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An application of the Φ-functions series
method to the integration of seismic modelling
A. Reyes1, J. A. Reyes2, M. Cortés-Molina2, Y. Villacampa2
& F. García-Alonso2
1Research Group Mathematical Modeling of Systems,
University of Alicante, Spain
2Department of Applied Mathematics, Escuela Politécnica Superior,
University of Alicante, Spain
Abstract
The interest to improve the response of structures in front of an earthquake has
increased in recent years, leading to the investigation of different calculation
methods, especially those based on static non-linear analysis to increase
accuracy. The non-linear calculation can be approached by means of discrete or
continuous models. The discrete models represent the structure by a finite
number of degrees of freedom; in this case the movement equations are ordinary
differential equations which are resolved by numerical methods.
This paper applies a new method for the numerical integration of SDOF and
2DOF, which is developed from the Scheifele methods. The algorithm integrates
the unperturbed problem without truncation error, which represents an advantage
in front of the Taylor series. The new method calculates the exact solution of the
perturbed problem through a series of functions, whose coefficients are obtained
by simple algebraic recurrences involving the perturbation function.
To illustrate the application of the algorithm the resolution of two linear
systems is shown; the first one with a single degree of freedom and the second
with two degrees of freedom.
Keywords: seismic response, SDOF, 2DOF, numerical solutions, perturbed
linear systems of ODEs.
Earthquake Resistant Engineering Structures IX 245
www.witpress.com, ISSN 1743-3509 (on-line)
WIT Transactions on The Built Environment, Vol 132, ©2013 WIT Press
doi:10.2495/ERES130201
1 Introduction
In recent years, interest in improving the response of structures to seismic
activity has increased dramatically which has in turn led to research into the
different methods of calculation. Historically speaking, structural calculations,
both in the field of building and civil engineering, have been carried out from a
static viewpoint, with a particular focus on those based on non-linear static
analysis to increase precision.
The need for non-linear calculation is due to the fact that elastic calculation
allows us to obtain the elastic capacity of the structure but not the failure
mechanism of the same and, therefore, the redistribution of stresses on the
sections.
Non-linear calculation also enables greater detail to be achieved as regards the
structural model and, particularly, in formulating a more precise equation to
model the movement caused by seismic activity.
Non-linear calculation can be approached from the point of view of
continuous or discrete models. Continuous models have distributed parameters,
where the movement equations are differential equations in partial derivatives,
models whose exact integration is only possible in the case of structures with
simple geometry.
Discrete models represent the structure through a finite number of degrees of
freedom. In this case, the movement equations are ordinary differential equations
that are resolved by numerical methods.
Typical civil engineering structures are always schematised as MDOF. For
example, buildings with several floors are analysed using these systems.
For n degrees of freedom, the equation is expressed in matricial fashion:
() () () ()Mt Ct Kt Mt++=−
xxx a
(1)
(0) = =
0
0xx
,
(0) = =
0
0
xx
where M, C and K are the mass, damping and stiffness matrices, respectively.
The vector column
()ta
contains the acceleration values.
These problems are currently resolved by using different software programs
that implement numerical methods to calculate the structural response to such
oscillatory movements as seismic activity.
This study applies a new method for the numerical integration of these kinds
of oscillators and systems, and is based on the methods used by Scheifele [1–5],
involving an extension of the Taylor series methods. Said method allows us to
express the solution of the system as a series of
Φ
-functions, which are real
functions with values in the ring of the matrices
( )
,m
[6], obtaining the
coefficients of the series by recurrences which involve the perturbation function.
The
Φ
-function series method is able to integrate the perturbed problem
exactly, which is an advantage over algorithms based on the Taylor series.
In order to achieve numerical integration of the IVP (1), a linear differential
operator is defined, D+B, with B being a suitable matrix that allows us to
annihilate the perturbation terms, transforming the system into a homogeneous
246 Earthquake Resistant Engineering Structures IX
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second-order system, and managing to integrate it exactly with only the two first
Φ
-functions of the series method.
To illustrate how the new algorithm is applied, we show the resolution of the
single degree of freedom and two degrees of freedom linear system, which
model earthquakes.
The precision and efficiency of the
Φ
-function series method is contrasted
with the results obtained by other well-known integrators, such as LSODE,
Rosenbrock, Gear, Newmark
β
-method and the Wilson
θ
-method.
2 Application of the Φ-functions series method to an
earthquake modelled by an SDOFa
The equation of motion (or dynamic equilibrium equation) of a Single Degree Of
Freedom (SDOF) is
() () () ()
e
mxt cxt kxt F t++=
where m is the floor’s mass, c
and k are the damping and stiff coefficients, respectively.
()
e
Ft
is the external
force [7].
The importance of a SDOF resolution lies in that is that best shows the
interdependence between structure and its properties and the duration of an
earthquake.
Whereas the given structure (Fig. 1), which is not subject to any external
force but a movement of the ground due to an earthquake, the elastic force of the
columns is expressed through
( )
() ()
sg
F k yt u t=−−
, where
()yt
and
()
g
ut
are
the absolute displacement of the mass and of the ground, respectively.
Figure 1: Single Degree of Freedom System (SDOF).
The expression
() () ()
g
xt yt u t= −
is the relative displacement between the
mass and the ground, therefore
()
s
F kx t= −
.
Analogously, the damping force is
( )
() () ()
dg
F c y t u t cx t=−−=−
and the
external force is zero.
Applying Newton’s second law,
()F my t=
∑
, is obtained
() () () ()
g
mxt cxt kxt mu t++=−
, in standardized form:
2
() 2 () () ()
nn g
xt xt xt u t
ζω ω
+ +=−
(2)
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where
/
n
km
ω
=
is the undamped natural frequency of vibration and
( )
/2
n
cm
ζω
=
is the critical damping ratio.
If
()
g
mu t
is a harmonic forcing function, i.e.
00
( ) sin( )
g
mu t F t
ω
=
equation (2)
can be expressed:
20
0
() 2 () () sin( )
nn
F
xt xt xt t
m
ζω ω ω
+ +=−
(3)
at the moment the earthquake occurs, it is very reasonable to assume that the
structure is at rest, i.e.
(0) 0x=
,
(0) 0x=
and
[ ]
0,tT∈
.
In order to apply the
Φ
-function series method, the change of variable is
affected:
1
xu=
,
2
xu=
where
12
u xu= =
22
00
2 0 21 0
2 sin( ) 2 sin( )
nn n n
FF
ux x x t u u t
mm
ζω ω ω ζω ω ω
==− −− =− − −
(4)
The IVP (3) can be expressed as:
11
00
2
22
01 sin( )
0
2
t
nn
uu
Ft
uu
m
ω
ω ζω
−
+=−
,
1
2
(0) 0
(0) 0
u
u
=
(5)
The variable is introduced in order to make easier the elimination the
disturbance’s function of the IVP (5), following the Steffensen’s techniques [8,
9].
0
30
sin( )
F
ut
m
ω
= −
, obtaining a new IVP.
11
200
2 2 0 00
33
() 0 1 0 ()
( ) 2 0 ( ) 0 sin( ) cos( ) ,
() 0 0 0 ()
t
nn
ut ut FF
ut ut t t
mm
ut ut
ω ζω ω ω ω
−
+=−
(6)
with
( )
(0) 000
t
=u
To invalidate the function of disturbance, the differential operator
( )
DB+
is
applied to (6), where B is the following matrix:
2
0
00 0
00 1
00
B
ω
= −
(7)
obtaining the extended IVP,
11 1
2
22 2
2 22 2
3 0 3 0 03
() 0 1 0 () 0 0 0 () 0
() 2 1 () 0 0 0 () 0
() 0 0 () 2 0 () 0
nn
nn
ut ut ut
ut ut ut
ut ut ut
ω ζω
ω ω ω ζω ω
−
+ −+ =
(8)
( )
(0) 000
t
=u
,
0
0
(0) 0 0
t
F
m
ω
= −
u
248 Earthquake Resistant Engineering Structures IX
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which is integrated exactly using the
Φ
-functions series algorithm described in
[6].
2.1 Resolution of a seismic model (SDOF) by Φ-functions series method
Choosing the following specific values for the structural variables [7]:
2
1.0 .
ks
min
⋅
=
,
5%
ζ
=
,
2
n
rad
s
ωπ
=
,
0
10 F kip=
,
04 rad
s
ω
=
and
1 Ts=
.
The IVP is:
( )
11
2
22
33
0 10
() ()
( ) 4 0 ( ) 0 10sin(4 ) 40 cos(4 ) ,
5
() ()
0 00
t
ut ut
ut ut t t
ut ut
π
π
−
+=−
1
2
3
(0) 0
(0) 0 .
(0) 0
u
u
u
=
(9)
By applying the differential operator,
( )
DB+
, is obtained the expanded IVP:
11 1
2
22 2
2
33 3
0 10
() () 0 0 0 () 0
() 4 1 () 0 0 0 () 0 ,
5
() () 16 () 0
0 16 0 64 0
5
ut ut ut
ut ut ut
ut ut ut
π
π
π
π
−
+ −+ =
(10)
( )
(0) 000
t
=u
,
( )
(0) 0 0 40
t
= −
u
,
which is integrated exactly by the following algorithm, particularized for this
problem.
( )
00
000
t
= =au
(11)
2
10
0 10 0
4 00
540
0 00
π
π
−
=−−
aa
from k = 1 up to n calculates:
0 011
() ()
k
hhΦ +Φu= a a
0k
=au
2
1
0 10 0
4 0 10sin(4 )
540cos(4 )
0 00
k
kh
kh
π
π
−
=−−
au
following k.
The results obtained using the
Φ
-functions series method, are compared with
the known codes:
LSODE methods, causes a numerical solution to be found using the
Livermore Stiff Ode Solver. It solves stiff and nonstiff systems. It uses Adams
methods (predictor-corrector) in the nonstiff case, and Backward Differentiation
Formula (BDF) in the stiff case.
ROSENBROCK the method finds a numerical solution using an Implicit
Rosenbrock third-fourth order Runge-Kutta method with degree three
interpolant.
Earthquake Resistant Engineering Structures IX 249
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WIT Transactions on The Built Environment, Vol 132, ©2013 WIT Press
GEAR causes a numerical solution to be found by way of Burlisch-Stöer
rational extrapolation method. The method has higher precision and calculation
efficiency, especially in solving stiff differential equations.
TAYLOR SERIES the method finds a numerical solution to the differential
equations, using a Taylor series method. This method can be used for high
accuracy solutions.
NEWMARK BETA-METHOD is a method of numerical integration used to
solve differential equations. In this method the constant average acceleration is
generally used in structural dynamics because it has been shown to have a high
degree of numerical stability.
WILSON THETA-METHOD assumes that the acceleration of the system
varies linearly between two instants of time,
t
to
th
θ
+
, where the value of
θ
need not be an integer and is usually greater than 1.0. the method is
unconditionally stable for linear dynamic systems when
1.37
θ
>
, and a value of
1.4
θ
=
is often used for nonlinear dynamic systems.
Figure 2:
The decimal logarithm
of module of the relative
error of the solution
()tu
.
Figure 3:
The decimal logarithm
of the relative error of
the solution
()xt
.
Fig. 2 shows the graph of the decimal logarithm of module of the relative
error of the solution
()tu
, vs t, calculate using (11), step size
0.01h=
and 50
digits, with the numerical integration codes LSODE with
25
10tol
−
=
,
ROSENBROCK with abserr = 10-30, GEAR with
25
10errorper
−
=
and
TAYLORSERIES with abserr = 10-25.
Fig. 3 shows the logarithm graph for the absolute value of the relative error of
solution
()xt
, vs. t, obtained with 50 digits, calculated by means of (11), with
two
Φ
-functions and step size
0.001h=
, compared with the numerical
integration codes NEWMARK BETA-METHOD with
1/2
δ
=
,
1/4
α
=
,
0.001h=
and WILSON THETA-METHOD with
1/2
δ
=
,
1/6
α
=
,
1.4
θ
=
,
0.001h=
. Analogous results are obtained for the velocity
()xt
.
250 Earthquake Resistant Engineering Structures IX
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3 Application of the Φ-functions series method to an
earthquake modelled by a 2DOF
A 2DOF can be represented in Fig. 4 and it is used to study the dynamic forces
acting on this system. Similarly at SDOF, four types of forces act on each floor
mass, the stiffness force, the damping force, the external force and inertial force
[7].
Figure 4: Two Degrees of Freedom System (2DOF).
The dynamic equilibrium equations of motion are:
1 1 12 21 12 21 1
22 2 22 2 22 2
0 ()
0 ()
m x c c c x k k k x Ft
m x c c x k k x Ft
+− +−
++ =
−−
(12)
defining
1
2
0,
0
m
Mm
=
12 2
22
,
cc c
Ccc
+−
=
−
12 2
22
,
kk k
K
kk
+−
=
−
1
2
()
() ()
xt
txt
=
x
and
( )
12
() () ()
t
Ft F t F t=
, when the symmetrical and positive definite matrices M, C
and K, are the mass, damping and stiffness matrix, respectively; the system (12)
can be expressed by
() () () ()Mt Ct Kt Ft++=
xxx
.
Considering that the structure is subjected to an earthquake ground motion,
where only horizontal translation of the earthquake ground motion is considered.
Applying the Newton’s second law and given that the external force is zero, are:
( )
( )
( )
( )
( ) ( )
11 21 2 11 21 2 11
22 22 1 22 1
0,
0,
gg
my c y y cy u k y y cy u
mycyy kyy
+ −+ −+ −+ −=
+ −+ −=
(13)
where
g
u
and
g
u
are the absolute ground displacement and the absolute ground
velocity, respectively.
If it define
11
() () ()
g
xt yt u t= −
and
22
() () ()
g
xt yt ut= −
, as relative
displacement between the mass and the ground, the equations (13) are:
1 1 12 21 12 21 1
22 2 22 2 22 2
00
1
00
1
g
m x cc c x k k k x m u
mx c c x k k x m
+− +−
++ =−
−−
(14)
If
1
2
01
01g
mu
m
is a harmonic matrix forcing function, i.e.:
00
1
00
2
sin( )
01
sin( )
01
g
Ft
muFt
m
ω
ω
=
(15)
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equation (14) is:
00
1 1 12 21 12 21
00
22 2 22 2 22
sin( )
0
sin( )
0
Ft
m x cc c x kk k x
Ft
mx c c x k k x
ω
ω
+− +−
++ =−
−−
(16)
at the moment that the earthquake occurs, it is very reasonable to assume that the
structure is at rest.
Therefore or normalized form, the IVP is:
( )
11
001002
() () () sin( )/ sin( )/
t
t MCt MKt F t m F t m
ωω
−−
++=−
xxx
(17)
( )
0
(0) 0 0 t
= =xx
and
( )
0
(0) 0 0 t
= =
xx
In order to apply the
Φ
-function series method, is effected the change of
variable:
11
xu=
,
13
xu=
,
13
xu=
,
22
xu=
,
24
xu=
,
24
xu=
.
The IVP (17) can be expressed as:
11
22
22 22 0000
11
33
12
44
sin( ) sin( )
00 ,
t
uu
uu
OI FtFt
uu
MK MC mm
uu
ωω
××
−−
+=−
0
0
(0) .
0
0
=
u
(18)
3.1 Resolution of a seismic model (2DOF) by
Φ
-functions series method
Consider the two-story frame subjected to an earthquake ground motion [7]
(Fig. 5).
Figure 5: Two-story frame.
The dynamic equilibrium equation of motion is:
11 1
22 2
20 3 4 2 201
()
0 2 23 0 1
g
xx x
m cc k k m ut
xx x
m cc k k m
−−
++ =−
−−
(19)
If
2 01 ()
01
g
mut
m
is a harmonic matrix forcing function, i.e.
00
00
sin( )
2 01 () sin( )
01
g
Ft
mut Ft
m
ω
ω
=
then equation (19) is:
00
11 1
00
22 2
sin( )
20 3 4 2
sin( )
0 2 23
Ft
xx x
m cc k k
Ft
xx x
m cc k k
ω
ω
−−
++ =−
−−
(20)
In notation more compact and normalizing the equation (20), is obtained:
252 Earthquake Resistant Engineering Structures IX
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11 0000
sin( ) sin( )
() () () 2
t
FtFt
t MCt MKt mm
ωω
−−
++=−
xxx
(21)
at the moment that the earthquake occurs, it is very reasonable to assume that the
structure is at rest.
To solve the IVP:
11
0000
sin( ) sin( )
() () () 2
t
FtFt
t MCt MKt
mm
ωω
−−
++=−
xxx
(22)
with
(0) 0=x
,
(0) 0=
x
,
[ ]
0,tT∈
,
using the methodology of the
Φ
-functions, the new expression for the IVP
11
22
22 22 00
00
11
33
44
() ()
() () 0 0 sin( ) sin( ) ,
() () 2
() ()
t
ut ut
ut ut
OI FF
tt
ut ut
MK MC mm
ut ut
ωω
××
−−
+=−
(23)
(0) =0u
where
1
21
23
k
MK m
−
−
=
−
and
1
31
24
2
c
MC m
−
−
=
−
.
The variable is introduced in order to make easier the elimination
the disturbance’s function of the IVP (23), following the Steffensen’s techniques
[8, 9].
0
50
sin( )
2
F
ut
m
ω
= −
, obtaining a new IVP.
11
22
22 22 21
11 0 0 00
33
21 0
44
12 12 11
55
() ()
() () sin( ) cos( )
() () 0 0 sin( )
22
() ()
() ()
t
ut ut
ut ut
O IO Ft t
ut ut
MK MC O t
m
ut ut
O OO
ut ut
ω ωω
ω
× ××
−−
×
× ××
+=−
(24)
with
(0) =0u
.
To invalidate the function of disturbance, the differential operator
( )
DB+
is applied to (24), where B is the following matrix
22 22 21
22 22 21
12 12 11
OOO
BO O
OO
×××
×× ×
×××
= Ω
Ω
, with
21
1
2
×
−
Ω=
−
and
( )
2
12 0 0
ω
×
Ω=
(25)
obtaining the extended IVP
11
22
22 22 21 22 22 21
11
33
21 2 2 22 21
11
44
12 12 11 12 12 11
55
() ()
() ()
() ()
() ()
() ()
ut ut u
ut ut
O IO O O O
ut ut
MK MC O O O
ut ut
O O MK MC O
ut ut
× ×× × × ×
−−
× × ××
−−
× ×× × × ×
+ Ω+
Ω ΩΩ
1
2
3
4
5
() 0
() 0
() ,
0
() 0
() 0
t
ut
ut
ut
ut
=
(26)
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( )
(0) 00000
t
=u
,
0
0
(0) 0000 2
t
F
m
ω
= −
u
,
which is integrated exactly using the
Φ
-functions series algorithm described in
[6].
Choosing the following values for the structural variables [7]:
2
1.0 .
ks
min
⋅
=
,
5%
ζ
=
,
2
n
rad
s
ωπ
=
,
010 F kip=
,
04 rad
s
ω
=
and
1 Ts=
.
The IVP is:
22
11 1 1
22
22 2 2
3
(0)
2 0 10sin(4 ) 0
16 8
10 5 , (0)
0 1 2 10sin(4 ) 0
8 12
55
xx x x
t
xx x x
t
ππ
ππ
ππ ππ
−
−
++ =− =
−
−
(27)
making the change of variable
11
xu=
,
13
xu=
,
13
xu=
and
22
xu=
,
24
xu
=
,
24
xu=
, the new IVP is:
11
22
22
33
22
44
55
0 0 1 00
() () 0
0 0 0 10
() () 0
3
84 0
() () 5sin(4 )
10 10
() ()
2 10sin(4 )
8 12 0
55
() () 20cos(4 )
0 0 0 00
ut ut
ut ut
ut ut t
ut ut t
ut ut t
ππ
ππ
ππ
ππ
−
−
−
−
+=−
−
−
,
1
2
3
4
5
(0) 0
(0) 0
(0) .
0
(0) 0
(0) 0
u
u
u
u
u
=
(28)
A matrix which annihilates the disturbance function is:
00000
00000
,
0000 1
0000 2
00160 0
B
=−
−
(29)
applying the operator
( )
DB+
to the system (28) we obtain the extended IVP:
11
22
22
33
22
44
22
55
0 0 1 00 0 0 0 00
() ()
0 0 0 10 0 0 0 00
() ()
30 0 0 00
84 1
() ()
10 10 0 0 0 00
() ()
2
8 12 2
55
() () 128 64
0 0 16 0 0
ut ut
ut ut
ut ut
ut ut
ut ut
ππ
ππ
ππ
ππ ππ
−
−−
++
−
−−
−
1
2
3
4
5
() 0
() 0
() ,
0
() 0
24 8 () 0
0
55
ut
ut
ut
ut
ut
ππ
=
−
(30)
( )
(0) 00000
t
=u
,
( )
(0) 0000 20
t
= −
u
,
that is integrated exactly by the following algorithm the
Φ
-functions series ,
applied to this problem
( )
00
00000
t
= =au
(31)
22
10
22
0 0 1 00
0
0 0 0 10
0
3
84 0 0
10 10
20
8 12 0
55 20
0 0 0 00
ππ
ππ
ππ
ππ
−
−
= −
−
−
−
aa
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from k = 1 up to n calculates:
0 011
() ()
khhΦ +Φu= a a
0k
=au
22
1
22
0 0 1 00
0
0 0 0 10
0
3
84 0 5sin(4 )
10 10
2 10sin(4 )
8 12 0
55 20cos(4 )
0 0 0 00
k
kh
kh
kh
ππ
ππ
ππ
ππ
−
−
= −
−
−
au
following k.
Figure 6:
The decimal logarithm
of module of the relative
error of the solution
()tu
.
Figure 7: The decimal logarithm
of module of the
relative error of the
position
()tx
.
Fig. 6 shows the graph of the decimal logarithm of module of the relative
error of the solution
()tu
, vs t, calculate using (31), step size
0.01h=
and 50
digits, with the numerical integration codes LSODE with
25
10tol
−
=
,
ROSENBROCK with abserr = 10-30, GEAR with
25
10errorper −
=
and
TAYLORSERIES with abserr = 10-25.
Fig. 7 shows the graph of the decimal logarithm of module of the relative
error of the solution
()tx
, vs t, obtained with 50 digits, calculated by means of
(31), with two
Φ
-functions and step size
0.001h=
, compared with the
numerical integration codes NEWMARK BETA-METHOD with
1/2
δ
=
,
1/4
α
=
,
0.001h=
and WILSON THETA-METHOD with
1/2
δ
=
,
1/6
α
=
,
1.4
θ
=
,
0.001h=
. Analogous results are obtained for the velocity
()t
x
.
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4 Conclusions
The method of numerical integration of
Φ
-functions series is based on the ideas
developed by Scheifele in his
Γ
-functions series method. The
Φ
-functions
series method has an advantage over the Scheifele method to integrate exactly
the perturbed problem, transforming it into second-order homogeneous problem
which is able to integrate exactly with two first
Φ
-functions.
The good performance and accuracy of the
Φ
-functions series method is
shown by comparing numerical results obtained in the resolution of a SDOF and
2DOF with the results calculated with other well known integrators, such as
ROSENBROCK, GEAR, TAYLORSERIES, NEWMARK BETA-METHOD
and WILSON THETA-METHOD.
Acknowledgement
This work has been supported by the project of the Generalitat Valenciana
GV/2011/032.
References
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256 Earthquake Resistant Engineering Structures IX
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