Discrepancy Analysis of Load–Displacement in the Combination Space for Concrete Box Girder Assessment
The combination space concept had its origin in the theory of subspace and combinatorics. The combination space may be treated as the space reconstruction through combination, as well as the subspace one. Several new analytical methods are proposed based on combination principles applied to some ope...
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Ye, X.F. Ogai, H. Kim, C.W. 2020-12-28T18:20:18Z 2020-12-28T18:20:18Z 2018 Discrepancy Analysis of Load–Displacement in the Combination Space for Concrete Box Girder Assessment / X.F. Ye, H. Ogai, C.W. Kim // Проблемы прочности. — 2018. — № 5. — С. 14-21. — Бібліогр.: 16 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173988 539.4 The combination space concept had its origin in the theory of subspace and combinatorics. The combination space may be treated as the space reconstruction through combination, as well as the subspace one. Several new analytical methods are proposed based on combination principles applied to some operations. Концепция пространства комбинаций возникла из теории подпространства и комбинаторики. Пространство комбинаций может рассматриваться перестраиванием как пространства, так и подпространства. Предложен ряд новых аналитических методов, основанных на принципах комбинации применительно к некоторым операциям. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Discrepancy Analysis of Load–Displacement in the Combination Space for Concrete Box Girder Assessment Анализ отклонений напряженно-деформированного состояния в пространстве комбинаций при оценке состояния бетонной коробчатой балки Article published earlier |
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Discrepancy Analysis of Load–Displacement in the Combination Space for Concrete Box Girder Assessment |
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Discrepancy Analysis of Load–Displacement in the Combination Space for Concrete Box Girder Assessment Ye, X.F. Ogai, H. Kim, C.W. Научно-технический раздел |
| title_short |
Discrepancy Analysis of Load–Displacement in the Combination Space for Concrete Box Girder Assessment |
| title_full |
Discrepancy Analysis of Load–Displacement in the Combination Space for Concrete Box Girder Assessment |
| title_fullStr |
Discrepancy Analysis of Load–Displacement in the Combination Space for Concrete Box Girder Assessment |
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Discrepancy Analysis of Load–Displacement in the Combination Space for Concrete Box Girder Assessment |
| title_sort |
discrepancy analysis of load–displacement in the combination space for concrete box girder assessment |
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Ye, X.F. Ogai, H. Kim, C.W. |
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Ye, X.F. Ogai, H. Kim, C.W. |
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Научно-технический раздел |
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Научно-технический раздел |
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2018 |
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English |
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Проблемы прочности |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Article |
| title_alt |
Анализ отклонений напряженно-деформированного состояния в пространстве комбинаций при оценке состояния бетонной коробчатой балки |
| description |
The combination space concept had its origin in the theory of subspace and combinatorics. The combination space may be treated as the space reconstruction through combination, as well as the subspace one. Several new analytical methods are proposed based on combination principles applied to some operations.
Концепция пространства комбинаций возникла из теории подпространства и комбинаторики. Пространство комбинаций может рассматриваться перестраиванием как пространства, так и подпространства. Предложен ряд новых аналитических методов, основанных на принципах комбинации применительно к некоторым операциям.
|
| issn |
0556-171X |
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https://nasplib.isofts.kiev.ua/handle/123456789/173988 |
| citation_txt |
Discrepancy Analysis of Load–Displacement in the Combination Space for Concrete Box Girder Assessment / X.F. Ye, H. Ogai, C.W. Kim // Проблемы прочности. — 2018. — № 5. — С. 14-21. — Бібліогр.: 16 назв. — англ. |
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| first_indexed |
2025-11-26T06:53:50Z |
| last_indexed |
2025-11-26T06:53:50Z |
| _version_ |
1850616074868883456 |
| fulltext |
UDC 539.4
Discrepancy Analysis of Load–Displacement in the Combination Space for
Concrete Box Girder Assessment
X. F. Ye,
a,1
H. Ogai,
a
and C. W. Kim
b
a Faculty of Science and Engineering, Waseda University, Tokyo, Japan
b Graduate School of Engineering, Kyoto University, Kyoto, Japan
1 yexiongfei@gmail.com (Xiong-Fei Ye)
The combination space concept had its origin in the theory of subspace and combinatorics. The
combination space may be treated as the space reconstruction through combination, as well as the
subspace one. Several new analytical methods are proposed based on combination principles applied
to some operations. Under progressing structural damage, simple raw data cannot adequately
characterize continuously changing processes and provide early prevention of considerable damage
or loss of resistance to hazardous failure. Therefore, the data need to be further processed to find
new criteria and factors of damage assessment. Three evaluation methods were experimentally
established with the combination space. For the damage assessment and its early prevention some
factors are also proposed such that the combination space method can find the structural change
earlier than the original data space. In the case of the the disparity between different conditions and
the initial state, this method can contribute to the structure recovery, preventing its failure.
Keywords: combination space, load–displacement, discrepancy analysis, concrete box
girder.
Introduction. The load–displacement (stress-strain) analysis in the research of
structural material is an old topic that through the change of the elastic modulus or
elasticity coefficient, the change of the structure of general dimensional load–displacement
will be intuitively demonstrated [1]; even in the fractal dimensional load–displacement, it
can also be known [2]. In the view of damage mechanics [3, 4], the change of the structure
has the growth of microcracks and cavities will lead the system to be dangerous [5]. The
damage happened in the structure is because of the static-dynamic loading, heating/
temperature-changing, rust/corrosion, etc. [6].
On the other hand, for the existence of damage, the resistance of the structure will go
down which can be an indicator in damage evaluation, this kind of research in the view of
fixed relationship is named as the structural stress analysis [7], which can be modeled by
the finite element method (FEM) for a complex system to find and evaluate the damage,
and this kind of research in the view of probability is often named as the reliability research
of structure [8]. In the evaluation of the local damages and the global damage, the local
damages will influence the global damage, and often strength the damage of global. The
investigation of local stiffness degradation can be used to detect the global damage [9].
Through FEM method, such kind of relationship will be more convenient to express [10].
Then, the damage development in the structural lifetime is from micro to macro, and
then destroy the structure (it is an ideal situation, but it need be avoided). The displacement
averaging can be used to show the change in this development [11]. In other words, this
method uses the relationship of local change and global change, through the displacement
averaging of a continuous system, the global damage is described, but it cannot be used for
local damage detection. Some case concentrates the damage detection of the local damage
[10] using discrete elements, but it is not proper for the global damage detection. For a
more comprehensive understanding of the damage development in the structure, there is a
© X. F. YE, H. OGAI, C. W. KIM, 2018
14 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5
need to development of local-global analysis. To propose a method for both local damage
and global damage, a combination method will be used. This method is based on the
combinatorics [12] and combinational analysis [13].
Since there is a notation for the combinational elements in this research is the
subspace [14], through the combination of these spaces and subspaces, there is a chance to
present new information for the structural damage detection. In this research, three kinds of
combination space are constructed which shows the structural damage detection based on
the combination space is efficient and sensitive. In the data analysis for experiments using
combination-space-based methods, the curves compared with the curve of original data will
also be obtained.
1. The Basic Concept and Research Method and Research Target. In the initial
research, the combination space is just the linear combination space.
For a series, a vector or a series of vectors whose data describes/measures the same
thing:
x x x x x x x� �{ , , , ... , , ... , } { },1 2 3 s n s
in which s i j k� { , , , ...}, and i j k n, , , ... { , , , ... , },� 1 2 3
x xnew � operation ( ). (1)
For example, using the combinatorics, the operation “�”, and its inverse operation
“�”, the original information can be transferred into the new series,
x x x x x x x x x xnew i j k i j i i i� � � � � � � �... , { }, { }, ( }, { }, { }, {0 j i j k}, { },...x x x� �
as well as xnew � { } (the empty set). So, there are 1 2 2 3
� � � �( ... )n n n kinds of
combinations. If the combination is limited by m M� , there is
S
n n
n
n
M n
n
M
�
�
�
�
�
� �
�
�
�
�
�
1 2
1
1
1 2 1
1
, ,
, .
For the same operation “�” in the combination space, there are
(i) for any two elements in combination space, the result of the operation of the two
elements is also in this combination space;
(ii) for any three elements, they meet the law of associativity;
(iii) there is an element { } for any other element having an operation with this
element, the result will be itself;
(iv) for any element, there exists another element that when having an operation with
this element, the result is { }.0
So, it is a group. If it meets the law of commutation, it will be an Abel group.
In the damage detection for a concrete structure, three kinds of combination space will
be used in this paper, i.e., combination structural information, information of difference/
similarity, and denominator information.
1) The combined structural information.
This kind of information will be obtained from x x x x x x� { , , , ... , , ... , }1 2 3 i n that
some data will be selected to describe some characteristic that
x x x x x xj j j j i j n j� { , , , ... , , ... , }., , , , ,1 2 3 (2)
For example, x xi j i, max( ).�
Discrepancy Analysis of Load–Displacement ...
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5 15
2) The information of difference/similarity.
For x x x x x x� { , , , ... , , ... , },1 2 3 i n there is a difference/similarity matrix:
� � � �
�
�
���
�
�
�
( , ) ( , ) ( , ) ( , )
, )(
1 1 1 2 1 1
1
� �j n
j � �
�
�
� ���
�
n
i i i j i n
n
)
( , ) ( , ) ( , )
,
� � � � � �
� �
� � � � � �
2
1 2) ( , ) ( , ) ( , )
( , )[ ],
n n j n n
i j
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
� i j n, , , ... , .� 1 2 (3)
3) The denominator information.
For x x x x x x� { , , , ... , , ... , },1 2 3 i n this kind of information often refers to not only
some operations,
x x xnew f� �operations ( ) ( ), (4)
but also some transforms after the operation,
y x xnew new newg� �transforms ( ) ( ), (5)
like the Fourier, Hilbert, and Laplace transforms. So, for the second case, after the
transform, there is
y x x x x x xnew i i j i j kf f f� � � �{ ( )}, { ( )}, { ( )}, ... . (6)
2. Experiment, Analysis, and Result for Artificial Damage on One Box Girder.
The experiment is to investigate the structural change when there is artificial damage, by
the discrepancy analysis of different loading steps. This experiment object is near all-scale
pre-stressed reinforced concrete open-section box girder (in short, box girder) in Public
Works Research Institute (Fig. 1), with 8500 mm length, 2300 mm width, and 1000 mm
height. On this girder, there are 4 SDP-200 of displacement meters, 8 SDP-100 of
displacement meters (Table 1).
In the experimental process (Fig. 2), there are 384 steps in total (Fig. 3), and the
sensors D3_F, D3_B, D4_F, D4_B can only test the step 1–360. For the data analysis, the
data includes the array and matrix in three kinds of combinations. For the original data,
using the style of the array, for every test step, some results like the average of the data, the
root-mean-square will be obtained. The first category of the information is the combined
structural information (Fig. 4).
X. F. Ye, H. Ogai, and C. W. Kim
16 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5
T a b l e 1
The Introduction of the Sensors
Sensor Type Number Purpose Direction and location
Displacement
meter
200 mm 4 For deflection Vertical Under the girder
100 mm 8 For deflection Vertical Both under and
up the girder
25 mm 4 Subsidence
at the fulcrum
Vertical Fulcrum of the girder
From the loading process in Fig. 2 and data processing in Fig. 4 both sub-figure (1) of
the differential derivations, and sub-figure (2) of the root mean square of the differential
derivations that near the step 350, there is a significant change, which can be used to
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5 17
Discrepancy Analysis of Load–Displacement ...
Fig. 1. The tested object, D1–D6 specify locations, while F and B mean “Front” and “Back”,
respectively.
Fig. 2 Fig. 3
Fig. 2. The loading process of this experiment.
Fig. 3. The experiment process of displacement of all meters from step 1–384.
Fig. 4. (1) The result of differential derivations of every meter, according to Appendix 1; (2) the
root-mean-square (all meters from step 1–360), according to Appendix 2.
indicate the most significant system change that there is a substantial damage in the
structure.
For the second category of combination space of the information of difference/
similarity, compared with the original displacement, there are two kinds of combination
spaces.
Even though Figs. 2 and 3 do not have the same tendency for all meters. But after the
data processing, both shapes of figures (Fig. 5) of difference matrices’ norm and distance
are in the same trend with the loading process in Fig. 2 which can be used to directly show
the system change evaluated by the difference matrix and the loading has the linear
relationship.
For the third category, here is an example using Andrews plot/curve [16] for [ , ]0 1 in
Fig. 6, which is used to visualize high-dimensional data. Suppose x x x x xi� { , , , ... , ,1 2 3
... , },xn the Andrews plot defines a finite Fourier series:
f t
x
x t x t x t x tx ( ) sin( ) cos( ) sin( ) cos( ) .� � � � � �
1
2 3 4 5
2
2 4 5 .. . (7)
18 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5
X. F. Ye, H. Ogai, and C. W. Kim
Fig. 5. (1) Norm of difference matrix, according to Appendix 3; (2) distance of difference matrices
between considered and initial state (all meters from step 1–360), according to Appendix 4.
Fig. 6. Divide all 360 step into nine groups, from top to bottom, from left to right: step 1–40, step
41–80, step 81–120, step 121–160, step 161–200, step 201–240, step 241–280, step 281–320, and
step 321–360.
From Fig. 6, one can see that in the statistics, from step 1 to step 360, the average of
the f t( ) in the Andrews plot becomes greater in the first maximum. When there is some
high damage, the extent of change for the Andrews plot will be greater. For more
information, all figures have the same tendency which shows the data relationship between
all meters are keep the same from the beginning to the end (from step 1 to step 360). For
more discussion, here is a comparison between combination space and the subspace.
In contrast to the concept of combination space, there is a notion of subspace. Here is
the definition of linear subspace [16]. Let K be a field (such as the real numbers), V be a
vector space over K , and let W be a subset of V. Then W is a subspace if (1). The zero
vector, 0, is in W. If u and v are elements of W, then the sum u v� is an element of W;
(3) If u is an element of W and c is a scalar from K , then the scalar product cu is an
element of W. In some sense, the combination space can also be treated as the combination
of the subspaces since the general linear space can be interpreted as the union of all
subspaces.
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 5 19
Discrepancy Analysis of Load–Displacement ...
Fig. 7. The frequency using FFT for both general data space and combination space: (1) original data
space in the form of multi-time series (left) vs combination space in the form of a sum of all time
series (right); (2) the frequency obtained by FFT with original data space in the form of original
multi-time series (left) and combination space in the form of a sum of all time series (right)
separately.
Moreover, in other application, for instance, the Fourier transform (fast Fourier
transform, FFT) for records of acceleration sensors on the same structure as the response to
the impulse action in Fig. 7. Even though some information, like some modes in stochastic
subspace identification, will miss, but the stationary natural frequency of structure will be
found.
Conclusions. The combination space can be treated as the information mining from
kinds of general investigation, through the analysis of the relationship of those data to find
some new information, from difference/similarity among kinds of general investigation,
characteristics reconstruction, or transformation. In the linear space, the combination space
can be regarded as the combination of not only the general space but also the combination
of subspace through some operation and then its inverse operation.
By considering the structural damage and the combination space together, the data
analysis of load–displacement can be investigated in 3 kinds as well, the difference
matrices for the data of different locations, the average of data measured in different
locations, and the transform of the data through Andrews plot. From the experiment result,
through these combinations, it adequately reflects the continuous changing of the system
and the early-warning for significant damage.
Appendix 1.
In each column, get new time series, x x x x xi j i j i j i j i j, , , , ,( ) ( ),� � �
� �1 1 the result
is Eq. (1), x x x x x xj j j j i j n j� { , , , ... , , ... , }, , , , ,1 2 3 in each column, then get a new
multidimensional time series.
Appendix 2.
According to the x j , then calculate the mean value in each row. Then, there is a
new column:
x x x x x xi i n� { , , , ... , , ... , },1 2 3 x xi i j� mean( ),, j� 1 2 3, , , ... .
Appendix 3.
According to Eq. (2),
� �[ ], , , , ... , ,( , )i j i j n1 2 and
( , ) ,i j i jx x� �
�
2
� max
*( ),
where
* is the conjugate transponce of
.
Appendix 4.
To calculate the distance (D) between two status matrices (initial status and considered
status), the method used in this paper is
D i j
j
i j
j
� �
�
�
�
�
�
�
�
�
� �
� �
( , ) ( , )initial considered
1 1
2
( , ) ( , ) .i j
i
i j
j
initial considered�
�
�
�
�
�
�
�
�
� �
� �
1 1
2
4
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Received 05. 03. 2018
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