210 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
e maximum bending stress of this beam will be on the stepped cross-section and can be
calculated per Equation (4.34):
max
D K
t
Mc
I
D K
t
M.h
1
=2/
bh
3
1
=12
D
6K
t
M
bh
2
1
: (b)
2. e limit state functions of the beam.
Since the beam material is brittle, we will use Equation (4.35) to establish the limit state
function.
When the resultant internal bending moment M is equal to M
1
D 1:6 (klb.in), the limit
state function of the beam is:
g
.
S
u
; K
t
; b; h
1
/
D S
u
6K
t
M
1
bh
2
1
D
8
ˆ
<
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(c)
When the internal resultant bending moment M is equal to M
2
D 2:0 (klb.in), the limit
state function of the beam is:
g
.
S
u
; K
t
; b; h
1
/
D S
u
6K
t
M
2
bh
2
1
D
8
ˆ
<
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(d)
ere are four random variables in the limit state functions (c) and (d). e stress concen-
tration factor K
t
can be treated as a normal distribution. Its mean can be obtained from
design handbook by using the nominal dimensions, that is, the heights h
1
D 1
00
; h
2
D 1:5
00
and the radius of the fillet r D 1=8
00
. e stress concentration factor under bending for this
example is 1.72. Its standard deviation can be estimated per Equation (4.10). e geomet-
ric dimensions can be treated as normal distributions per Equation (4.1). e distribution
parameters of four random variables in the limit state functions (c) and (d) are listed in
Table 4.35.
Table 4.35: e distribution parameters of random variables in Equations (c) and (d)
S
u
(psi) K
t
(in)
b (in) h
1
(in)
μ
lnS
u
σ
lnS
u
μ
K
t
σ
K
t
μ
b
σ
b
μ
h
1
σ
h
1
4.10 0.196 1.72 0.086 0.5 0.00125 1 0.00125
4.9. RELIABILITY OF A BEAM UNDER BENDING MOMENT 211
3. e reliability of the beam.
e reliability of the beam in this example will be calculated per Equation (4.37) because
the internal bending moments are specified by a PMF:
R D p
1
R
1
C p
2
R
2
D 0:72R
1
C 0:28R
2
; (e)
where R
1
is determined by the limit state function (c) and R
2
by the limit state func-
tion (d). e limit state functions (c) and (d) contains one lognormal distribution random
variable and three normal distributions. We can follow the procedure of the R-F method
in Section 3.7 and the flowchart in Figure 3.7 to create a MATLAB program. e iterative
results for the limit state function (c) are listed in Table 4.36. e reliability index ˇ and
corresponding reliability R
1
of the beam in this example are:
ˇ D 2:9826 R
1
D ˆ
.
2:9826
/
D 0:9986:
Table 4.36: e iterative results of Example 4.18 for the limit state function (c)
Iterative #
S
u
*
K
t
*
b
*
h
1
*
β
*
|∆β
*
|
1 61.51051 1.72 0.5 0.732724 3.510922
2 19.32361 1.794619 0.499946 1.335415 3.359916 0.151006
3 28.96702 1.788618 0.499948 1.08888 3.017535 0.342381
4 33.58777 1.781819 0.499953 1.009281 2.983043 0.034492
5 34.19932 1.781332 0.499954 1.000079 2.982618 0.000425
6 34.20802 1.781339 0.499954 0.999954 2.982618 7.77E-08
e iterative results for the limit state function (d) are listed in Table 4.37. e reliability
index ˇ and corresponding reliability R
2
of the beam in this example are:
ˇ D 1:87774 R
2
D ˆ
.
2:9826
/
D 0:9698:
erefore, the reliability of the beam in this example per Equation (e) is
R D 0:72R
1
C 0:28R
2
D 0:9905:
..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.
Reset