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10NCEE
Tenth U.S. National Conference on Earthquake Engineering
Frontiers of Earthquake Engineering
July 21-25, 2014
Anchorage, Alaska
NONLINEAR STRUCTURAL VIBRATION
UNDER BI-DIRECTIONAL RANDOM
EXCITATION WITH INCIDENT ANGLE θ
BY TAIL EQUIVALENT LINEARIZATION
METHOD
Mohsen Ghafory-Ashtiany 1 and Reza Raoufi2
ABSTRACT
In this paper Tail Equivalent Linearization Method (TELM) has been extended to cover
independent bi-directional excitation that acts with different angle from the major axes of
structure. The developed method has been applied on a 3D structure with a rigid diaphragm
supported by four different columns with bi-axial Bouc-Wen non-linear behavior model. After
finding Tail Equivalent Linear System which is defined by two unit impulse response functions
in the directions of independent components of excitation, probability density function and
cumulative distribution function, average rate of crossing and first passage probability of the
response of the structure have been calculated. The comparison of TELM results with Monte
Carlo simulation results shows good agreement. The effects of angle of incident of independent
components of bi-directional excitation with structural axis for different nonlinearity are
investigated and the most of the critical angle related to the minimum reliability index is found.
1
Professor, International Institute of Earthquake Engineering and Seismology (IIEES). [email protected]
PhD candidate, Department of Civil Engineering, Science and Research Branch, Islamic Azad University, Tehran,
Iran. [email protected]
2
Mohsen Ghafory-Ashtiany, Reza Raoufi. Nonlinear structural vibration under bi-directional random excitations with
incident angle
by Tail Equivalent Linearization Method. Proceedings of the 10th National Conference in
Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014.
10NCEE
Tenth U.S. National Conference on Earthquake Engineering
Frontiers of Earthquake Engineering
July 21-25, 2014
Anchorage, Alaska
Nonlinear structural vibration under bi-directional random excitation
with incident angle θ by Tail Equivalent Linearization Method
Mohsen Ghafory-Ashtiany 1 and Reza Raoufi2
ABSTRACT
In this paper Tail Equivalent Linearization Method (TELM) has been extended to cover
independent bi-directional excitation that acts with different angle from the major axes of
structure. The developed method has been applied on a 3D structure with a rigid diaphragm
supported by four different columns with bi-axial Bouc-Wen non-linear behavior model. After
finding Tail Equivalent Linear System which is defined by two unit impulse response functions in
the directions of independent components of excitation, probability density function and
cumulative distribution function, average rate of crossing and first passage probability of the
response of the structure have been calculated. The comparison of TELM results with Monte
Carlo simulation results shows good agreement. The effects of angle of incident of independent
components of bi-directional excitation with structural axis for different nonlinearity are
investigated and the most of the critical angle related to the minimum reliability index is found.
Introduction
Structures behave nonlinear due to extreme dynamic loads caused by the occurrence of natural
hazards such as large Earthquake, storm or wind. Because of high uncertainty and low
probability of occurrence these types of loading are modeled as random processes, and their
structural responses are studied using nonlinear random vibration theories. Considering that
superposition principle cannot be used for nonlinear systems, the nonlinear system need to be
transformed to equivalent linear system in order to be solved by common random vibration
analysis. In the conventional Equivalent Linearization Method (ELM) which is widely used
because of its simplicity and applicability to different systems the equivalent system is selected
by minimizing the mean-square error between the responses of nonlinear and equivalent linear
systems based on the assumption of Gaussian response for nonlinear system. Since the Gaussian
assumption is not valid for high nonlinear systems, the probability distribution of the desired
1
Professor, International Institute of Earthquake Engineering and Seismology (IIEES). [email protected]
PhD candidate, Department of Civil Engineering, Science and Research Branch, Islamic Azad University, Tehran,
Iran. [email protected]
2
Mohsen Ghafory-Ashtiany, Reza Raoufi. Nonlinear structural vibration under bi-directional random excitations with
incident angle
by Tail Equivalent Linearization Method. Proceedings of the 10th National Conference in
Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014.
response can be far from correct, particularly in the tail region.
To overcome the shortcomings of the conventional ELM, in 2007 Fujimura and Der Kiureghian
[1] presented Tail Equivalent Linearization Method (TELM) which uses the advantages of First
Order Reliability Method (FORM). In this method stochastic excitation is discretized and
represented in terms of a finite set of independent standard normal random variables. In this
normal random variables space, the nonlinear limit state of the specified response threshold at a
specified time instant is linearized at the nearest point to the origin. Based on the rotational
symmetry and exponential decaying of the standard normal probability density function, this
point which is called design point in FORM has the maximum likelihood among all points on the
limit state surface and has the most contribution to the probability of failure. The Tail Equivalent
Linear System (TELS) is defined based on the linearized limit state surface, because its tail
probability is equal to the tail probability of the nonlinear system. The TELS can be defined in
terms of unit Impulse Response Functions (IRFs) for nonlinear system for each direction of
independent excitation, and then be used to obtain the statistical properties of the response by
linear random vibration methods.
This method has been applied to single and multi-degrees of freedom 2D shear beam
frames and stick like models by Fujimura and Der Kiureghian in 2007 [1] and Der Kiureghian
and Fujimura in 2009 [2]; and to 3D structures with rigid diaphragm subjected to independent
bi-directional excitation in alignment with the structural axes for uni-axial Bouc-Wen material by
Broccardo and Der Kiureghian in 2012 [3] and with bi-axial Bouc-Wen material by Raoufi and
Ghafory-Ashtiany in 2013 [4].
In general the components of bi-directional excitation in the alignment with the structural
axes are dependent and correlated. But in practice by using the concept of the orthogonal set of
principal axes these components can be stated as uncorrelated and statistically independent
components along the principal directions [5,6]. In this paper TELM has been applied to a 3D
structure subjected to bi-directional independent components of excitation which act with
direction θ with respect to the major axes of structure. Thus the optimization problem should be
solved in the space of independent standard normal random variables which are defined
excitation in the alignment with the principal directions of excitation not along the structural
axes. In addition to the probability density function (PDF), cumulative density function (CDF),
mean level of crossing rate and first passage probability, the most critical angle which is related
to the minimum reliability index is also obtained. For modeling the nonlinear material behavior
the bi-axial Bouc-Wen model is used [7]. In order to verify the results the Monte Carlo
simulation on TELM results has been used.
TELM algorithm
TELM is a linearization method which uses the advantages of the FORM, where random
variables are defined in terms of independent standard normal random variables . In the space
of these normal random variables, nonlinear limit state surface, ( ) is linearized at the nearest
point to the origin ∗ . This point which is called design point in reliability has the most
contribution in the probability of failure because of exponential decaying of the PDF of standard
normal random variables. Assuming the reliability index of the desired response as β = ‖ ∗ ‖,
the first order approximation of its tail probability is Φ(−β). , see Fig. 1.
Figure 1. Nonlinear limit state surface, design point and linearization.
To obtain ∗ which is the nearest point to the origin with constraintG( ∗ ) = 0; an optimization
problem should be solved as:
∗
‖ ‖ |G( ) = 0
= arg
(1)
Usually linear search methods are used for solving this optimization problem as:
(
)
( )
=
+ λ( ) .
( )
(2)
Where m is iteration number, λ( ) and ( ) are step length and direction vector at m iteration
respectively. Design point ∗ should satisfy two convergence criteria: 1) ∗ must be located on
the limit state surface or in numerical solution it must be close to this surface; and 2) ∗ should
be the closest point on the limit state surface to the origin. To satisfy the second criteria, the
gradient vector of the limit state surface at ∗ i.e. ∇ G( ∗ ) should be pointing to the origin.
These two conditions can be stated as:
|G( ∗ )| ≤
∗
‖
−
(3)
∗
(4)
‖≤
where
is negative of the normalized gradient vector i.e. = − G( ∗ )⁄‖ G( ∗ )‖. A
common choice for and is10 . The difference between different line search algorithms is
in defining the direction vector and step length. Using improved HL-RF algorithm [8], ( ) and
λ( ) defined as:
( )
λ(
)
( )
=
=
( )
( )
+
( )
−
( )
(5)
(6)
where constant
and
usually are equal to 0.5 and is an integer number that is equal to the
smallest value that satisfies the following condition for a merit functionm( ):
m
(
)
−m
( )
≤ a. λ(
)
m
( )
.
(7)
in which a is a constant and usually set equal to 0.5 and the merit function is defined as:
m( ) = 0.5‖ ‖ + |G( )|
(8)
( ) + 10.
where c = 2 ‖u‖⁄
In stochastic dynamic for deterministic systems subjected to m-directional random
excitation, dimensional vector of excitation in the j direction, ( ) can be defined as:
= .
where
( )
(9)
is time variant deterministic
( )
×
Jacobean matrix of excitation and
=
( )
is the vector of standard normal random variables representing uncertainty
u ,u ,…,u
in the j direction of excitation with n dimension. Equation (9) can be stated for i element of
the vector of discrete excitation at time t = i × ∆t (∆t is the interval or resolution of
discretization) as:
() ()
(t) =
where
(t) = s
u s
(t) = s (t)u j = 1, … ,
( )
( )
(t), s
(t), … , s
( )
(t)
(10)
is deterministic basis function vector related
toj input which can be calculated based on the presented method in Ref. [1] and [6].
Based on the above definition, the nonlinear limit state surface of a nonlinear system
related to responseχ and threshold X at time point t in the standard normal space is defined as:
(11)
G(X, t , u) = X − χ(t , u)
where =
, ,…, ,…,
is a vector with m × n elements containing the randomness of
the excitation in all directions. After solving the optimization problem with the above mentioned
procedure that requires a finite element based algorithm for finding the response and a direct
differentiation method (DDM) based algorithm for calculating the gradient (sensitivity) of the
response with respect to the input loads, the non-Gaussian response will be replaced by a
Gaussian one which is defined by the based function vector (t ) = ∇ χ(u, t )| ∗( , ) as [1]:
(t ) =
∗ (X, )
X
t
∗
∗
‖ (X, t )‖ ‖ (X, t )‖
Separating to m vectors
to
()
(12)
, the TELS can be obtained as:
()
h (t − t )s (t ) Δt ≅ a (t ) ;
i = 1, … , n
(13)
This relation represents a set of n equations for each direction, j = 1, … , , which can be solved
for the values of the IRFs at time points in that direction. The obtained IRFs indicates TELS for
the specified threshold X and time pointt and defines a linear system in the space of variables
that has an identical design point with the nonlinear system. By obtaining the IRFs or FRFs (the
Fourier transform of IRFs) of the equivalent linear system, linear random vibration methods can
be used to determine the desired statistics of the nonlinear response for specified threshold with
first order approximation. The TELM algorithm or process has been presented in Fig. 2, which
can simplify the programing and application of this method.
TELM for incident angle
Dependent bi-directional excitation
= f , f in alignment with the major axes of structure x
and y in terms of independent components of the input excitation i.e. f , f with incident angle
in alignment with the principal axes of excitation p and q can be written as:
f (t)
cos θ
=
(t)
f
sin θ
− sin θ
cos θ
f (t)
f (t)
(14)
To find design point by improved HL-RF algorithm, the gradient of reliability response surface
χ and
χ variables should be calculated. Thus
with respect to all dependent random load i.e.
the gradient of response with respect to dependent loads must be calculated first with DDM
algorithm. Using Eq. 14, the gradient of response with respect to random variables in p and q
directions can be written as:
χ=
χ . cos θ −
χ . sin θ
(15. a)
χ=
χ . sin θ +
χ . cos θ
(15. b)
where and
are the Jacobean matrices of excitation in Eq. 9 and defined based on the
selected discretization method of excitation.
The gradient of reliability surface by considering Eq. 11 can be stated as:
G=
G;
G =−
χ;
χ
(16)
This vector is used in optimization algorithm to find design point in 2n dimensional
=
,
space. After finding the design point, by using Eq. 12 and Eq. 13 the IRFs along the
independent components of excitation i.e. h and h will be found. These IRFs which define
TELS can be used to calculate the desired statistics of response. The following example shows
the application of TELM for different input angle of incidence.
Figure 2. Tail Equivalent Linearization Method algorithm for multi-directional excitation.
Figure 3. 3D structural model; a rigid diaphragm supported by four different massless columns
with bi-axial Bouc-Wen material model, degrees of freedom and input excitations.
Numerical Analysis
The proposed method has been applied to a 3D structural model with a rigid diaphragm
supported by four different massless columns and material with bi-axial Bouc-Wen model [7] as
shown in Fig. 3 and Table 1. The desired response is displacement of column C in x direction,
i.e. χ = d . The bi-directional excitation is white-noise with spectral intensity 1 m ⁄sec and
0.5 m ⁄sec in p and q directions respectively with the duration of t = 10sec. In Table 1, K
and C are related to initial stiffness and damping respectively and σ is the root mean square
response of the linear system where for the mentioned properties is equal toσ = 0.129m.
Table 1.
Roof Diaphragm’s Properties
Diaphragm’s
Roof Mass
dimension
b = d = 20m
m=1
KN. sec
m
α = 0.1 for highly nonlinear case
α = 0.5 for mildly nonlinear case
α = 1 for linear case
Structural and material properties.
Structural Properties
Column’s Properties in x and y directions
Column A
K (KN⁄m)
C (KN. sec⁄m)
π⁄30
π
Bouc-Wen Parameters
Column A
γ =β
n=2
= 0.125(1⁄(2σ ))
Column B
Column C
Column D
K
C
K
C
K
C
2K
2C
3K
3C
4K
4C
Column B
γ =β
= 2γ
Column C
γ =β
= 4γ
Column D
γ =β
= 8γ
Fig. 4 shows FRFs of TELS for threshold = 4 and θ = 0and θ = 30for highly
nonlinear system(α = 0.1). These results can be used for random vibration analysis of the
system subjected to independent bi-directional excitation in p and q directions.
The complementary CDF, Φ −β(X, t ) has been obtained for 20 different threshold
levels from 0.25σ to 5σ with intervals 0.25σ for highly nonlinear system(α = 0.1) and
incident angles θ = 0 andθ = 30; the results are shown vs. threshold values X in Fig. 5a. The
PDF of TELS has been calculated from ϕ −β(X, t ) ⁄‖(a(X, t ))‖ for linear and nonlinear
systems, and the results are shown in Fig. 5b. These results show good agreement with the
results of 20000 Monte Carlo simulations. The difference between probability values for incident
angle θ = 0 and θ = 30 with increasing threshold and therefor intensity of nonlinear behavior is
evident. Furthermore it is seen that the probability of failure for θ = 0 is higher than for θ = 30
in all thresholds.
Figure 4.
(a)
(b)
FRF of displacement response (d ) for threshold level of 4σ for highly nonlinear
system(α = 0.1), a)p direction. b)qdirection.
(a)
(b)
Figure 5. Complementary CDF (a) and PDF (b) of the responsed for highly nonlinear
system (α = 0.1) with input angle of incidence ofθ = 0 and θ = 30 calculated by
TELM and simulation.
Using the obtained FRFs and linear random vibration methods [1], the mean level crossing rate
and first passage probability for highly nonlinear system(α = 0.1) for incident angles of θ = 0
and θ = 30 has been obtained and the results are shown in Fig. 6. Comparison of the results with
Monte Carlo simulations shows good agreement.
The advantage of using TELM for different incident angles is finding the critical incident
angle of excitation. Fig. 7 shows the probability of exceeding of the response from the specified
threshold levels 2σ and 3σ withθ for linear(α = 1), moderate nonlinear(α = 0.5) and highly
nonlinear(α = 0.1) systems at specified time instant t = 10sec for θ = −90 to θ = +90 with
5 degree intervals. The critical angle of incidence is related to the minimum value of reliability
index for different θ. As it could be seen, the critical angle of incidence for the two considered
threshold levels in the three considered degrees of nonlinearity is−5 degrees. Furthermore +85
degrees incident angle is related to the minimum probability of failure.
As It is seen in Fig. 7a the values of probability for linear system are higher than the two
considered nonlinear systems for all θ values and 2σ threshold level, but it is not valid for 3σ
threshold level as it can be seen in the Fig. 7b. This means that if the probability of exceeding of
the response from a specified threshold is larger than the related probability for the other degree
of nonlinearities, this may not be true for the other level of thresholds.
(a)
(b)
Figure 6. Mean level crossing rate (a) and first passage probability (b) for responsed of
highly nonlinear system(α = 0.1) and angle of incidence of θ = 0 and θ = 30 with
TELM and simulation.
(a)
(b)
Figure 7. Probability of failure for responsed and different θ values and different degrees of
nonlinearity. a) threshold 2σ b) threshold3σ .
Conclusions
TELM for 3D structures subjected to independent bi-directional excitation with incident angle of
θ by major axes of structure are presented. The considered 3D model example is an asymmetric
structure in two directions and is a rigid diaphragm supported by four columns with different
properties. After finding TELS for this nonlinear system, the PDF, CDF, mean rate of upcrossing and first passage probability of the desired response are obtained for coincidence and
un-coincidence of principal axes of excitation and structural axes. The most critical incident
angle is found by considering different angles of incident the minimum reliability index.
The proposed method as shown in the Fig. 2 can be used in different purpose such as
investigating the effects of incident angle and intensity of earthquake components on nonlinear
behavior of structures, nonlinear structures with secondary systems, base isolated structures,
obtaining fragility curves and performance based design of structures and many other
applications. In this paper, only two horizontal components of excitation have been considered.
Application of this method with defining proper model for torsional component of earthquake
excitation which could have significant effects on response of structures is also interesting.
Acknowledgments
The Authors express their appreciation to Prof. A. Der Kiureghian for his valuable help and
comments for this study.
References
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2.
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engineering, Earthquake Engineering and Structural Dynamics 2009; 38:719-738.
3.
Broccardo M, Der Kiureghian A. Multi-Component Nonlinear Stochastic Dynamic Analysis Using TailEquivalent Linearization Method Proceeding of 15WCEE; September 2012; Lisbon, Portugal.
4.
Raoufi R and Ghafory-Ashtiany M. Nonlinear bi-axial structural vibration under bi-directional random
excitations by Tail Equivalent Linearization Method. Under preparation.
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Penzien J, Watabe M. Characteristics of 3-dimensional earthquake ground motions. Earthquake Engineering
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motion, Earthquake Engineering and Structural Dynamics 1986; 14:543-557
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California, Berkeley, CA; 2004.