3.3. DIMENSION OF A COMPONENT UNDER STATIC LOADING 169
erefore, the thickness of the vessel with the required reliability 0.999 under the specified
loading is
d D 0:358 ˙ 0:030
00
:
Example 3.17
A schematic of the critical cross-section of a rectangular column is subjected to a combined load-
ing which can be simplified as a compressive force and bending moment, as shown in Figure 3.9.
e compression force F
z
is along the z-axis and through the centroid of the cross-section, and
the bending moment M
y
is about the neutral y-axis. e compression force is F
z
D 87 ˙ 8 (klb).
e bending moment is M
y
D 610 ˙ 40 (klb.in). e width b of the column is b D 3 ˙ 0:010
00
.
e column is made of brittle material. Its ultimate tensile strength S
ut
follows a normal dis-
tribution with a mean
S
ut
D
22:0
(ksi) and a standard deviation
S
ut
D
1:6
(ksi). e ultimate
compression strength S
uc
follows a normal distribution with a mean
S
uc
D 43:0 (ksi) and stan-
dard deviation
S
uc
D 3:30 (ksi). Use the maximum normal stress (MNS) theory with the mod-
ified Monte Carlo method to design the height h of the column with a reliability 0.95 when the
height h has a tolerance ˙0:010
00
.
M
y
F
z
h
b
y
x
z
A
B
Figure 3.9: Schematic of a segment of a column under compressions and bending.
Solution:
In this example, there is no dimension-dependent parameter.
(1) e maximum tensile normal stress and normal compressive stress.
e maximum tensile stress on the critical cross-section is at the top line
A
and is equal
to
A
D
bending
compression
D
M
y
h=2
bh
3
=12
F
z
bh
D
6M
y
bh
2
F
z
bh
: (a)
170 3. THE DIMENSION OF A COMPONENT WITH REQUIRED RELIABILITY
e maximum compression stress on the critical cross-section as shown in Figure 3.9 is the
bottom line B and is equal to
B
D
bending
C
compression
D
M
y
h=2
bh
3
=12
C
F
z
bh
D
6M
y
bh
2
C
F
z
bh
: (b)
(2) e limit state functions.
e stress at line A is the maximum tensile stress. e limit state function of the column
in line A by using the maximum normal stress theory [1] is
g
S
ut
; F
z
; M
y
; b; h
D S
ut
6M
y
bh
2
F
z
bh
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(c)
e stress in line B is the maximum compression stress. e limit state function of the column
at line B by using the maximum normal stress theory [1] is
g
S
uc
; F
z
; M
y
; b; h
D S
uc
6M
y
bh
2
C
F
z
bh
D
8
ˆ
ˆ
<
ˆ
ˆ
:
> 0 Safe
0 Limit state
< 0 Failure:
(d)
ere are five random variables in these two limit state functions (c) and (d). e loading F
Z
and M
y
can be treated as normal distributions. eir distribution parameters can be determined
by Equation (1.2). Geometric dimensions b and h can be treated as normal distributions. eir
distribution parameters can be determined per Equation (1.1). e distribution parameters for
these two limit state functions are listed in Table 3.39.
Table 3.39: e distribution parameters of random variables in Equations (c) and (d)
S
ut
(ksi) S
uc
(ksi) F
z
(klb) M
y
(klb.in) b (in) h (in)
μ
S
ut
σ
S
ut
μ
S
uc
σ
S
uc
μ
F
z
σ
F
z
μ
M
y
σ
M
y
μ
b
σ
b
μ
h
σ
h
22 1.6 43 3.3 87 2 610 10 3 0.0025
μ
h
0.0025
(3) Use the modified Monte Carlo method to determine the mean
h
of the height h.
Following the modified Monte Carlo method discussed in Section 3.2.5 and the flowchart
shown in Figure 3.5, we can make MATLAB programs for the limits state functions (c) and
(d).
e iterative results for the limit state function (c) are listed in Table 3.40.
3.3. DIMENSION OF A COMPONENT UNDER STATIC LOADING 171
Table 3.40: e iterative results of Example 3.17 for the limit state function (c)
Iterative #
μ
h
*
R
*
∆R
*
1 5.971437 0.94755 -0.00245
2 5.972437 0.947906 -0.00209
… … … …
8 5.978437 0.950026 2.63E-05
9 5.979437 0.950382 0.000382
According to the result for the tension strength obtained from the program, the mean of
the height h with a reliability 0.99 is
h
D 5:980
00
: (e)
e iterative results for the limit state function (d) are listed in the Table 3.41.
Table 3.41: e iterative results of Example 3.17 for the limit state function (d)
Iterative #
μ
h
*
R
*
∆R
*
1 6.098313 0.948051 -0.00195
2 6.099313 0.948417 -0.00158
… … … …
6 6.103313 0.949867 -0.00013
7 6.104313 0.950227 0.000227
According to the result for the compression strength obtained from the program, the
mean of the height h with a reliability 0.99 is
h
D 6:105
00
: (f)
e height h of the rectangular column with the required reliability 0.95 under the specified
loading is the larger one of (e) and (f). erefore, the height h of the rectangular column with
the required reliability 0.95 is
h D 6:105 ˙ 0:010
00
:
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