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MINISTRY OF EDUCATION AND TRAINING
NATIONAL UNIVERSITY OF CIVIL ENGINEERING
Chu Thanh Binh
AN APPROXIMATE METHOD TO DETERMINE THE
RELIABILITY OF VIBRATING STRUCTURES UNDER
THE EFFECT OF RANDOM LOADING TAKING INTO
CONSIDERATION THE RANDOM DISCREPANCY OF
MATERIAL AND GEOMETRICAL PARAMETERS
Major: ENGINEERING MECHANICS
Code: 62.52.01.01
SUMMARY OF DOCTORAL THESIS
HA NOI-2014
This thesis is completed at the National University of Civil Engineering.
Supervisors:
1. Professor Nguyen Van Pho, PhD – National University of Civil
Engineering.
2. Associate Professor Le Ngoc Thach, PhD – National University
of Civil Engineering.
Peer reviewer 1: Associate Professor Phan Y Thuan, PhD –
National University of Civil Engineering.
Peer reviewer 2: Associate Professor Tran Duc Nhiem, PhD –
University of Transport and Communications.
Peer reviewer 3: Nguyen Vi, PhD – University of Transport
Technology.
The thesis is to be defended before the University-level Board of
examiners at the National University of Civil Engineering.
At: ……………(hours)………….(date)……….(month) 2014 (year).
This thesis can be looked up at:
- National University of Civil Engineering.
- National Library.
LIST OF PUBLISHED RESEARCH PAPERS OF THE AUTHOR.
1. Le Ngoc Thach, Chu Thanh Binh (2004), “Reliability following the safety condition
of bar and plate structures and its application in technical diagnosis”, Proceedings of the
7th National Scientific Conference on Deformable Bodies Mechanics, Do Son, pp. 790796.
2. Ta Thanh Van, Chu Thanh Binh (2004), “Some reliability problems of elastic
rectangular panels following safety standard for stability”, Proceedings of the 7th National
Scientific Conference on Deformable Bodies Mechanics, Do Son, pp. 978-986.
3. Chu Thanh Binh, Le Ngoc Thach (2006), “On some bending vibration problems of
beams with the stochastic perspective”, Proceedings of the 8th National Scientific
Conference on Deformable Bodies Mechanics, Thai Nguyen, pp. 59-67.
4. Nguyen Van Pho, Le Ngoc Thach, Chu Thanh Binh (2007), “On the reliability
problems of structures subjected to dynamic loads”, Vietnam Journal of Mechanics,
VAST, Vol.29, N03, pp. 427-439.
5. Chu Thanh Binh, Le Ngoc Thach (2007), “The problem of checking and designing
planar frames under the effect of harmonic excitation following reliability”,
Proceedings of the 8th National Conference on Mechanics, Ha Noi, pp. 47-56.
6. Chu Thanh Binh (2008), ”Reliability problems of Plates on Oscillation”, Proceedings of
the International Conference on Computational Solid Mechanics, Ho Chi Minh City,
Vietnam, pp. 36-44.
7. Nguyen Van Pho, Le Ngoc Thach, Chu Thanh Binh (2010), “The method to
analyze the reliability of vibrating structures under the effect of stochastic processes”,
Journal of Structure and Construction Technology, Vol.3-2010, pp. 27-34.
8. Nguyen Van Pho, Le Ngoc Thach, Chu Thanh Binh (2011), “Time-dependent
reliability of structures”, Scientific Journal of Construction Technology – National
University of Civil Engineering, Vol.9-2011, pp. 5-16.
9. Chu Thanh Binh (2012), “Assessing the reliability of single-degree-of-freedom
vibrating system”, Journal of Structure and Construction Technology, Vol.10-2012,
pp. 13-24.
1
`
INTRODUCTION
Scientific and evident basis: Reliability is an important and general criterion to assess the
safety of a structure. In dynamic problems involving inertial force and time t, external loadings and
characteristics of the system are random, making them more complicated. Existing literatures on
stochastic differential equation mainly consider random excitations, while random characteristics of
the system itself are less concerned. Therefore, the thesis titled ”An approximate method to
determine the reliability of vibrating structures under the effect of random loading taking
into consideration the random discrepancy of the material and geometrical parameters” has
firm scientific and evident basis.
Research objective of the thesis: Study the existing methods to assess the reliability of structures,
comment on the advantages and disadvantages, thus construct a method to analyze the reliability of
vibrating structures under the effect of stochastic process, with the material and geometrical parameters
being stochastic values.
Object and limit of research: Objects of research are beam, frame and plate structures, in which the
material works within the linear elastic limit. The loadings that exert on the structure are modeled as either
predetermined values, stochastic values or stochastic process.
Methodology and content of research: The thesis employs theoretical method of research
combined with numerical method. Convert the random inputs into a set of equivalent predetermined
inputs (possible combinations). Determine the weight of each predetermined input. Subsequently,
perform a “computer test” by solving the problem of predetermined vibration corresponding to each
predetermined input. Finally, process the results from “computer tests” to determine the reliability,
which is the occurrence frequency of the event of safety.
New results of the thesis:
1. Analyze the advantages and disadvantages of some common methods to determine
reliability, then derive a method to determine the reliability of vibrating structures.
2. Propose an approximate method to determine the reliability of vibrating structures under the
effect of loadings which are stochastic processes, taking into consideration the random discrepancy
of the material, geometrical and initial parameters.
3. Construct a program to determine reliability.
4. Apply the proposed method to analyze the reliability of some structural dynamic problems (beam,
frame and plate).
Structure of the thesis: The thesis consists of the Introduction, 4 chapters, Conclusions and
Appendix.
MAIN CONTENT OF THE THESIS
Chapter 1. OVERVIEW OF THE RELIABILITY THEORY OF STRUCTURES UNDER
STATIC AND DYNAMIC LOADING
1.1 Introduction
1.2 Overview of the reliability theory of structures under static loading
Reliability theory is an applied science major, consists of many other scientific fields such as
mathematics, physics, mechanics and engineering. Reliability analysis of structures has almost fully
developed [8], [43], [48], [49], [54], [55], [84] … and has been specified in design codes [21], [72]. In
1935, the application of mathematical statistical methods in structural mechanics have been studied
by A.M.Freudenthal. The basis of reliability theory was founded by Russian academician Волотин
В.В. In his research [102], [103], he presented the problems of reliability in general form and apply
it to a number of important problems. Besides, there are similar research by Western mechanicians
[54], [55], [83], [84].....
In Vietnam, the teaching and researching of reliability have been taken into consideration since
the 80s of the last century [26], [27], [28]… Especially, in recent years, an increasing number of
theoretical research as well as applications in structures has been observed (National University of
2
Civil Engineering, Water Resources University, Vietnam Institute for Building Science and
Technology, Institute of Mechanics, Institute of Transport Science and Technology…) [18], [26],
[50], [51]… Many doctoral theses on reliability of structures have been conducted [8], [13], [43],
[48], [49], [50]… Many research on reliability at different levels have also yielded results.
1.3 Overview of the problem of determining the reliability of vibrating structures
In dynamic problems, determining the reliability encounters many more difficulties in
comparison with static problems. Two distinguished difficulties are:
- Solving the equation of state: it is required to solve a single or a set of stochastic differential
equations, in which not only the right-hand-side are stochastic processes, but the coefficients on the lefthand-side are also functions of stochastic values.
- Calculating the probability such that a stochastic process in multi-dimension space is in a certain
determined domain within a provided time frame.
Studying stochastic differential equations has become a quickly-developed field in mechanics.
Local mechanicians have also gained many merits in this field [1], [53],... However, since building
structures are complicated systems, mechanical results are insufficient to apply into analyzing
building structures. Regarding the exceedance of a stochastic process, there have been many
research [76], [95], [97], [100]… particularly by USSR mechanicians, who obtained important
results.
In [90], V.A. Svetlitsky introduced familiar stochastic processes and considered the random
vibration of single-degree-of-freedom (SDOF) system or multi-degree-of-freedom system (MDOF)
for beam. Subsequently, a chapter was dedicated to reliability (Fundamentals of Reliability
Analysis). However, these results were insufficient to apply into building structures.
In [76], Jie Li và Jian-Bing Chen dedicated one chapter (Dynamic Reliability of Structures), but only
mentioned some theoretical problems as well as introduced some additional mathematical assumptions to
prove some related clauses. To calculate the reliability of structures, there must be more research.
In [89], a doctoral thesis defended and published in India contained many basic problems when
considering reliability. Nevertheless, results were only applied to simple examples.
In [88], Robert E. Melchers wrote chapter 6 (Time dependent reliability) to introduce the problem,
though mainly focused on exceeded stochastic processes, did not consider the problem of solving the
equation of state, and did not solve all difficulties in calculating building structures.
There were many papers on foreign journals consider time-dependent reliability, in which
vibration problems were concerned.
In [69], Hector A. Jensen, Marcos A.Valdebenito introduced a method to analyze the reliability of
linear systems with random parameters and under the effect of random excitations. When considering
the dynamic behavior of the system, some assumptions were made to simplify the problem, which are
hardly applicable to building structures.
In [75], [77], Jian-Bing Chen and Jie Li, after having studied the stochastic equation of
vibration, considered the probability density of behavior of the building structure, which is
mathematically complicated. However, due to mathematical requirements to accept a number of
assumptions to simplify, it is hard to apply to structures.
In [80], Lin-lin Zhang, Jie Li, Yongbo Peng studied the random spectrum of wind and the
probability density of the behavior, then assessed reliability. The results obtained were clear, but by
accepting a specific form of the one-way behavior, the application in calculating structures is limited.
In [60], B.Y. Moon và B.S. Kang analyzed the seismic spectrum and extracted the results to consider the
exceedance of the state variable applying the exceedance theory that was used by V.V. Bolotin.
In [13], Pham Khac Hung introduced how to “Determine the reliability of orthogonal bar systems under the
dynamic effect of random loading”. In his study, the author based on the method of constructing quality space and
boundary surface by V.V Bolotin as well as the seed theory of stochastic processes to determine the reliability.
In conclusion, from the published results of the reliability of vibrating structures, some distinguished
problems can be observed as follows:
3
- The studies have thoroughly applied mathematical tools which are statistics and stochastic
processes. These tools require a lot of statistical data (normal, independence, white noise, stationary,
ergodic, v.v…), during the calculation process, some other mathematical assumptions are also accepted.
- Some studies have started from density spectral functions of loading, then applied the Monte-Carlo
method to modelize the loading as expressions to calculate the reliability [13], [14].
- For the majority of published papers, when determining the reliability of the system, the random
characteristics of the system itself are not well-concerned. Rather, the papers focused on the behavior of the
system under random loading, with random safety criterion. Therefore, it is useful if the problem has a closedform (analytical) solution, while employing numerical solutions to calculate the reliability would encounter many
difficulties.
1.4 Objective of the thesis
- Find out about the existing methods to analyze the reliability of structures. Assess the advantages,
disadvantages of the methods, select the method to study vibrating structures.
- Propose an approximate method to determine the reliability of vibrating structres under the effect
of loading which are stochastic processes, and the material, geometrical characteristics are
stochastic values.
- Construct an algorithm and program to calculate the reliability.
- Apply the obtained results to analyze the reliability of some structural dynamic problems for
beam, frame and plate.
Chapter 2. SOME COMMON METHODS TO CALCULATE THE RELIABILITY OF
STRUCTURES AND THEIR ADVANTAGES, DISADVANTAGES
2.1 Introduction
2.2 Method of determining the factor of reliability M
M
2.2.1 Linear limit state function
n
g(X1, X2,...Xn ) a0 a1X1 a2 X2 ...an Xn a0 ai Xi
(2.1)
i1
n
a0 ai X i
The factor of reliability is determined as follows:
i 1
(2.2)
2
n
(a
i
i 1
Xi
)
2.2.2 Nonlinear limit state function
Apply the Taylor's expansion for g X at the expectation x* to the first-order [55].
n
g(X1, X2...Xn ) g(x1, x2,....,xn) (Xi xi)
i1
Factor of reliability : g ( X , X ,.... X ) ; a g
i
2
1
X i
n
i
(x1,x2,...,xn)
Xi
)
(2.5)
(2.7)
n
2
(a
g
Xi
Xi
i 1
In the case of applying the Taylor's expansion for
g( X
1, X2 ,....,X n
g(X , X ,..., X ,)
)
1
2
n
1 n 2g
2 i1 Xi2
gX
2g
2
i j Xi
X2
i
X
i
at x* to the second-order [96].
Kij
X
i
(2.8)
4
In which Kij is the correlative moment between Xi and Xj. In the case which X1,X2,…,Xn are
uncorrelated, the expectation of g is:
1 n 2 g
g ( X , X ,...., X ) g ( X , X ,..., X n ) 2
X2 i
(2.9)
1 2
n
1
2
2 i 1 X i
Xi
The variance of g would be :
2
2
2g
G
1 n 2g
2
4
Dg
(
[
X
]
)
Xi
4
X
i
4 i 1 X i2
i 1 X i
i j X i X j
n
i
i
2
n
g 2 g
2
2
3[ X i ]
X X
2
i
j
i 1 X i X i
i
i
i
(2.10)
2.2.3 Advantages and disadvantages of the method of determining the factor of reliability
using FOSM
2.2.3.1 Advantages
- Simple calculation, easy to apply.
- Not required to know the type of distribution (or density) function of random variables, only require the
expectation and variance of the safety domain.
2.2.3.2 Disadvantages
- Inaccurate results due to the omission of nonlinear quantities in the Taylor's expansion.
- Difficult and inaccurate in valculating M. M is the function of the state variables Xi, while Xi are
functions of the input variables. Only the numerical characteristics of the input random variables are
known, it is required to calculate the expectation and standard deviation of the state variables Xi.
From the state variables Xi calculate the expectation and standard deviation of M. In case the
solution of the state equation is nonlinear, or there is only numarical solution (which is applicable to
building structures) then determining M is difficult. The way to overcome this difficulty is still via
Taylor's expansion around the average value only to the first-order and approximately replace
derivatives by the ratio between the function increment and the variable increment. By doing so,
errors are expected, which are hard to assess.
- The value of the factor of reliability depends on the obviousness of the equation of the limit state
surface. On the other hand, one limit state surface can be mathematically expressed in different
forms. For instance, consider a basic case that consists of teo random variables R and S, the
R
S
equation of the limit state surface is g=R–S=0, or also is g1 1 0 or g 2 1 0 ,
S
R
( R, S 0) . For different mathematical forms, the values of M/M are generally different, thus the
difficulty in determining .
2.3 Iteration method to determine the factor of reliability by Hasofer-Lind[55].
2.3.1 Content of the method
Consider the equation of the limit state surface g( X1,X2,…,Xn)=0, in which the random
variables are uncorrelated. The state function written in the standard form of simplified variables.
Z
i
X
i
X
X
i
; g ' ( Z 1 , Z 2 , ... .Z n ) 0
(2.17)
i
Hasofer-Lind's factor of reliability is defined as the shortest distance from origin of the
space of simplified variables to the limit state surface g’=0. If the limit state function is nonlinear
then iteration steps must be carried out to determine the design point [Z1*,Z2*….,Zn*] in the space of
simplified variables such as corresponds to the shortest distance. The iteration procedure is
actually solving a system of (2n+1) equations simultaneously with (2n+1) unknowns: , 1, 2,…
n, z1*, z2*,…, zn*. That system of equations is:
5
g
z i at the design point
i
2
n
g
k=1 z k at the design point
Z i* i
*
*
*
*
g ( Z 1 , Z 2 , Z 3 ,..., Z n ) 0
(2.18)
(2.21)
(2.22)
2.3.2 Advantages and disadvantages of the iteration method
2.3.2.1 Advantages
- Does not make errors from linearization. Nowadays, the use of computers makes it easy and
effective to carry out the iteration process.
- The design point is adjusted during the iteration process.
2.3.2.2 Disadvantages
- The problem of selecting the appropriate initial value of such that the results converge.
- Similar to FOSM, to end up with the standard space, it is required to do the transformation
Xi X
i , i.e. calculating X . To calculate X basing on input values is difficult, especially in
Z
i
i
X
i
i
case the state equation only has numerical (approximate) solutions, resulting in errors.
- The iteration procedure is carried out by utilizing the results of previous step as inputs for the next
step, thus the accumulation of errors.
2.4 Some methods to determine the reliability of structures under random excitations, taking into
consideration the random characteristic of the system.
2.4.1 Stochastic finite element method (FEM).
General form of the equation of the stochastic finite element method:
(2.23)
M 1 X C 2 X K 3 X F 4,t
In which M, C, K are mass, damping, stiffness matrices (respectively), i are random vectors.
Due to the complication of the algorithm, there is no commercial software to analyze structures.
2.4.2 Monte Carlo method to determine the reliability of structures [55], [76]
Monte Carlo method is an approximate method (numerical) in applied mathematics. The general idea of
Monte Carlo method is setting a relation between a numerical problem with a certain probability scheme,
from which derive a calculating procedure. By strict mathematical proofs, tables of pseudo-random numbers
of a uniformly-distributed value within the [0,1] interval are constructed. Probability characteristics
(probability distribution functions) are employed to generate random numbers representing random inputs,
i.e. transforming random inputs into predetermined inputs. “Computer tests” are then carried out using
predetermined calculations. Subsequently, process the test results according to the requirements of the
problem. The reliability is approximated basing on frequency
Monte Carlo method is advantageous due to the simplicity, solving random problems by predetermined
calculations. However, its disadvantages are: the number of calculations is immense, and it is required to
determine the inverse function of the probability distribution function.
2.4.3 Determining the reliability of structures in some specific cases
Random loading (input of the random vibration problem) is restricted to being a stationary stochastic
process, and is determined basing on the spectrum S() of the random loading. To transform a problem in
the frequency domain into time domain, discretize the frequency with N intervals (N sufficiently large) for
an expression in the form of a sum of (N) harmonic functions with amplitudes ai depending on S(i) and the
phase i which is a uniformly-distributed random number within [0,2] and is determined by Monte-Carlo.
This method has been applied in analyzing offshore structures, in which it is firstly required to select a
6
spectrum of the loading (a specific formula), then determine the probability characteristics of the structural
behavior. The reliability is subsequently calculated.
2.4.4 Seeding method (or exceedance method) [12], [80], [89], [103]…
For building structures, the safety probability is usually high, i.e. failure probability (exceedance
probability) is low. Therefore, it is possibile to use the assumption of Poisson's flow [95]. In this case, the
probability such that after time t, the limit a of the process (expressions) is calculated by the formula:
P(t ) e Na (t )
(2.24)
In which Na(t) is the average number of exceedance of limit a within time t.
For stationary process: P(t ) e na (t )
In which na is the average number of exceedance of limit a within a unit of time. Expand e Na (t ) using
Taylor's series with the average number of exceedance to the first-order, we have: P(t ) 1 N a (t )
with
1
the condition t
(2.25)
N a (t )
This method also only apply to one inequality (limit) and it is required to know the probability density of both v and
v
2.5 Method to determine the reliability basing on the probability of occurrence of the safety
event of the structure [36], [37], [38].
The method consists of 3 main steps (similar to Monte Carlo method)
Step 1: Transform the initial random input into an equivalent set of predetermined input values.
Step 2: “Computer tests” are setup according to each predetermined input value.
Step 3: Statically process the sets of output values according to the requirement of the problem
Chapter 3. THE METHOD ANALYSIS RELIABILITY OF VIBRATING STRUCTURES
3.1 Introduction
In this chapter, after having inherited and expanded some existing results [36], [37], [38], the author
proposes a method to determine the reliability of vibrating structures by employing a number of popular
assumptions and concepts widely applied in calculating the reliability of building structures. The main idea
of the method is as follows::
- Characteristics of the system are stochastic values, loadings are stochastic processes, these
processes are modeled into expressions or a function of time and stochastic values (by spectral
expression, normal expression or Fourrier's expansion) [12], [95], [96].
- Instead of directly solving the differential equation of random vibration, solve the
corresponding differential equation of predetermined vibration.
- Construct “computer tests” with the set of predetermined input.
- Verify the safety criterion to calculate the frequency (reliability).
3.2 Definition of reliability by Волотин В.В
In the papers [88], [89], [90] the problem of time-dependent reliability was considered.
However, for a general definition of time-dependent reliability, it is worth to mention the research
by Russian academician Волотин В.В [102], [103] .
Let
T
u( x, , t) ui ( x, , t)
loading, the state equation of the system is: L u ( x, , t) q( x, , t)
T
be the vector of state variable, q( x, , t) qi ( x, ,t) be the vector of external
T
(3.1)
in which x x1 , x2 , x3 T are spatial variables, i are random parameters, t is time, L is the
differential (algebraic) operator. Russian academician Волотин В.В [101], [102], [103] introduced
the general definition of reliability as follows:
7
f v ( x , , ) 0
P (t ) P x V
0, t
L
u
(
x
,
,
)
q ( x , , )
(3.4)
G
u
(
x
,
,
)
v ( x , , )
. P(t ) P f v ( x, , )
0
For problems that cannot be divided into two steps:
x V
solve the state equation and calculate the exceedance
0, t
probability, such as limit equilibrium problems,
adaptation problems of elasto-plastic system, then the
state equation and the transformation from u into v
are within the reliable probability. It has the form (3.5)
(3.5)
in which V is the domain of the system in
three-dimensional space x x1 x 2 x 3 T .
(3.5) is the probability of simultaneously satisfying a system of equations and inequations. The
condition 0 , t needs to be satisfied at any points before and at the considered moment. The
condition
x V
needs to be satisfied at any point in the structure. The safety criterion is f v 0
for structural member or for whole structure. The stochastic process q ( x , , t ) is usually denoted
q ( x, t ) . Here it is written as q ( x , , t ) to distinguish from random functions that only depend on x and
, independent of t.
3.3 Approximate method to determine the reliability of vibrating structures through the
occurrence probability of the safety event.
3.3.1 Basic assumptions of the proposed method
The proposed method in the thesis accepts the following assumptions:
1. The structure is a linear elastic system.
2. The vibration equation of the structure is a linear stochastic differential equation that has the
following characteristics:
- The right-hand-side consists of stochastic processes (excitations).
- The coefficients on the left-hand-side are functions of stochastic values (random functions).
- Initial condition, boundary condition can be random.
3. Stochastic values of the input are independent and are given all characteristics (expectation,
variance, correlation coefficient). In case of dependence, transform into equivalent independence
by ISO 2394-1998 [72].
4. Random loadings are stochastic processes which can be modeled by either:
- A family of expressions.
- A function of time t and some stochastic values. Therefore, the stochastic process can be
stationary or non-stationary, as long as it can be modelized.
5. Corresponding problem of predetermined vibration (predeterminization) already has solutions,
method of solution – the majority of which are numerical solutions (numerical method).
3.3.2 Flowchart to calculate the reliability of vibrating structures
Step 1: Determine the input parameters.
- Loadings are predetermined values, stochastic values and stochastic processes.
- Material parameters are predetermined values or stochastic values.
- Geometrical parameters are predetermined values or stochastic values.
- Initial condition and boundary condition can be stochastic values.
Step 2: Preliminary process of the input.
8
- Limit the domain of determinancy of the random parameters (only consider the domain with
reasonably large density function) and discretize values in the interval of determinancy.
- Apply the modeled results of loading (stochastic process) by either:
From the given spectral density, modelize as a family of expressions.
By statistics theory, modelize the stochastic process by functions of stochastic values and time t
(a family of predetermined functions).
Step 3: - Establish the differential equation of vibration of the structure, initial condition and
boundary condition (equation in exact form or asymptotic form).
- Establish the safety criterion according to the problem, Mi0 for all inequalities (Mi is the ’’safety
interval’’, i=1, 2,… n).
Step 4: Establish the set of possible predetermined inputs equivalent to the initial stochastic input.
- Construct the set of possible predetermined inputs by listing all combinations of inputs
that can occur, corresponding to discrete values of stochastic values and expressions.
Step 5: Determine the weight of each predetermined input basing on the values of the density function
at discrete points.
Step 6: Perform predetermined analitic of the structure according to each possible predetermined input
to form a set of predetermined outputs (computer test).
Step 7: Statistically process the obtained test results. Verify each test against the safety criterion. Since individual test
is considered, calculation and determination of safety follow familiar predetermined method. Subsequently,
N
determine the approximate value of reliability by the occurence frequency of the safety event Ps s , in which N is
N
the total number of tests, Ns is the number of tests resulted in safety (weight considered), safety is defined as satisfying all
safety criteria set by the problem
Flowchart of the method
Determine the input (predetermined+stochastic)
Preliminary process of the input
Establish the DE of vibration, initial condition,
boundary condition & safety criterion of structure
Establish the set of predetermined input equivalent
to the initial set of stochastic input
Determine the weight of each predetermined input
Analyze the structure according to each recentlyestablished predetermined input to form a set of
predetermined output
Statistically process the obtained results, verify
Figure 3.2 Flowchart of the proposed method
against the safety criterion Calculate frequency
3.3.3 Detailed content of each step of the method
9
3.3.3.1 Determine the input parameters
Input of the vibration problem considered in the thesis consists of 2 types: predetermined and
stochastic.
Predetermined parameter is defined by one value, which is clear, with no error or small error
that can be ignored during calculation.
Stochastic parameters include stochastic value or stochastic process.
- For stochastic value, its value is defined within a certain domain, usually the domain of
observed, measured results. Therefore, the mathematical expression of the domain of determinancy
of the density function can be infinity. However, when discretizing to perform calculations on
computers, it is finite, one only need to calculate on a considerably large domain of value of the
density function.
- For stochastic process, they are random functions depend on space and time t such as wind
load, seismic load, wave load, etc. From experimental data, it is mathematically modelized as
functions of time t and some stochastic values or approximated by a family of expressions, these
expressions are predetermined functions of time t.
3.3.3.2 Preliminary process of the input: Discretize stochastic parameters.
- Each stochastic value is defined within a domain. That domain corresponds to the domain of
determinancy of the probability density function. However, for reasonably small domains of density
function, stochastic events rarely happen; therefore, in approximate calculation, we only consider a
finite domain within which the density function is considerably large. For instance, consider a
stochastic value with normal distribution with standard deviation , the domain of determinancy is
chosen as 3 or 4 depending on the requirement of accuracy of the problem.
- For stochastic process (loading): The thesis only considers processes that can be modelized
(stationary or non-stationary) in 2 forms.
A family of expressions (by different method of modeling).
Express in the form of a function with respect to time t and some stochastic values (by the
methods of Fourrier's transformation, normal expression, etc.).
Substantially, the above 2 methods are mathematical expressions of experimental results, the
only difference is the involvement of time. An expression is regarded as “one discrete value”. Thus,
the inputs (stochastic values and processes) are discretized.
3.3.3.3 Establish the differential equation of random vibration of the structure, determine the
initial condition, boundary condition and safety criterion of the structure.
Basic equation of the FEM method established basing on the principle of virtual work takes the
form of:
(3.9)
M U C U K U F ( t )
With the initial condition U (0 ) U 0 , U (0 ) U 0 .
in which [M] is the mass matrix of the whole system.
T
T
M Te M e Te với M e N e e N e dVe
e
(3.10)
e
[C] is the damping matrix of the whole system.
T
T
C Te Ce Te với Ce N e e N e dVe
(3.11)
[K] is the stiffness matrix of the whole system.
T
T
K Te K e Te với K e Be De Be dVe
(3.12)
e
e
e
e
[Me], [Ce], [Ke] are mass, damping, stiffness matrices of the finite element e.
3.3.3.4 Establish the set of predetermined input equivalent to the initial set of stochastic input
After discretizing the stochastic values and modeling the stochastic processes by expressions,
divide the input into 3 groups: predetermined, stochastic values (discrete values), stochastic
processes (expressions). Establish the set of predetermined input by:
+ Predetermined group appears in all combinations.
10
+ Group of stochastic values: each discrete value of a stochastic value only appears once in each
combination.
+ Group of stochastic processes: consider the expressions as “discrete value” of stochastic value to
combine.
Principle to establish predetermined inputs: Consider all possibilities, in any combination, each
parameter only appears once. Take the following particular cases for example:
Case 1: Predetermined group + 1 stochastic value + 1 stochastic process (Fig. 3.6)
Take the predetermined group combine with one discrete value of the stochastic value and
one expression of the stochastic process, we have one predetermined input. Since the discrete
value of the stochastic value or one expression of the stochastic process only appears once in a
predetermined input, for l1 discrete values of the stochastic value and l2 expressions of stochastic
processes we have l1l2 combinations, i.e. we have a set of l1l2 predetermined inputs.(Since it
is unknown which stochastic value is taken in the domain of determinancy and it is also
unknown which expression of the stochastic process is taken, the input set is established with all
possibilities).
Predetermined
group
Nhãm tÊt ®Þnh
§LNN
®îc
l1 numbers
of®·
discretized
l1 sè
rêi r¹c values
stochastic
QTNN
®· ®îc
m« pháng
l numbers
of modelized
2
l2 thÓ
hiÖn
thµnh
stochastic
processes
Figure 3.6 Input combinations of predetermined variables + 1 stochastic value + 1 stochastic
process
Case 2: Predetermined group + n stochastic values + m stochastic processes (Fig. 3.7)
Nhãm tÊt ®Þnh
Predetermined
group
q1
ln
l1
n §LNN
n stochastic
values
qm
QTNN
m stochasticmprocesses
Figure 3.7 Input combinations of predetermined variables + n stochastic values + m
stochastic processes
Step 1: Take the predetermined group + l1 discrete values of the first stochastic value, we
have l1 combinations.
Step 2: Take each combination in step 1 to combine with l2 discrete values of the second
stochastic value, we have l1l2 combinations. Perform similarly until the nth random value,
we have the number of combinations: l1l2…ln.
Step 3: Perform similarly for stochastic processes. The final number of combinations is:
n
m
l1.l2 ....ln .q1.q2 ...qm li . q j
i 1
(3.17)
j 1
3.3.3.5 The weight of each recently-established predetermined input
To reflect their equivalent role in calculation, each discrete value of the random variable is attached to a
weight, which represents the number of its appearance in the experimental results, i.e. proportional to the
frequency or the probability density function. Predetermined values appear in every possible combination,
thus no weight. For stochastic values, to determine the interval of discrete values [a,b], the initial and end
points of that interval are points with minimum density function. Denote those points as x1(0) and x2(0). Let
f(x) be the density function, then f ( x 1( 0 ) ) f ( x 2( 0 ) ) m i n f ( x ) . Consider that at x1(0) and x2(0) there
x[a ,b ]
11
is one occurence, other discrete points xi have the number of occurrence being a real number times the ratio:
f ( xi)
(3.18)
, i 1, 2 ....
f ( x 1( 0 ) )
Each predetermined input is a combination of determined values (or determined functions) of initial
random input parameters, in which each initial input parameter (including loading) only appears once.
Denote a recently-established predetermined input as: a1 a2…..an b1 b2…bm c1c2…cq in which ai are
predetermined values, bj are discrete values of stochastic values with weight rj , ck are expressions of
m
stochastic processes, then the weight of predetermined input is: L
j
r1 . r2 . . . rm ri
i 1
(3.19)
3.3.3.6 Analyze the structure according to each recently-established predetermined input to
form a set of predetermined output
Having had predetermined inputs, perform structural analysis using common softwares such as Etab,
Sap2000…. Such softwares usually employ, which is an approximate method but has been tested in many
cases, thus trust-worthy. With a predetermined input, via analysis algorithm we have a predetermined output.
Statically process the set of outputs (test results) we have desired results.
3.3.3.7 Determine the reliability from frequency
To calculate the frequency, it is required to determine whether the test result is safe, then establish the
ratio:
NS
,
(3.20)
Ps P M 0
N
in which: M M i , Mi>0 is the safety criterion, Ns is the number of safe tests, N is the total number
of tests (taking weight into consideration).
Note: Since the weight is the number of occurrence of that value, it is needed to utilize weight during
statistical process, consider the number of safety events (or non-safety event) equal to the weight. Thus, the
number of tests in statistical process is considerably larger than that on computer.
3.3.4 Scientific basis of the proposed method.
3.3.4.1 From the definition of probability
Reliability or probability of safety is the occurence frequency of safety event when the number of
tests approaches infinity. In reality, however, it is not possible to conduct infinite number of tests.
Therefore, depending on the accuracy requirement when calculating reliability, a finite number of
tests is accepted. It is also possible to estimate the required number of tests and assess errors [55],
[96]... There are 2 important points in the definition of reliability:
- Tests are conducted in the same condition, i.e. it is regulated that experiments or observations, measurements
must have identical or equivalent condition.
- The number of tests must be considerably large to maintain accuracy.
These two requirements are satisfied in the previous proposed flowchart to determine reliability.
Because:
- Regarding tests in the same condition, which are computer tests in this case, the input for the
program must be equivalent (this was reflected by the combinations of all possibilities of the random
event and the determination of weights of discrete values).
- Regarding considerably large number of tests: results in [55] is applied.
3.3.4.2 Maintaining the equivalence between the initial random input with a set of
predetermined inputs
One determined input cannot be equivalent to one random input, but a set of predetermined
input can be equivalent to one random input. It is the nature of random event (stochastic value is
modelized from predefined experimental results, stochastic process is modelized from expressions).
- For stochastic value: Stochastic is not interpreted as “arbitrary, without rule”. Stochastic value is
contained within a defined interval (the domain of results of tests), in which the occurrence
frequency is proportional to the density function. Actually, density function is constructed from the
12
frequency chart (organization chart) [55] (Fig. 3.8). Mathematical formula of the probability density
function is the approximation of the organization chart. Thanks to the modelization into
mathematical formula, the calculation is simplified. Therefore, it is common to utilize the
organization chart to calculate the approximation.
Normal distribution of density function fX(x) is: f X ( x )
1
X
2
e
1 xx 2
- (
)
2 x
(3.21)
In which X is the average value, X is the standard deviation, fX(x) is determined in the infinite
domain (-,) by mathematical expression. The experimental results (organization chart), however, is
determined in a finite domain, such as the interval [xA,xB]
f(x)
0
x
A
xB
x
Figure 3.8 Organization chart, density function of stochastic value
- For stochastic process: To determine a stochastic process, initialize from expressions Xi(t), which are
observed or measured results. Subsequently, mathematically modelize by expectation function (average
value) X(t), density function fX(t) and correlation function family, spectral density, etc. [76]. With a
determined value t=t0 we have one stochastic value (Fig. 3.9)
Xx(t)
hiÖn
expression
j(t) --thÓ
j
x(t)
Cross-section
at tot 0
mÆt
c¾t t¹i
(t)
X
X (t) -thÓ hiÖn
i
xi(t) - expression
0
t0
t
Figure 3.9 Expressions of stochastic processes around the expectation
3.3.4.3 Maintaining the equivalence between recently-established predetermined inputs with
each other.
Because the number of occurrences in the test results of discrete values of the stochastic value is
different, it is possible to attach a weight, the value of which equals to the number of occurences in
the test results, thus maintaining the equivalence.
3.3.4.4 Error and method of rectification
There are 3 causes of error:
Error by discretization
Error in each analysis
Error by the finite number of tests
Regarding the first cause, it is possible to rectify by increasing the number of discrete points or
distributing the points differently: create more points at the region where the considered quantity has
higher gradient, similar to FEM.
Regarding the second cause, although commercial softwares are expected to generate errors, they
are acceptably small (this matter was considered when selecting the algorithm).
13
Regarding the third cause, the formula to estimate the required number of tests was introduced in
[55].
3.3.4.5 Reliability of the method
To express the reliability of the proposed method, inheriting the tradition of mechanical theses, this
thesis was structured as follows: after having proposed, verify the reliability of the method by comparing the
obtained results with existing literatures (or exact results). Subsequently, apply the new method in a number
of more complicated problems to prove the applicable capability of the method. In that manner, the thesis
considers simple examples of SDOF system in 4 cases to compare, then considers more complicated
examples (4-storey frame, 20-storey frame) to compare results. It is observed that the difference between the
2 results is not significant. Therefore, the proposed method is reliable.
3.3.5 Limit of application of the method
The proposed method considers a wide range of linear random vibration problems, with the
right-hand-side of the equation of vibration being stochastic process, coefficients of the left-handside are random functions, initial condition and boundary condition can be random. On the other
hand, the method can also be applied for problems of reliability that depend on many inequalities
(safety criterion) without having to determine the margin.
3.3.6 Advantages and disadvantages of the proposed method.
3.3.6.1 Advantages:
- Instead of directly solve the state equation which is a stochastic differential equation to find the behavior
of the system, solve a system of predetermined vibration problems. Especially, for problems which the
coefficients of the left-hand-side of the equation of vibration are random functions, initial condition and
boundary condition are random, then although finding a solution is difficult, the proposed method still can
solve.
- The solution process does not require additional mathematical assumptions such as: white noise, ergodic,
stationary, etc. but only requires that it is possible to modelize stochastic process.
- The safety probability of the structure is the probability of simultaneously satisfying a system of
random inequalities. The method to determine frequency only requires to verify the satisfactory of a
system of predetermined inequalities, since it is the result of each predetermined test, which is easy to
conclude whether each structural test is safe or not. It is not necessary to assess upper and lower margins
of the reliability through the reliability of each component or each inequality [55], [76], [83], [84],…
- The process of determining frequency also does not require to satisfy additional mathematical
assumptions such as Poisson's flow, Markov's process, etc. like other methods.
- Capable of avoiding the errors by linearization.
- Especially, capable of avoiding accumulation errors by iteration process (still perform structural
analysis multiple times, but with different input data).
- Able to calculate the probability with safety criterion following any standard.
3.3.6.2 Disadvantages:
- Large amount of calculation. Can be rectified by a few methods, such as Monte-Carlo method or
utilizing weight. Since weights are attached to predetermined inputs, the number of tests is fewer than
that in statistical process, yet a hinderance in application.
- Error by discretizing stochastic values and by determining weights. This disadvantage is common in
analyzing structures, and is usually accepted. Nowadays, the majority of structural analysis softwares
employ approximate algorithm by discretizing continuous values. Even FEM is an approximate method
since only results at nodes are calculated, other points in the element are approximated by a shape
function, subjectively chosen by the user (satisfying some conditions). Nevertheless, error assessment is
still necessary.
3.3.7 Comparison between Monte-Carlo method to determine reliability with the proposed method
14
3.3.7.1 Monte-Carlo model to determine reliability
Monte-Carlo method is usually applied in the following three scenarios:
a. To solve complicated problems, closed-form solution is not available or is hard to obtain. For instance,
statistical problems, complicated models containing nonlinear element.
b. To solve complicated problems, which can be solved (or best-approximated) by closed-form solution, if
accepting many simplification assumptions.
c. Can be applied to verify the results of other techniques of solution.
Content of Monte-Carlo method
Essentially, Monte-Carlo model is forming a set of independent expression values xi for each random
variable by generating pseudo-random numbers following their known probability distribution function
(or density function), from which model the probability distribution of safety margin Z(X1,X2,…,Xn) of
the reliability problem. In general, the probability distribution of Z does not follow any familiar form, even
when the variables follow normal distribution (except when Z is a linear function of independent random
variables).
Unreliability probability can be solved by either:
- Determine the failure event probability by the formula:
Pf P
Z
0 li m
N
N
f
(3.22)
N
In which Nf is the number of times Z<0, N the total number of tests. Safety probability is Ps=1-Pf.
- Determine the failure event probability knowing the probability density function of performance function
fZ. Generally, it is hard to find fZ; in case accepting some simplification assumptions, errors are expected.
Basic procedures of Monte-Carlo method are as follows:
Step 1: Generate pseudo-random numbers of the load-resistance capability.
Step 2: Generate pseudo-random numbers of the load effect.
Step 3: Generate pseudo-random numbers of the safety margin (following the pseudo-uniformdistribution on [0,1] and distribution function).
Step 4: Calculate to form a storage of values of M by repeating steps 1 to 3 above multiple times, until
the required quantity following the accuracy requirement is reached.
Step 5: Assess the failure probability by frequency. Pf
Number of events M 0
Total number of tests
3.3.7.2 Comparing Monte-Carlo method with proposed method
The two methods are different, yet ther are similar points. Similar points: performing computer tests,
and determining the reliability by frequency. Different point: the method to transform stochastic input into
predetermined input, leading to different methods to determine the total number of tests. By Monte-Carlo
method, the number of tests equals to the real number of computer tests; by proposed method, thanks to
the weights, the number of computer tests is significantly less than the number of tests when calculating
frequency. Before going into specific comparisons, it is noted that both methods rely on the following 3
important information:
- Probability characteristics of the random variable (numerical characteristics, domain of determinancy, density
function,etc.)
- Utilizing the modeled values of the uniform-distribution in [0,1] to maintain similar test conditions.
- According to the law of great numbers in statistics theory (as the number of tests approaches infinity,
frequency reaches probability) to approximate probability by frequency.
The procedure of Monte-Carlo method consists of 5 main steps as mentioned. The procedure of the
proposed method consists of the following main steps:
15
Step 1: Transform the random input into an equivalent set of predetermined inputs (including
discretization, generating possible combinations, determining weights,…)
Step 2: Calculate the system response following each predetermined input in step 1.
Step 3: Verify the safety of the structure following each test to find the frequency.
It is observed that:
- Steps 1, 2 and 3 of Monte-Carlo method is equivalent to step 1 of the proposed method.
- Step 4 of Monte-Carlo method is equivalent to step 2 of the proposed method.
- Step 5 of Monte-Carlo method is equivalent to step 3 of the proposed method.
To compare reults between the two methods, introduce a simple example which was solved by
Monte-Carlo method in [55].
Consider a wooden cantilever beam under the
loading as shown in Fig. 3.11
In which q, F are normal stochastic values with
µF=18.14(kN), F=1.814(kN), µq=0.744(kN/m),
q=0.0744(kN/m). Utilize the Monte-Carlo method,
determine the value of moment at the cross-section at
a distance of L=1.8288 m (6 ft) from the free-end.
F
q
A
B
L
Figure 3.11 Beam layout
Perform Monte-Carlo modeling with 50 expressions. It is determined that the average value and the
standard deviation (approximately) are µM=31.785 (kNm) and M=3.224 (kNm).
Following the proposed method, using the same data: Discretize F into 10 values, q into 5 values, thus the
set of predetermined inputs has 50 values. After calculation, we have: µM=31.826 (kNm) and M=3.290
(kNm). Exact values are µM=31.936 (kNm) and M=3.318 (kNm).
It is evident that both methods well-reflect the exact values.
3.4 Example, comparison.
3.4.1 Free vibration of linear SDOF system with random initial condition [90].
z
2
o
Equation of vibration y 2 y y 0
y
(3.37)
The solution of equation (3.37) takes the form
y
y (cos t+ sin t )+ 0 sin t
0
(3.38)
L
ye
t
Assume the initial data y0, y 0 are stochastic values with
known statistical characteristics (expectation y0 , ,
y0
variance Dy0 , D and correlation moment K
y0
).
y0 y 0
y
Figure 3.12 Model of calculation.
Reliability is the probability P max 0 (3.42)
3.E.I
3.E.I
in which: max max 2 x y (t ) 2 x max y (t )
0 t T L W
L Wx 0 t T
x
a.Following the method in [90]:
Reliability following the strength criterion is: P( max 0 ) . Perform numerical substitution, we
have: =2.5403P1s=0.9945.
16
b.Following the proposed method: Discretize the random variables y0 and y 0 in the domain of
determinancy. Then, determine the weight of each predetermined input. Finally, calculate
n 33750
Ps 2
0.9956 .
N 33900
- Results of both methods are well-comparable ( PS1 0.9945 and PS2 0.9956 ).
3.4.2 Free vibration of linear SDOF system with random system characteristics [90]
Consider the same problem as 3.4.1, assume E and 0 are random, y0, y 0 are predetermined.
a.Following the method in [90]:
Reliability
following
the
strength
criterion
3EI
M E , 0 0 2 x y ( t ) . Determine the reliability:
L Wx
is:
P( max 0 ) .
Safety
margin:
1.7403 Ps 0.9591 .
b. Following the proposed method:
- Safety criterion max0.
- Discretize the random variables 0 and E in the domain of determinancy. Reliability following
n
the safety criterion is approximated by the formula: Ps 0.9503 .
N
- The reliability results of both methods are well-comparable ( PS1 0.9591 and PS2 0.9503 ).
3.4.3 Free vibration of linear SDOF system with random initial condition and system
characteristics.
Consider the same problem as 3.4.1, assume y0, y 0 , E and 0 are random.
a. Following the method in [90]:
Reliability
M
following
E , 0 0
the
strength
criterion
is:
P( max 0 ) .
Safety
margin:
3EI x
y (t ) .
L2 W x
Reliability factor: 1.4551 Ps 0.9272
b. Following the proposed method: - Safety criterion max0.
- Discretize the random variables y0, y 0 ,0 and E in the domain of determinancy. The reliability
n
is determined by Ps 0.9214 .
N
- The reliability results of both methods are well-comparable ( PS1 0.9272 and PS2 0.9214 ).
3.4.4 Random vibration of linear SDOF system under the effect of random excitation F=Fosin(t+)
with random system characteristics.
Consider a SDOF system under the effect of random excitation F=Fosin(t+) at the free-end,
in which F0 is the amplitude of the excitation, is the frequency of the excitation, is the initial
phase angle. is a random number, uniformly-distributed within (0,2), D is the cross-sectional
diameter, L is the span length (ignore the self-weight of the system). Assume F0 and E are
stochastic values, the others are predetermined. The safety criterion (stiffness criterion) is ymax[y]
L
( y
)
200

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