Topic 4-4 Approximation methods
Solved problems
Problem 4-4-1
In the following engineering formula
Z = X Y2 W1/2
X:
Y:
W:
uniformly distributed between 2 and 4
beta distribution between 0 and 3 with parameters q = 1 and r = 2
exponentially distributed with a median of 1
X, Y and W are statistically independent.
(a) Determine the mean values and variances of X, Y and W respectively. (Hint: you may use Ang & Tang
Table 5.1 to facilitate your calculation)
(Ans. 3, 1/3; 1, 1/2; 1.443, 2.081)
(b) Estimate the mean, variance and c.o.v. of Z using the first order approximation.
(Ans. 3.6, 29.7, 1.51)
Solution:
(a) Using Ang & Tang Table 5.1,
E(X) = (4 + 2) / 2 = 3, Var(X) = (4 - 2)2 / 12 = 1/3;
E(Y) = 0 + (3 - 0) / (1 + 2) = 1, Var(Y) = 12(3 - 0)2/(1+2)2(1+2+1) = 1/2;
Also, for W, since we know its median is 1, i.e.
1
 e  dx  0.5  1  e   0.5    ln 2 , hence
 x

0
E(W) = 1/  1.443, Var(W) = 1/2  2.081
(b) Since Z = XY2W0.5, we compute the approximate values
E(Z)  E(X) [E(Y)]2 [E(W)]0.5 = 3121.4430.5  3.6, and
 Z 
 Z 
 Z 
Var ( X )    Var (Y )  
Var (W )
Var(Z)  

 X  
 Y  
 W  
2

2

2

2

2


2
= Y 2W 0.5  Var ( X )  2 XYW 0.5  Var (Y )  0.5 XY 2W 0.5  Var (W )
where  denotes "evaluated at the mean values X = 3, Y = 1, W = 1.443", hence
Var(Z)  29.7,
29.7
Thus the c.o.v. of Z is
 1.51
3.6
Problem 4-4-2
A clay liner is constructed to reduce the amount of pollutants migrating from the landfill to the surrounding
environment.
Land Fill X
CLAY
LINER
Y
Suppose X is the flow rate of pollutants from the landfill without the installation of the clay liner; L is the
effectiveness of the clay liner, such that the flow rate of pollutants ejected to the environment is given by
Y = (1 - L) X
X follows a log-normal distribution with a mean of 100 units per year and a c.o.v. of 20%. Acceptable
performance requires that Y should not exceed 8 units per year.
(a) Suppose the quality of construction of the clay liner can be assured to have an effectiveness of 0.95. What is
the probability of acceptable performance? (ans. 0.993)
(b) Suppose the contractor cannot guarantee the effectiveness to be 0.95. Instead he offers to provide a liner
with a mean effectiveness of 0.96 but with a standard deviation of 0.01.
(i)
Estimate the mean and variance of Y using the first order approximations procedure. (ans. 4, 1.64)
(ii)
Determine if this offer from the contractor will yield a larger probability of acceptable performance.
You may assume Y to be log-normal with values of mean and variances estimated in part (i). (ans.
no)
Solution:
(a) Let X’  ln(X). Since X is log-normal, X’ is normal with parameters
X’  X = 0.20, and
X’  ln 100 – X2 / 2
= ln 100 – 0.202 / 2
= 4.585170186
When L is a constant (0.95), we have
Y = (1 – 0.95)X = 0.05 X
Taking the log of both sides and letting Y’  ln(Y)
 Y’ = ln 0.05 + X’,
i.e. Y’ is a constant plus a normal variate, hence Y’ is normal with parameters
Y’ = ln 0.05 + X’ 1.589437912, and
Y’ = X’  0.20
Hence
P(acceptable) = P(Y  8)
= P(Y’  ln 8)
Y 'Y ' ln 8  1.589437912
= P(
)

Y'
0.20
= (2.450018146)
 0.993
(b)
(i)
When L is also a random variable, Y = (1 – L)X is a non-linear function of two random variables,
hence we need to estimate its mean and variance by
E(Y)  Y(L = L, X = X)
= (1 – L)(X) = (1 – 0.96)(100)
= 4, and
 Y 
 Y 
 Var ( L)  
 Var ( X )
 L  
 X  
2
2
Var(Y)  
=  X  X 100, L 0.96 (0.012 )  1  L  X 100, L 0.96 (100  0.2) 2
2
2
= 100000.0001 + 0.0016400
= 1.64
(ii)
Assuming a log-normal distribution for Y and using its estimated mean and variance from
(i), we have the approximate parameter values
Y 
1.64
, hence
4
Y =
ln( 1   Y2 )

ln( 1  1.64 / 16) = 0.312378502;
Y  ln 4 – 0.3123785022 / 2 = 1.337504197, thus
ln Y  Y ln 8  1.33750419 7
P(Y  8) = P(
)

Y
0.31237850 2
= (2.375122938)
 0.991
No, this offer from the contractor does not yield a larger performance acceptance probability.
Exercises
Exercise 4-4-1
Manning's formula frequently used for determining flow capacity of a circular storm sewer is
Qc = 0.463 n-1 D2.67 S0.5
in which Qc is the flow rate (in ft3/sec); n is the roughness coefficient; D is the sewer diameter (in ft.); and S is
the pipe slope (ft/ft).
Due to manufacturing imprecision and construction error, the sewer diameter and pipe slope, together with the
roughness coefficient, are subject to uncertainty. Consider a section of sewer in a storm sewer system. The
statistical properties of n, D and S in the above Manning's formula are given in the following table.
Variable
n
D (ft.)
S (ft/ft)
Mean
0.015
3.0
0.005
Coefficient of variation
0.10
0.02
0.05
Suppose that all three random variables are statistically independent. Please perform the following analyses.
(a) Use first-order second-moment method to estimate the mean and variance of the sewer flow carrying
capacity. (ans. 41.0 ft3/sec, 22.7 (ft3/sec)2)
(b) Calculate the percentage of contribution of each random variable to the total uncertainty of the sewer flow
capacity and identify the relative importance of each random variable. (ans. n: 74.2%, D: 21.2%; S: 4.6%)
(c) Assuming that sewer flow capacity follows the log-normal distribution with the mean and variance obtained
from Part (a), determine the reliability that the sewer capacity can accommodate an inflow of 30 ft3/sec.
(ans. 0.996)
Exercise 4-4-2
The amount of rainfall R in each rainstorm is independently log-normally distributed with a mean of 1 inch and
a c.o.v. of 100%.
(a)
Suppose 50 rainstorms have occurred last year and other contributions to rainfall is negligible. Estimate
the probability that last year’s annual rainfall exceeded 60 inches?
(b)
Suppose the amount of water collected from a single rainstorm is predicted by the following equation:
V = N R ln(R + 1)
Where V is the volume of water in thousand gallons and N is a random variable (independent of R) with a
mean value 1 and a c.o.v. of 50% representing the error of the prediction model.
(i)
Determine the mean and c.o.v. of V using the first order approximate formulas.
(ii)
Are the approximations good in this case? Why?
Exercise 4-4-3
A cantilever beam is subjected to a concentrated load F applied at a distance X from the fixed end. The bending
moment caused by this load at the fixed end is
M = FX
Suppose F is log-normally distributed with a mean value of 1 and c.o.v. of 30%; and X follows a uniform
distribution between 0 and 10.
(a)
Determine the mean and variance of M using the first order approximate formulas. (Hint: Table 5.1 on p.
224 of Ang & Tang may be used to determine mean and variance of X.)
(b)
Suppose the beam is subjected to 20 similar loads whose magnitudes and locations are statistically
independent. Determine the mean and variance of the total bending moment caused by all the loads.
(c)
What is the probability that the total bending moment will exceed 120?
(d)
The moment capacity of the beam is the amount of bending moment it can support without failure.
Suppose the moment capacity C is a discrete random variable which can take on the values 100, 120 and
140 with respective probabilities 0.2, 0.3 and 0.5. Determine the probability of failure of the beam
subjected to the 20 loads.