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).