4.7. RELIABILITY OF A COMPONENT UNDER DIRECT SHEARING 195
When the shear force on a direct shearing section is described by a PMF, we can use
the total probability theorem Equation (2.24) in Chapter 2 to calculate the reliability of the
components. In this case, the shearing force is expressed by a PMF as:
P
.
V
D
V
i
/
D p
i
i D 1; 2; : : : ; n; (4.24)
where p
i
is the PMF when the shearing force V is equal to V
i
. Since it is a PMF, the reliability
of the component under such the shearing force will be:
R D
n
X
iD1
.
p
i
R
i
/
; (4.25)
where R is the reliability of component under such shearing forces. R
i
is the reliability of a
component when the direct shearing force V is equal to V
i
. Even V
i
can be a constant value; we
still need to use Equations (4.22) or (4.23) to calculate R
i
because other variables in the limit
state function are still random variable.
We will use two examples of direct shearing to demonstrate how to calculate the reliability
of a component under direct shearing.
Example 4.12
A bar under a tensile loading P D 0:90 ˙ 0:0:08 (klb) is connected to the ground supporter
by a single shearing pin. e pin material is brittle. e ultimate shear strength S
su
(ksi) of
the pin follows a log-normal distribution with the distribution parameters
ln S
su
D 3:25 and
ln S
su
D 0:181. e pin has a diameter d D 0:25 ˙ 0:005
00
. Calculate the reliability of this pin.
Solution:
1. e limit state function of the pin.
e limit state function of this single shearing pin per Equation (4.22) will be:
g
.
S
su
; d; P
/
D S
su
P
d
2
=4
D S
su
4P
d
2
D
8
ˆ
<
ˆ
:
> 0 Safe
0
Limit state
< 0 Failure:
(a)
e loading can be simplified as a normal distribution per Equation (4.2). e pin diameter
will be treated as a normal distribution per Equation (4.1). e type of distributions and
corresponding distribution parameters of random variables in the limit state function (a)
are listed in Table 4.26.
2. e reliability of the single shearing pin.
ere are three random variables in the limit state function (a). One is a lognormal distribu-
tion, and two are normal distributions. We can follow the procedure of the R-F method in