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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, Нигерия, тел. +2340833819640, e -mail geosystems_2004@yahoo.com
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,e-mail: 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:
rP = f(xp, yP, zp > t (1)
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,Уp,zp,t)~f(x0’y»zo,to) (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:
a b Dti
AS = ae + c,
(3)
where AS, - 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)
b 0 A 12
b 0 A tm
a 0 A 11 eb Ah 1
a 0 A 12 eb °A H 1
a 0 A t3 eb °A t3 1
a A te
(5)
The misclosure vector L will have the form:
L =
( m ,1 )
AS1 - (a 0 eb 0 A h + c °)
A S 2 - (a 0 eb 0 A 12 + c °)
A S 3 - (a 0 eb 0 A 13 + c °)
A S m - (a 0 eb 0 A tm + c 0 )
(6)
The corrections to the approximated values will be determined by:
a
b
c
= (AT A)-1 (AT L)
(7)
Then the adjusted values of parameters a, b and c
a adjust. a0 a
b adjust. = b0 + b
c adjust. c0 c
b A t
e
b A t
e
1
And the accuracy of these parameters can be calculated by:
ml a £ m ac
mba ml m bc = (AT A)
mca mcb mC
(9)
Below are the Velocity values at each stud with respect to time
Table - 1: Velocity and time Value
Monitoring point 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
STUD1 3.84 3.68 2.87
STUD9 5.82 7.08 4.43
STUD16 4.67 4.75 3.69
STUD8 4.14 4.6 3.52
STUD2 3.69 3.99 3.18
STUD10 5.6 6.97 4.46
STUD4 0 0.64 1.24
STUD12 5.44 7.14 4.41
STUD3 0 0.76 1.32
STUD11 5.6 7.07 4.47
STUD5 1.33 2.35 2.07
STUD13 4.97 6.6 4.26
STUD7 1.3 2.35 2.2
STUD15 3.46 5.84 3.88
STUD6 1.07 2.19 2.04
STUD14 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 At1 = 0, At2 = 3 At3 = 4.5 At4 = 8 respectively.
a
L =
1.02429 0.93659
1.03458 0.92222
^ 1.06609 0.89411
r -1.008 ^
3.74978
3.86405
v 2.8993 ,
1
1
1
The solution of the Normal equation given by: n = aT x a is presented thus:
r 4.25609 3.86666 4.12497 ^
n = 3.86666 3.52712 3.75292
v 4.12497 3.75292 4 J
The correction for the approximated value is given as: ga =
9.92148
8.65981
V 9.50514 J
T T ga := a • L
From equation (7), the correction to the approximated values is:
x1 =
152.12864 131.55298
V -282.68464 J
From equation (8), the adjusted values of parameters a, b and c xf =
152.13664
131.56098
V -282.67664 J
The inverse of the normal equation is given as:
-1 n =
r 4890 2850 -7717 N
2850 1827 -4654
V-7717 -4654 12325 j
Errors of parameter are given as
ma
(0, 0)
mb mb = 43
(1, 1)
ma = 70
Equation (7) becomes,
152 ± 70
X 1 = 132 = ± 43
- 283 ± 111
Equation (3) becomes:
mc
mc = 111
(2, 2)
AS16 = 152 x e132 xAli - 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. 0.(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. 0.(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.
© B.A. Cepedoern, P. Эхигиатор-Hругхе, O.M. Эхигиатор, X. OpuaKXU, 2012
