r - The Hong Kong Polytechnic University

Example 3
6
A portal frame is shown in Figure 1. A , A , P are random variables. Modulus of elasticity is E = 2.0 × 10
1 2
KPa. Moments of inertia of beams and columns are I = A 2
/12
1 1 and I 2
= A /6
2 2
. The limit state function
is defined as in [20]:
g( x) = 0.01 −u
( x )
3
(20)
Where, u ( x ) is the horizontal displacement of point 3. The statistical parameters are shown in Table 4. All the
3
variables are assumed to be uncorrelated.
P
2 A 2 3
A1 A1
4
1 4
4
Figure 1 The diagram of the portal frame (unit:m)
Table 4 The statistical parameters of variables of example 3
Coefficient of
Variables Uint Mean μ
variation
Distribution Type
A1
m 2 0.36 0.036 Log-normal
A2
m 2 0.18 0.018 Log-normal
P kN 20 5 Extreme-I
There are 3 random variables. In accordance with the distribution, samples were distributed uniformly within
μ − 3 σ, μ + 3σ
. In order to ensure the accuracy of response surface, uniform distribution experimental
[ ]
2
design table ( 30
30 )
U is used to decide 30 tests. The failure probability is 2.299×10 -3 and the reliability
index is 2.8339 calculated by UDM-SVM RSM proposed in this paper. The failure probability obtained by
Monte-Carlo simulation based on importance sampling using 2000 simulations is 2.3223×10 -3 (Deng et al.
2005) and the reliability index is 2.8307 . It is evident that these results correspond quite well.
Example 4
Figure 2 is a calculation diagram of plane frame structure with three spans and 12 layers. Elasticity Modulus of
7
each element is E = 2.0× 10 Pa . The relationship between inertia moment and area of element section
is I = α A i2 ( i = 1,2,3,4,5) . The area A
i i i
of element section and external load P are chosen as random
variables. To consider the condition of normal use, the maximum allowable deformation [ U ] is equal to 0.096
according to the requirements of the specification. Therefore, the limit state function can be expressed as:
G = 0.096 − U ( A, A , A , A , A , P)
(21)
max 1 2 3 4 5
The statistical characteristics of variables are shown in tableⅡ.
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