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