1.25 20.37968 1.641913
2 26,977,766 10.50873 0.499887 1.24983 20.01083 1.681462 0.039549
3 26,931,076 10.50869 0.499881 1.249822 20.00203 1.681553 9.11E-05
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