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Deformation prediction using exponential polynomial functions.

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УДК 502
ПРОГНОЗ ДЕФОРМАЦИЙ С ИСПОЛЬЗОВАНИЕМ ФУНКЦИЙ ПОКАЗАТЕЛЬНОГО
МНОГОЧЛЕНА
Владимир Адольфович Середович
Сибирская государственная геодезическая академия, 630108, Россия, г. Новосибирск,
ул. Плахотного, 10, профессор, кандидат технических наук, проректор по науке и
инновациям, тел:+79139865680, e-mail: sva@ssga.ru
Р. Эхигиатор-Иругхе
Сибирская государственная геодезическая академия, 630108, Россия, г. Новосибирск,
ул. Плахотного, 10, аспирант, тел: +79130605666, e-mail:raphehigiator@yahoo.com
О.М. Эхигиатор
Отдел физики и энергетики, Университет Бенсон Айдахоза, Бенин Сити,
+2340833819640, e -mail geosystems_2004@yahoo.com
Нигерия, тел.
Х. Ориакхи
Кафедра геодезии и геоинформатики, Эдо Государственный институт Технологический и
Управления Усен, Нигерия, тел. +2348054574255, e-mail:oriakhihenry@yahoo.com
Под деформацией понимается изменение формы любого сооружения от его исходного
состояния. При геодезическом мониторинге сооружений может быть выявлено изменение
формы, размера и динамика изменения в целом. Таким образом, на основе серии измерений,
полученных при мониторинге сооружений, можно прогнозировать время существования,
возникновение чрезвычайных ситуаций. Целью данной работы является прогноз деформаций
резервуаров для хранения сырой нефти посредством серии измерений с использованием
показательных многочленов. Прогнозные значения сравнивались с данными, приведенными
в литературных источниках, и затем выполнялся анализ. Также кратко рассмотрены вопросы
применения рассчитанной модели.
Ключевые слова: показательный многочлен, деформация сооружения, прогноз,
резервуар.
DEFORMATION PREDICTION USING EXPONENTIAL POLYNOMIAL FUNCTIONS
Vladimir А. Seredovich
Professor, Siberian State Academy of Geodesy (SSGA), Novosibirsk, Russia, tel. +79139865680,email: sva@ssga.ru
R. Ehigiator – Irughe
PhD Student, SSGA, Novosibirsk, Russia, tel. +79130605666, e-mail:raphehigiator@yahoo.com
O.M. Ehigiator
Department of Physics and Energy, Benson - Idahosa University, Benin City, Nigeria, tel.
+2348033819640, e-mail geosystems_2004@yahoo.com
H. Oriakhi
Department of Surveying and GeoInformatics, Edo State Institute of Technology and Management,
Usen, Nigeria. email:oriakhihenry@yahoo.com, tel. +2348054574255
By Deformation, we mean change of shape of any structure from its original shape and by
monitoring the structure over time using Geodetic means, the change in shape, size and the overall
structural dynamics behaviors of structure can be detected. Prediction is therefor based on the
epochs measurement obtained during monitoring of structure, the life time, failure and danger
period of the structured may therefore be forecast. The aim of this study is to predict the
Deformation experience by crude oil Tank under continuous loading with data obtained in four
epochs of measurement using Exponential polynomial technique. The predictions were compared
with measured data reported in literature and the results are discussed. The computational aspects of
implementation of the model are also discussed briefly.
Key words: exponential polynomial, structural deformation, prediction, tank.
INTRODUCTION
In many civil structures like bridges, vertical oil storage tanks, tunnels and
dams; the deformations are the most relevant parameters to be monitored. Monitoring
the structural deformation and dynamic response to the large variety of external
loadings has a great importance for maintaining structures safety and economical
design of man-made structures.
Prediction is therefor based on the epochs measurement obtained during
structural monitoring, from the data obtained, the life time, failure and danger period
of the structured may be forecast. The main purpose of structural deformation
monitoring scheme and analysis is to detect any significant movements of the
structure. The knowledge of behavior of Tank Structure under uniaxial/biaxial tensile
loads is necessary to predict the changes in perform geometry of the structure. The
aim of this study is to predict the deformation experience by the structure under
continuous loading with data obtained in four epochs of measurement using
Exponential polynomial technique. The predictions are compared with measured data
reported in literature and the results are discussed. The computational aspects of
implementation of the model are also discussed briefly.
Prediction of the deformation values of circular oil storage tanks
Deformation structures can be fully determined by the movement of points
which are measured on the structure. Let the vector position of point P in threedimensional coordinate system (X, Y, Z) before and after deformation is equal to rp
and r/p respectively. Then r/p may be expressed as:
(1)
rP/ = f ( x p , y p , z p , t ),
where t - time variation between two cycles (epochs) of observations.
From equation (1), the position of points on the object observed depends on their
initial position and time. The displacement vector dp at the point P is defined as:
dp = r/ −rp = f (xp, yp, zp,t) − f (x0, yo, z0,t0)
(2)
Prediction with exponential function
In mathematics, the exponential function is the function ex, where e is a base of
natural logarithm. The exponential function is used to model phenomena when a
constant change in the independent variable gives the same proportional change ((i.e.
increase or decrease) in the dependent variable. The exponential function is often
written as exp(x), especially when the input is an expression too complex to be
written as an exponent. In calculus a branch of mathematics, the derivative is a
measure of how a function changes as its input changes.
For predicting structural deformation values with exponential function, we
suggested applying the following equation form:
b ∆ti
∆Si = a e
+ c,
(3)
where ∆Si – the deformation values at time i in vertical or horizontal
dimensions; a, b, c – coefficients of proposed equation; and i = 1, …, m.
Using the observational data and least square method, the three coefficients a, b
and c can be estimated using the general equation form:
A(m, 3) X(3,1)+ L(m, 1) = V(m, 1) .
(4)
where m – the number of epochs of observations.
It is important to note that the first step of solution is approximating values of
unknowns’ a0, b0 and c0. Matrix A will be determined by differentiation the equation
(3) with respect to parameters a, b and c. So matrix A, in this case, has the form:
A
( m ,3 )
 e

 e

=  e

 ....
 ...

 e
b 0 ∆ t1
a 0 ∆ t1 e
b 0 ∆ t1
b ∆ t2
a 0∆ t2 e
b ∆ t2
b 0 ∆ t3
a 0 ∆ t3 e
b 0 ∆ t3
0
b 0 ∆ tm
1 

1 

1 ,




1 
0
a 0∆ tm e
b 0∆ tm
(5)
The misclosure vector L will have the form:
L
( m ,1 )
∆ S1

∆ S 2

∆S3
= 
 ....
 ....

 ∆ S m
− (a
0
− (a
0
− (a
0
− (a
) 

e b ∆ t2 + c 0 ) 

0
e b ∆ t3 + c 0 ) 




0
e b ∆ t m + c 0 ) 
e
0
b
0
∆ t1
+ c
0
0
(6)
The corrections to the approximated values will be determined by:
 a 


T
−1
T
 b  = ( A A) ( A L) .
 c 


(7)
Then the adjusted values of parameters a, b and c
 a adjust . 


b
 adjust .  =

c
 adjust . 
a 0   a 

 0 
b
+
b

.
 
c 0   c 
  

(8)
And the accuracy of these parameters can be calculated by:
 m a2 m ab m ac

 m ba m b2 m bc

 m ca m cb m c2



(9)
 = ( A T A ) −1 .



Below are the Velocity values at each stud with respect to time
Table - 1: Velocity and time Value
Monitoring
point
STUD1
STUD9
STUD16
STUD8
STUD2
STUD10
STUD4
STUD12
STUD3
STUD11
STUD5
STUD13
STUD7
STUD15
STUD6
STUD14
Velocity, mm/year
Vertical values, mm/year
t= 3 years
t= 4.25 year
t= 8 years
from 5/2000
from 5/2000
from 5/2000
to
to
to
5/2003
8/2004
May-08
3.84
3.68
2.87
5.82
7.08
4.43
4.67
4.75
3.69
4.14
4.6
3.52
3.69
3.99
3.18
5.6
6.97
4.46
0
0.64
1.24
5.44
7.14
4.41
0
0.76
1.32
5.6
7.07
4.47
1.33
2.35
2.07
4.97
6.6
4.26
1.3
2.35
2.2
3.46
5.84
3.88
1.07
2.19
2.04
4.1
6.42
4.15
Using Mathcad, the solution to the unknowns is presented below. Our initial
approximation was 0.008 for a0, b0 and c0, while ∆t1 = 0, ∆t2 = 3 ∆t3 = 4.5 ∆t4 = 8
respectively.
1
 1
1.02429 0.93659
a=
 1.03458 0.92222
 1.06609 0.89411

 −1.008 
3.74978 
L=
 3.86405 
 2.8993 



1
1

1
1
The solution of the Normal equation given by: n = a T × a is presented thus:
 4.25609
n =  3.86666

 4.12497

3.75292 

4

3.86666 4.12497
3.52712
3.75292
The correction for the approximated value is given as:
 9.92148 
ga =  8.65981 


 9.50514 
T
ga := a ⋅ L
From equation (7), the correction to the approximated values is:
 152.12864 
x1 =  131.55298 


 −282.68464 
From equation (8), the adjusted values of parameters a, b and c
 152.13664 
xf =  131.56098 


 −282.67664 
The inverse of the normal equation is given as:
n
−1
 4890 2850 −7717 
=  2850 1827 −4654 


7717
4654
−
−
12325


Errors of parameter are given as
ma :=
( n − 1) ( 0 , 0)
mb :=
ma = 70
mb = 43
Equation (7) becomes,
±
152
X 1 =
132
− 283
=
Equation (3) becomes:
70
± 43
± 111
( n − 1) ( 1 , 1)
mc :=
( n − 1) ( 2 , 2)
mc = 111
∆ S 16 = 152 × e 132 × ∆ t i − 283
Below is the graph of prediction plotted time against deformation values for tank
6 stud 16
Fig. 1. Plot of Velocity against time for Stud 16
ANALYSIS OF RESULTS AND CONCLUSION
Table 1 vertical deformation values while fig 1.0 is the plot of time against
velocity for monitoring point stud 16 for tank № 6. From the above, the predicted
deformation graph and the observed value intersected at two points with time equal to
4.0yr with a velocity of 4.8mm/yr and time 9.8yr with velocity of 2mm/yr
respectively.
A further projection of the prediction graph and the observed values will may
not reveal uniformity. It is important to note that no observation was carried out in
year 2001, 2002, 2005, 2006 and 2007 because of the unrest in the Niger delta of
Nigeria.
The results obtained in this study may however be acceptable to the structural
Engineer depending on the tank specifications and its properties at the design stage.
REFERENCES
1. Ehigiator – Irughe, R. and Ehigiator M. O.(2010)
“Estimation of the centre coordinates and radius of Forcados Oil Tank from Total Station data
using least square Analysis” International Journal of pure and applied sciences. A pan – African
Journal Series 2010
2. Ehigiator-Irughe, R. Environmental safety and monitoring of crude oil storage tanks at
the Forcados terminal. M. Eng. Thesis. - Department of civil engineering, University of Benin,
Benin City. Nigeria. – 2005.
3. Gairns, C. Development of semi-automated system for structural deformation monitoring
using a reflector less total station. M.Sc. Thesis. – Department of Geodesy and Geomatics
Engineering – University of New Brunswick, 2008. –
4. Ehigiator – Irughe, R. Ashraf A. A. Beshr, and Ehigiator M. O.(2010)
“Structural deformation analysis of cylindrical oil storage tank using geodetic observations”
(Paper Presented at Geo –Siberia 2010, International Exhibition and scientific conference VI page
34 - 37, Novosibirsk Russia Federation).
5. Chrzanowski S., A. M. Massiera, A. Chrzanowski, (2003). “Use Of Geodetic Monitoring
Measurements In Solving Geomechanical Problems In Structural And Mining Engineering”,
Proceedings of the 11th Int. Symp. On Deformation Measurements, Santorini, Greece, May25-28.
© В.А. Середович, Р. Эхигиатор-Иругхе, О.М. Эхигиатор, Х. Ориакхи, 2012
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