226 4. RELIABILITY OF A COMPONENT UNDER STATIC LOAD
3. Reliability of the column under combined stresses.
We can use the Monte Carlo method to calculate the reliability of this example based
on two limit state functions (f) and (g). e Monte Carlo method has been discussed
in Section 3.8. We can follow the procedure of the Monte Carlo method and the pro-
gram flowchart in Figure 3.8 to create a MATLAB program. Since this problem is no
very big and complicated, we will use the trial number N D 1,598,400 from Table 3.2 in
Section 3.8. e estimated reliability R of this column at the critical point B is:
R D 0:9983:
e estimated reliability R of this column at the critical point A is:
R D 1:0000:
erefore, the reliability of this complement in this example will be
R D 0:9983:
Example 4.24
A plane stress element of a component of a brittle material at the critical point is shown in
Figure 4.22. e normal compressive stress
x
(ksi), the normal tensile stress
y
, and the shear
stress
xy
follows normal distributions. eir distribution parameters are listed in Table 4.48.
Table 4.48: e stresses in a plane stress element
σ
x
(ksi) σ
y
(ksi) τ
xy
(ksi)
μ
σ
x
σ
σ
x
μ
σ
y
σ
σ
y
μ
τ
xy
σ
τ
xy
31.2 2.51 1.80 0.085 15.0 2.31
e component’s ultimate tensile strength S
ut
follows a normal distribution with a mean
S
ut
D 22:00 (ksi) and standard deviation
S
ut
D 1:80 (ksi). Its ultimate compression strength
S
uc
follows a normal distribution with a mean
S
uc
D 82:00 (ksi) and standard deviation
S
uc
D
10:50 (ksi). Calculate the reliability of the component by using the BCM theory.
Solution:
1. e two principal stresses in a plane stress.
As shown in Figure 4.22,
x
is normal compressive stress,
y
is normal tensile stress, and
xy
is negative shear stress. In the following calculations,
y
,
x
, and
xy
will be the values