# Loads and Structural Design Methods

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```					Loads and Structural Design Methods
– Structural Reliability Analysis and Design

Dr. Lu Dagang

Professor
School of Civil Engineering
Harbin Institute of Technology
Contents
Chapter 1 Introduction

Chapter 2 Basic Concepts of Structural Reliability Theory

Chapter 3 Component Reliability Analysis of Structures

Chapter 4 System Reliability Analysis of Structures

Chapter 5 Statistical Analysis of Loads

Chapter 6 Statistical Analysis of Structural Member Resistance

Chapter 7 Reliability-Based Design Methods of Structures
Chapter 1

Introduction
Chapter 1: Introduction

Contents
1.1 Uncertainties in Structural Engineering

1.2 Objectives and Tasks of Structural Design

1.3 Development of Structural Design Method

1.4 Probability Levels of Structural Design

1.5 Probability Foundations for Structural Reliability Theory
Chapter 1 Introduction

1.1 Uncertainties in Structural Engineering
1.1 Uncertainties in Structural Engineering …1

There are three kinds of uncertain information in
Structural Engineering, which include

   Random Information
–   Probability Theory, Statistics, Random Process Theory,
Random Field Theory
   Fuzzy Information
–   Fuzzy Mathematics, Fuzzy System Theory

   Unascertain or Incomplete Information
–   Unascertain Mathematics, Grey Mathematics, Rough
Set Theory, Evidence Theory, etc.
1.1 Uncertainties in Structural Engineering …2

We can divide the above three uncertainties into
two categories

   Objective Uncertainty
–   Randomness

   Subjective Uncertainty
–   Fuzziness
–   Unascertainty or Incompleteness
1.1 Uncertainties in Structural Engineering …3

The uncertainties in structural engineering can
also be put into two major categories with regard
to causes
   Natural causes of uncertainty
– Load uncertainties such as wind, earthquake, snow, ice,
– Resistance uncertainties such as material strength,
geometry, fabrication error, etc.

   Human causes of uncertainty
–   Intended human errors
–   Unintended human errors
1.1 Uncertainties in Structural Engineering …4

There are three types of uncertainties as to the
randomness in structural engineering
   Physical Uncertainty
–   Variation of load and resistance that is inherent in the
quantity being considered
   Statistical Uncertainty
–   Uncertainty arising from estimating parameters based
on a limited sample size

   Model Uncertainty
–   Uncertainty due to simplifying assumptions, unknown
boundary conditions, and unknown effects of other
variables
1.1 Uncertainties in Structural Engineering …5

Conclusions
1. Many sources of uncertainty are inherent in structural
planning, design, construction, and operation.

capacities of structural members are not deterministic
quantities, they are random variables, and therefore
absolute safety of structures cannot be achieved.

3. Due to uncertainty, structures must be designed to serve
their function with a finite probability of failure.
Chapter 1 Introduction

1.2 Objectives and Tasks of Structural Design
1.2 Objectives and Tasks of Structural Design …1

1.2.1 Main Objectives of Structural Design
The main objectives of structural design is to satisfy the specified
function requirements with the optimal economic ways
   The specified function requirements
1. To carry all kinds of actions during the stage of proper
construction and operation
2. To have the favorable working performance during the
stage of proper operation
3. To have the adequate durability during the stage of
proper maintenance
4. To keep the essential integrity when encountered
occasional actions such as earthquakes, tornadoes,
explosions, fires, etc.
1.2 Objectives and Tasks of Structural Design …2

   As to the above four items of requirements
1.    The items 1 is called structural safety
2.    The item 2 is called structural serviceability
3.    The item 3 is called structural durability
4.    The items 4 is called structural integrity

Structural safety

Structural serviceability
Structural reliability
Structural durability

Structural integrity
1.2 Objectives and Tasks of Structural Design …3

1.2.2 Basic Tasks of Structural Design
– Load effect calculation (material mechanics, structural
mechanics, elastic mechanics, plastic mechanics, etc)

2. Structural resistance
– Structural testing of resistance
– Statistical analysis of resistance

3. Balance between load effect and structural resistance
– Structural design
– Structural reliability
Chapter 1 Introduction

1.3 Development of Structural Design Method
1.3 Development of Structural Design Method …1

1. Method of Allowable Stress Design (ASD)
– elastic theory
– design formula:  ≤[ ]
where,  is the stress of an arbitrary point of structural element
[ ] is the allowable stress of material.
u   where,  u is the ultimate stress of material;
[ ] 
K           K is the factor of safety.

–     all uncertainties in both load and resistance are all
considered by a single factor
–     the factor of safety K is estimated by designer’s
experience
1.3 Development of Structural Design Method …2

2. Method of Damage Phase Design (DPD)
– plastic theory
– also named Load Factor Design (LFD)
– design formula: KS F ≤ R p
where, S F is the load effect of the critical section of structural
R p is the load-carrying of structural element when it
reaches the state of damage;
–     all uncertainties in both load and resistance are all
considered by a single load factor
–     the factor of safety K is also estimated by designer’s
experience
1.3 Development of Structural Design Method …3

3. Method of Multiple Factor Limit State Design (MFLSD)
– the concept of Structural Limit State is proposed
– the partial factor of safety is adopted
– Multiple factor limit state design formula:
n
1     f1k f 2k
 G SGk +   Li SLki ≤         R( ,       , )
i=1           R      f f
1      2

load which are calculated by the standard value;
R      is the member resistance;
f ik   is the standard value of material strength;
G      and  Li is the partial factor of loads;
 fi    is the partial factor of material strength;
R      is the partial factor of member resistance.
1.3 Development of Structural Design Method …4

–     Single factor limit state design formula:
1
K1  Si ≤        R
K 2 K3
where, S i is the load effect;
R is the member resistance;
K1 is the load factor, K1  1.2 ;
K 2 is the member strength factor;
K 3 is the additional safety factor considering structural
importance.
K j  K1 K 2
K j is the basic safety factor, which is determined by both load factor K1
and member strength K 2
1.3 Development of Structural Design Method …5

4. Method of Probabilistic Limit State Design (PLSD)
– based on structural reliability theory
– the load and resistance factor design (LRFD) safety-
checking format is used
– the partial factor of safety is determined by structural
reliability theory
– represents the direction of structural design development
– has four levels of probability
   level Ⅰ: semi-probability and semi-experience
   level Ⅱ：approximate probability
   level Ⅲ：full probability
   level Ⅳ：optimal probability
Chapter 1 Introduction

1.4 Probability Levels of Structural Design
1.4 Probability Levels of Structural Design …1

1. Level 0: Empirical Design
– allowable stress design (ASD)

2. Level Ⅰ: Semi-Probability Design
– multiple factor limit state design (MFLSD)
– single factor limit state design (SFLSD)

3. Level Ⅱ: Approximate Probability Design
– load and resistant factor design (LRFD)
– first order and second moment (FOSM) method
– using reliability index as the measure of reliability
– determining target reliability index by way of
calibrating the level Ⅰ codes
1.4 Probability Levels of Structural Design …2

4. Level Ⅲ: Full Probability Design
– using failure probability as the measure of reliability
– exact probability analysis of the overall structural system
– directly using reliability checking as design format

5. Level Ⅳ: Optimal Probability Design
– reliability based optimum design (RBOD)
– using the total life-cycle cost (LCC) minimization as the
optimization criterion
– using reliability as the global constraint
– the optimal target reliability is determined by the utility
analysis or cost-effect analysis
Chapter 1 Introduction

1.5 Probability Foundations for
Structural Reliability Theory
1.5 Probability Foundations for Structural Reliability Theory …1

1.5.1 Definition of Random Variables

Definition: random variable
Let E be a random experiment,  is its sample space, for
  , there exists a real single-value function X ( ) .
If for x  R , { | X ( ) ≤ x} is a random event on event field,
then we can call X ( ) a random variable.
X ( )             X
{ | X ( ) ≤ x}            { X ≤ x} or ( X ≤ x)

discrete random variable
random variable
continuous random variable
1.5 Probability Foundations for Structural Reliability Theory …2

1.5.2 Description of Random Variables

There are three kinds of functions for description of
random variables as follows:
1. Probability Mass Function (PMF)
– for discrete random variable

2. Cumulative Distribution Function (CDF)
– for discrete random variable
– for continuous random variable

3. Probability Density Function (PDF)
– for continuous random variable
1.5 Probability Foundations for Structural Reliability Theory …3

1. Probability Mass Function (PMF)
The probability mass function (PMF) is defined for discrete
random variables as follows: p X ( x) represents probability
that a discrete random variable X is equal to a specific
value x , where x is a real number. Mathematically,
p X ( x)  P( X  x)

2. Cumulative Distribution Function (CDF)
The cumulative distribution function (CDF) is defined for
both discrete and continuous random variables as follows:
FX ( x) represents the total sum (or integral) of all probability
functions (continuous and discrete) corresponding to values
less than or equal to x . Mathematically,
FX ( x)  P( X ≤ x)
1.5 Probability Foundations for Structural Reliability Theory …4

3. Probability Density Function (PDF)
For continuous random variables, the probability density
function (PDF) f X ( x) is defined as the first derivative of
the cumulative function FX ( x) . Mathematically,
dFX ( x)               x
f X ( x)             FX ( x)   f X ( )d 
dx                  

Properties of CDF, PDF and PMF
1.   The CDF is a positive, nondecreasing function whose value is
between 0 and 1: 0 ≤ FX ( x) ≤1
2.   If x1  x2 , then FX ( x1 ) ≤ FX ( x2 )
3.   FX ( )  0 , FX ()  1
4.   For continuous random variable,
b
P(a ≤ x ≤ b)  FX (b)  FX (a)   f X ( )d 
a
1.5 Probability Foundations for Structural Reliability Theory …5

1.5.3 Moments of Random Variables

1. Mean or Expected Value (First Moment)
• The mean value of a random variable X is denoted by  X
–   For a continuous random variable, the mean value is defined as

 X   xf X ( x)dx


–   For a discrete random variable, the mean value is defined as
 X   xi pX ( xi )
all xi

• The expected value of X is commonly denoted by E ( X )
and is equal to the mean value of X
E( X )  X
1.5 Probability Foundations for Structural Reliability Theory …6

• The expected value of X n is called the nth moment of X
–   For a continuous random variable, the nth moment is defined as

E( X )  
n
x n f X ( x)dx


–   For a discrete random variable, the nth moment is defined as
E ( X n )   xin pX ( xi )
all xi

• The mathematical expectation of an arbitrary function g ( X )
of the random variable X is defined as

 g ( X )  E[ g ( X )]   g ( x) f X ( x)dx

1.5 Probability Foundations for Structural Reliability Theory …7

2. Variance and Standard Deviation (Second Moment)
• The variance of a random variable is a measure of the
degree of randomness about the mean value:
  ( x   ) 2 f ( x)dx
             X      X            continuous variable
var( X )  E ( X   X )  
2

       ( xi   X ) 2 p X ( xi )   discrete variable
 all xi

• An important formula
var( X )  E ( X 2 )   X
2

• The standard deviation of X is defined as the positive
square root of the variance:
 X  var( X )
1.5 Probability Foundations for Structural Reliability Theory …8

• The nondimensional coefficient of deviation VX is
defined as the standard deviation divided by the mean:
X
VX 
X
3. Moments of Sample
• If a set of n observations {x1 , x2 , , xn } are obtained for a
particular random variable X , then
–   the true mean  X can be approximated by the sample mean x
1 n
x   xi
n i 1
–   the true standard deviation  X can be approximated by the sample
standard deviation s X
n
 n 2
 ( xi  x )     xi   n( x )
2                     2

s X  i 1           i 1 
n 1             n 1
1.5 Probability Foundations for Structural Reliability Theory …9

1.5.4 Standard Form of Random Variables
• Let X be a random variable. The standard form of X ,
denoted by U , is defined as
X  X
U
X
–   the mean value of U is calculated as follows:
X  X        1                               1
U  E (            )        [ E ( X )  E (  X )]         ( X   X )  0
X           X                              X

–   the deviation of U is calculated as follows:
X  X                1                   X
2
var(U )  E (U 2 )  U  E[(
2
) 2 ]  0  2 E[( X   X ) 2 ]  2  1
X                X                    X
U  1
1.5 Probability Foundations for Structural Reliability Theory …10

1.5.5 Common Random Variables

1. Uniform Random Variables
f X ( x)
 1
         a ≤ x ≤b
1                                       f X ( x)   b  a
ba                                                 0
         otherwise

0
a       X   b        x
ab
FX ( x)                                          X 
2
1                                                  (b  a ) 2
X 
2

0.5                                                    12
0
a        b        x
1.5 Probability Foundations for Structural Reliability Theory …11

2. Normal Random Variables

–   For a general normal RV X
f X ( x)

1           1 x  X 2
f X ( x)                exp[ (      ) ]
X       2      2 X

0               X      x      –   For a standard normal RV U
   The PDF of U is denoted by  (u )
FX ( x)
1                                              1       1
 (u )       exp[ (u ) 2 ]  fU (u )
2      2
0.5
   The CDF of U is denoted by  (u )

0
(u)  1  (u)
x
1.5 Probability Foundations for Structural Reliability Theory …12

Standard Normal Random Variables

–   Relationship between general normal
 (u)
RV X and standard normal RV U
X  X
U
X

 x  X   
0            u       FX ( x)               FU (u )
 (u )
 X       
1
d           d  x  X 
f X ( x)     FX ( x)          
0.5                                  dx          dx   X 
1  x  X 
             
X  X 
0        u
1.5 Probability Foundations for Structural Reliability Theory …13

–   Properties of distribution function of a normal RV
   The PDF f X ( x) is symmetrical about the mean value  X
f X (  X  x)  f X (  X  x)
   The sum of FX (  X  x) and FX (  X  x) is equal to 1
FX (  X  x)  FX (  X  x)  1

–   The inverse CDF of a normal RV
p ≤ 0.5                                                p > 0.5
1         c0  c1t  c2t 2
z   ( p)  t 
1  d1t  d2t 2  d3t 3               p*  1  p
t   ln( p 2 )                                 z   1 ( p)  1   1 ( p* )
c0  2.515517       d1  1.432788
c1  0.802853       d 2  0.189269
c2  0.010328       d3  0.001308
1.5 Probability Foundations for Structural Reliability Theory …14

3. Lognormal Random Variables

f X ( x)                                           –   Definition of Lognormal RVs
If Y  ln( X ) is normally distributed ,

then X is a lognormal RV ( 0  x   ) .

–   CDF &PDF lognormal RVs
FX ( x)  P( X ≤ x)  P(ln X ≤ ln x)
0                                                        P(Y ≤ y)  FY ( y)
x
y  ln x     Y  ln x         Y   ln x

 y  Y                  d         d  ln( x)  ln( x ) 
FX ( x)  FY ( y )                    f X ( x)  FX ( x)                     
 Y                      dx           
dx      ln( x )     

 ln( x)  ln( x )                       1       ln( x)  ln( x ) 
                                                                   
       ln( x )    
                                     x ln( x ) 
      ln( x )     

1.5 Probability Foundations for Structural Reliability Theory …15

–   Moments of Lognormal RVs

var[ln( X )]   ln( X )  ln(1  VX )
2               2

1 2               X
ln( X )  ln( X )   ln( X )  ln
2                1  VX2
 ln( X )  ln(1  VX2 )

If VX  0.2 , then  ln( X )  VX   ln( X )  ln(  X )

–   Properties of Lognormal RVs

 ln(b)  ln( X )     ln(a)  ln( X ) 
P ( a ≤ X ≤ b)                                        
     ln( X )             ln( X )     
                                       
1.5 Probability Foundations for Structural Reliability Theory …16

4. Gamma Distribution

f X ( x)                                            –   PDF of Gamma RVs
 1
 ( x) k 1 e   x
f X ( x) 
(k )
for x≥ 0

 , k are distribution parameters

–   Gamma Function
x

(k )   e  u u k 1du
0
–   Moments of Gamma RVs
k                k               (k )  (k  1)!
X                
2

           X
2              (k  1)  (k )k
1.5 Probability Foundations for Structural Reliability Theory …17

5. Extreme Type Ⅰ(Gumbel Distribution)
–   CDF & PDF of ExtremeⅠ RVs
f X ( x)            a 1
u 1
FX ( x)  exp  exp   ( x  u) 
                     
for  ≤ x ≤ 
 e a ( xu )  a ( x u )
f X ( x)   e               e
u , a are distribution parameters

x   –   Moments of ExtremeⅠ RVs
0.5772
1.2825                    X  u 
                                          
X
 1 1.2825
X        
u   X  0.45 X                      6    
1.5 Probability Foundations for Structural Reliability Theory …18

6. Extreme Type Ⅱ
f X ( x)                                                –   CDF & PDF of ExtremeⅡ RVs
u 1
k 2                                              u k 
FX ( x)  exp      for 0 < x ≤ 
  x 
        
k u
k 1
  u k 
f X ( x)    exp     
u x           x 
        
u , k are distribution parameters
x
–   Moments of Extreme Ⅱ RVs
     1
 X  u  1             for k  1
     k

     2                   1 
 X  u 2  1     2 1   
2
for k  2
      k           
k         
1.5 Probability Foundations for Structural Reliability Theory …19

7. Extreme Type Ⅲ (Weibull Distribution)

f X ( x)                                         –       CDF of the Largest Values
w5
k 3                                                           w  x k 
u  0.5                                        FX ( x)  exp         
 
     wu  
for x ≤ k

w , u , k are distribution parameters

x        –       Moments of the Largest Values

   1
 X  w  ( w  u )  1  
   k    

         2             1 
 X  ( w  u ) 2  1     2 1   
2

        k           
k     
1.5 Probability Foundations for Structural Reliability Theory …20

7. Extreme Type Ⅲ (Weibull Distribution) …
–    CDF of the Smallest Values
f X ( x)
  5
k 3                                           x   k 
FX ( x)  1  exp         
u  0.1                                            w  
 
            
for x ≥ 

w ,  , k are distribution parameters

x
–    Moments of the Smallest Values

   1
 X    (u   ) 1  
   k    

        2           1 
 X  (u   )2  1    2 1   
2

       k         k    
1.5 Probability Foundations for Structural Reliability Theory …21

8. Poisson Distribution
–   Properties of Poisson Distribution
 It is a discrete probability distribution
 It can be used to calculate the PMF for the number of occurrence of a
particular event in a time or space interval (0, t)

–   Assumptions of Poisson Distribution
 The occurrence of events are independent of each other
 Two or more events cannot occur simultaneously

–   PMF of Poisson Distribution
( t) n  t
pN (n)  P( N  n in time t ) =        e       n  0,1, 2,   ,
n!
 N represents the number of occurrences of an event within a prescribed
time (or space) interval (0, t)
  represents the mean occurrence rate of the event which is usually
obtained from statistical data
1.5 Probability Foundations for Structural Reliability Theory …22

8. Poisson Distribution …
–   CDF of Poisson Distribution
n
( t) r  t
FN (n)  P ( N ≤ n in time t ) =           e
r 0 r !

–   Moments of Poisson Distribution
N   t        N   t

–   The Return Period of Poisson Distribution
1


–   The Annual Occurrence Probability of Poisson Distribution
( 1)0  1
P( N ≥ 1) = 1  P( N = 0)=1         e  1  e 
0!
1.5 Probability Foundations for Structural Reliability Theory …23

1.5.6 Random Vectors

1. Definition of Random Vectors
A random vector is defined as a vector (or set) of random variables
{X1, X 2 ,     , X n}

2. The Joint CDF and PDF of Random Vectors
–   The Joint Cumulative Distribution Function
FX1 X 2      Xn   ( x1 , x2 ,       , xn )  P( X 1 ≤ x1 , X 2 ≤ x2 ,        , X n ≤ xn )

–   The Joint Probability Distribution Function
 For   continuous RVs
 n F ( x1 , x2 , , xn )
f X1 X 2        ( x1 , x2 ,   , xn ) 
x1 xn
Xn

 For   discrete RVs
p X1 X 2    Xn    ( x1 , x2 ,       , xn )  P( X 1 = x1 , X 2 = x2 ,      , X n = xn )
1.5 Probability Foundations for Structural Reliability Theory …24

3. Marginal Density Function of Random Vectors
For continuous random variables, a marginal density function (MDF)
for each X i is defined as
            
f X i ( xi )                     f X1 X 2   Xn   ( x1 , x2 ,   , xn )dx1dx2          dxi 1dxi 1   dxn
           

4. Cases of Joint CDF and PDF of Two Continuous RVs
–   The Joint CDF of X and Y
FXY ( x, y )  P( X ≤ x, Y ≤ y )

–   The Joint PDF of X and Y
 2 FXY ( x, y )
f XY ( x, y ) 
xy
–   The MDF of X and Y
                                                          
f X ( x)              f XY ( x, y )dy                         fY ( y )          f XY ( x, y )dx
                                                        
1.5 Probability Foundations for Structural Reliability Theory …25

5. Conditional Distribution Function of Random Vectors
For continuous random variables, the conditional distribution function
for a random vector (X,Y) is defined as
f XY ( x, y )
f X |Y ( x | y ) 
fY ( y )

6. Statistical Independence of Random Vectors
If the random variables X and Y are statistical independent, then

f X |Y ( x | y )  f X ( x )
f XY ( x, y )  f X ( x) fY ( y )
fY | X ( y | x )  fY ( y )
1.5 Probability Foundations for Structural Reliability Theory …26

1.5.7 Correlation of Random Variables

1. Covariance of Two RVs
Let X and Y be two random variables with mean values  X and Y
and standard deviations  X and  Y . The covariance of X and Y
is defined as
CoV( X , Y )  E[( X   X )(Y  Y )]
(1) CoV( X , Y )  CoV(Y , X )
(2) For two continuous variables X and Y
       
CoV( X , Y )                  ( x   X )( y  Y ) f XY ( x, y )dxdy
       

2. Coefficient of Correlation
–   The formula of correlation coefficient
CoV( X , Y )
 XY 
 XY
1.5 Probability Foundations for Structural Reliability Theory …27

–   Properties of correlation coefficient
(1) 1 ≤  XY ≤ 1
(2) The values of  XY indicates the degree of linear dependence between
the two random variables X and Y
 If |  XY | is close to 1, then X and Y are linearly related to each other

 If    XY is close to 0, then X and Y are not linearly related to each other

–   Difference between Uncorrelated and Statistical Independent

• X and Y are uncorrelated
 XY  0        CoV ( X , Y )  0                   E[ XY ]   X Y

• X and Y are statistical independent
f XY ( x, y )  f X ( x) fY ( y )
• Statistical independent is a much stronger statement than uncorrelated
statistical independent                     uncorrelated
1.5 Probability Foundations for Structural Reliability Theory …28

3. Covariance Matrix of Random Vectors
–   For a random vector with n random variables, the covariance
matrix is defined as

 CoV ( X 1 , X 1 ) CoV ( X 1 , X 2 )            CoV ( X 1 , X n ) 
CoV ( X , X ) CoV ( X , X )                     CoV ( X 2 , X n ) 
         2     1           2     2                                
[C]                                                                    
                                                                  
                                                                  
CoV ( X n , X 1 ) CoV ( X n , X 2 )
                                                CoV ( X n , X n ) 

–   The matrix of correlation efficient is defined as
 11    12         1n 
        22        2n 
 21                     
[ ]                           
                         
                         
  n1
        n 2         nn 

1.5 Probability Foundations for Structural Reliability Theory …29

–   Properties of [C ] and [  ]
(1) Symmetric matrices
CoV ( X i , X j )  CoV ( X j , X i )
ij   ji
(2) The diagonal terms

diag ([C])  [ X1 , X 2 ,
2     2
, Xn ]
2

diag ([  ])  [1,1,   ,1]
(3) If all n random variables are uncorrelated, then

 X1
2
0                0 
                               
 0      X
2
2
0 
[C]                                 
                               
                               
                            2 
 0
          0                Xn 

1.5 Probability Foundations for Structural Reliability Theory …30

–   Statistical Estimate of the Correlation Coefficient
Assume that there are n observations {x1 , x2 ,       , xn } of variable X and n
observations { y1 , y2 , , yn } of variable Y
1 n                      1 n
sample mean x   xi                 y   yi
n i 1                   n i 1
n                           n

 (x  x )
i
2
( y  y)
i
2

sample standard deviation        sX    i 1
sY    i 1

n 1                        n 1

sample estimate of the correlation coefficient
n
 n       
 ( xi  x )( yi  y ) 1   xi yi   nx  y
1 i 1                            i 1    
 XY
ˆ                                  
n 1        s X sY           n 1        s X sY
1.5 Probability Foundations for Structural Reliability Theory …31

1.5.8 Functions of Random Variables

1. Linear Functions of Random Variables
Let Y be a linear function of random variables X 1 , X 2 ,                    , Xn :
n
Y  a0  a1 X 1  a2 X 2            an X n  a0   ai X i
i 1
where, the ai (i  0,1,        , n) are constants.

–   Moments of Linear Functions of Random Variables
n
Y  a0  a1 X  a2  X 
1       2
an  X n  a0   ai  X i
i 1

 Y2  E[(Y  Y ) 2 ]  E[Y 2 ]  Y2
n    n
  ai a j CoV ( X i , X j )
i 1 i 1
n    n
  ai a j  X i X j  X i  Xj
i 1 i 1
1.5 Probability Foundations for Structural Reliability Theory …32

–   Variance of Linear Functions of Uncorrelated Random Variables
If the n random variables are uncorrelated with each other, then
CoV ( X i , X j )  0   for i  j
n
  a 
n


2            2   2
Y            i   Xi      Y        ai2 X i
2
i 1                         i 1

–   Properties of Linear Functions of Random Variables
(1) The probability distributions of the random variables X 1 , X 2 ,   , Xn
are not needed.
(2) The linear function Y of uncorrelated normal random variables is a
normal random variable with distribution parameters Y and  Y .
(3) The constant a 0 does not affect the variance, but it does affect the
mean value.
1.5 Probability Foundations for Structural Reliability Theory …33

2. Product of Lognormal Random Variables
Let Y be a function involving the products of several random variables X i
X1 X 3
For example,     Y K                where K is a constant.
X2
Assume that these random variables are statistical independent , lognormal
random variables.

ln Y  ln K  ln X 1  ln X 3  ln X 2
 (constant)   (1)(normal random variables)

–   The above formula represents the sum of normally distributed
random variables ln X i .
–   The quantity lnY is a normally distributed random variable
–   Y is a lognormally distributed random variable
1.5 Probability Foundations for Structural Reliability Theory …34

–   Moments of the lognormally distributed random variable Y
n
ln Y  ln K   (1 as appropriate) ln X                 i
i 1
n
 2   2
ln Y               ln X i
i 1

  Xi
2

   2
 ln 1  2                 ln(1  VX i )
2
ln X i         X                 
      i             

   ln X i
 1 2
 ln  X i   ln X
2       i

X
 ln                i

1  VX2i
1.5 Probability Foundations for Structural Reliability Theory …35

3. Nonlinear Functions of Random Variables
Let Y be a general nonlinear function of the random variables
X i (i  1, 2, , n) . Mathematically
Y  f ( X1, X 2 ,                   , Xn)
–   The First Order Taylor Series Expansion of Y
n
f
Y  f (x , x ,
*    *
, x )   ( X i  xi* ) 
*
|( x* , x* ,
X i 1 2
1    2          n                                                          *
, xn )
i 1

Where, ( x1 , x2 ,
*    *        *
, xn ) is called “expansion point” which is denoted by P* .

–   Moments of Nonlinear Function Y
n
f
Y  f ( x , x ,
*    *
, x )   (  X i  xi* ) 
*
| P*
X i
1    2           n
i 1

2
n
    f       
 Y    Xi 
2
| P*                          for X i are uncorrelated .
i 1     X i 
1.5 Probability Foundations for Structural Reliability Theory …36

4. Central Limit Theory
(1) Sum of Random Variables
– Assumptions
Let the function Y be the sum of n statistically independent random
variables X i (i  1, 2, , n) whose probability distributions are arbitrary.
– Theorem
The central limit theory states that as n approaches infinity, the sum of
these independent random variables approaches a normal probability
distribution if none of the random variables tends to dominate the sum.
– Conclusions
• If we have a function defined as the sum of a large number of
random variables, then we would expect the sum to be
approximately as a normally distributed.
• The sum of variables is often used to model the total load on a
structure. Therefore, the total load can be approximated as a
normal variable.
1.5 Probability Foundations for Structural Reliability Theory …37

(2) Product of Random Variables
– Assumptions

Let Y be a product of n statistically independent random variables of
the form: Y  X 1 X 2 X n
–   Transformation and Theorem
ln Y  ln X 1  ln X 2     ln X n
By using the central limit theorem, we can conclude that as n approaches
infinity, lnY approaches a normal probability distribution . If lnY is
normal, then Y must be lognormal.
–    Conclusions
• If we have a product of many independent random variables,
then the product approaches a lognormally distribution.
• The product of variables is often used to model the resistance (or
capacity) of a structure or structural component. Therefore, the
resistance can be approximated as a lognormal variable.
1.5 Probability Foundations for Structural Reliability Theory …38

1.5.9 Simulation of Random Variables

1. Generation of Random Numbers
1)   Generation of a uniformly distributed random number U
between 0 and 1
u  rand (m, n)    MATLAB Function: rand
2)   Generation of a uniformly distributed random number X
between any any two values a and b ( a ≤ x ≤ b )
x  a  (b  a)u
3)   Generation of standard normal random numbers
z   1 (u )
4)   Generation of general normal random numbers
x   X  z X
5)   Generation of lognormal random numbers
x  exp[ ln X  z ln X ]
1.5 Probability Foundations for Structural Reliability Theory …39

6)   Generation of an arbitrary distributed random numbers
x  FX1 (u )

2. Simulation of Correlated Normal Random Variables
Correlated normal random variables X 1 , X 2 ,                , Xn
{ X }  { X1 ,  X 2 ,   , Xn }     [C X ]  [CoV ( X i , X j )]nn

eigenvalue and eigenvector analysis
Step1:   [C X ]                                                           [T ]
Step2: {Y }  [T ] { X }
T

Step3: [CY ]  [T ] [CX ] [T ]  diag[Y1 , Y2 ,        , Yn ]
T                     2     2              2

Step4: Generation of the uncorrelated normal random number vector {Y }
with {Y } and [CY ]
Step5: Generation of the correlated normal random number vector { X }
{ X }  [T ] {Y }
1.5 Probability Foundations for Structural Reliability Theory …40

3. Random Number Generators in MATLAB
1)    rand: Uniformly distributed random number
2)    random: Parameterized random number routine
3)    normrnd: Normal (Gaussian) random numbers
4)    lognrnd: Lognormal random numbers
5)    weibrnd: Weibull random numbers
6)    exprnd: Exponential random numbers
7)    gamrnd: Gamma random numbers
8)    betarnd: Beta random numbers
9)    raylrnd: Rayleigh random numbers
10)   geornd: Geometric random numbers
11)   poissrnd: Poisson random numbers
12)   binornd: Binomial random numbers
Chapter1: Homework 1

Homework 1

Plot the CDFs and PDFs of the following common random
variables in the environment of MATLAB or EXCEL:
(1) Uniform RV         Note:
(2) Normal RV
(1)   The parameters of these RVs can be
(3) Lognormal RV             assumed according to your willing.
(4) Gamma RV           (2)   These figures should be plotted by
using numerical method.
(5) Extreme Ⅰ
(3)   These figures as well as their
(6) Extreme Ⅱ                subroutines should be printed
(7) Extreme Ⅲ
formally.
End of
Chapter 1

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