Research on safety assessment method for bridge structure based on variable weight synthesis method

==Summary==

Variable weight synthesis method is a safety assessment method which is using the varied weight principle, based on analytic hierarchy process and combining the theory of grey relevancy, it reduce influence that secondary cause is not obvious in assessment process, and make the result more correct and dependable. In the paper, variable weight principle and the assessment method were expounded in detail.

==Keywords==

Variable weight synthesis method ; Bridge structure ; Safety assessment

==Variable weight principle and variable weights method==

===Variable weight principle===

In the literature ([[#bib0005|Bai, 2007]]  and [[#bib0025|Liu et al., 2004]] ), variable weight synthesis method is the definition of:
* Assume that a m-dimensional vector, <math display="inline">W^0=(W_1^0,W_2^0,\ldots ,W_m^0)</math> , if present <math display="inline">W_j^0\in (0,1)</math>  makes <math display="inline">\sum _{j=1}^mW_J^0=1</math>  correct for arbitrary variables ''j''  ∈ {1, 2, …, ''m'' };                          
* Given a ''m'' –''d''  vector of parts keep weight ''W'' (''X'' )Δ[''W''<sub>1</sub> (''X'' )2''W''<sub>1</sub> (''X'' ), …, ''W''<sub>''m''</sub> (''X'' )], for given mapping: ''W''  : [0, 1]<sup>''m''</sup>  → (0, 1], it makes that:
* <math display="inline">\sum _{j=1}^mW_j(X)=1</math>
* Assume that ''α''<sub>''j''</sub> , ''β''<sub>''j''</sub>  ∈ [0, 1], and ''α''<sub>''j''</sub>  ≤ ''β''<sub>''j''</sub> , for arbitrary variables ''j''  ∈ {1, 2, …, ''m'' }, present ''W''<sub>''j''</sub>  in regard to ''x''<sub>''j''</sub>  in the internal [0, ''α''<sub>''j''</sub> ] and [''β''<sub>''j''</sub> , 1] respectively decrease and increase progressively.                                      
* Assume that the vector ''S'' (''X'' )Δ[''S''<sub>1</sub> (''X'' ), ''S''<sub>2</sub> (''X'' ), …, ''S''<sub>''m''</sub> (''X'' )] is a ''m'' –''d''  vector of parts variable weight, for the given mapping ''S''  : [0, 1]<sup>''m''</sup>  → (0, + ∞)<sup>''m''</sup> , for arbitrary variables ''j''  ∈ {1, 2, …, ''m'' }, assume ''α''<sub>''j''</sub> , ''β''<sub>''j''</sub>  ∈ [0, 1], and ''α''<sub>''j''</sub>  ≤ ''β''<sub>''j''</sub>  present that:
* For arbitrary variables ''j''  ∈ {1, 2, …, ''m  '' } for fixed weights vector <math display="inline">W^0=(W_1^0,W_2^0,\ldots ,W_m^0)</math> , <math display="inline">W_j(X)=((W_j^0S_j(X))/(\sum _{k=1}^mW_k^{(0)}S_k(X)))</math>  in the internal [0, ''α''<sub>''j''</sub> ] and [''β''<sub>''j''</sub> , 1] respectively decrease and increase progressively.                                                  
* If 0 ≤ ''x''<sub>''i''</sub>  ≤ ''x''<sub>''k''</sub>  ≤ ''α''<sub>''i''</sub>  ∧ ''α''<sub>''k''</sub> , ''S''<sub>''i''</sub> (''x'' ) ≥ ''S''<sub>''k''</sub> (''x'' ), and ''β''<sub>''i''</sub>  ∨ ''β''<sub>''k''</sub>  ≤ ''x''<sub>''i''</sub>  ≤ ''x''<sub>''k''</sub>  ≤ 1, then

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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| 
{| style="text-align: center; margin:auto;" 
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| <math>\frac{w^0\cdot S(x)}{\sum _{j=1}^mw_j^0S_j(x)}\triangleq \left\{\frac{w_1^0\cdot S_1(x)}{\sum _{j=1}^mw_j^0S_j(x)},\frac{w_2^0\cdot S_2(x)}{\sum _{j=1}^mw_j^0S_j(x)},\cdots ,\frac{w_m^0\cdot S_m(x)}{\sum _{j=1}^mw_j^0S_j(x)}\right\}</math>
|}
| style="text-align: right;" | ( 1)
|}

be called ''m'' –''d''  vector of parts variable weight。                                          

===Amendment of variable weights method===

In the variable weights method, when synthesis assessment the assessed value of hundred-mark system, it will result data with denominator larger, and produce larger round-off error. For convenience of calculations and considering the required precision, to transform hundred-mark system into percentage of system, and then, by adjusting coefficients in the formulas, getting the synthesis assessed value of hundred-mark system. in addition, for some sequences of inspection data, it is impossible to collect all the data, but it is needed to compare the curve of sequences of inspection data, so in this paper, considering base on the grey theory related principle to offset the malconformation and incomprehensive of data ([[#bib0015|Lan and Shi, 2001]] ). The paper makes the improvement.

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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| 
{| style="text-align: center; margin:auto;" 
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| <math>V=\sum_{j=1}^m100w_j(x_1,\ldots ,x_m,w_1^{(0)},\ldots ,w_m^{(0)})x_j</math>
|}
| style="text-align: right;" | ( 2)
|}

In formula: ''V  ''  is the assessed value; <math display="inline">w_j^{(0)}</math>  is initial weight value of the ''j  '' th indicators, and <math display="inline">\sum _{j=1}^nw_j^{(0)}=1</math> ; <math display="inline">w_j</math>  is weight value after variable weight; ''x''<sub>''j''</sub>  is the percentage of assessment value of the ''j'' th indicators, and

<span id='eq0015'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\begin{array}{l}\displaystyle w_j(x_1,\ldots,x_n,w_1^{(0)},\ldots\\\displaystyle w_n^{(0)})=\frac{w_j^{(0)}((\partial B(x_1,\ldots,x_n))/\partial x_j)}{\sum _{k=1}^nw_k^{(0)}((\partial B(x_1,\ldots,x_n))/\partial x_k)}\end{array}</math>
|}
| style="text-align: right;" | ( 3)
|}

In formula, ''B'' (''x''<sub>1</sub> , …, ''x''<sub>''n''</sub> ) is balanced function, according to experience of engineering practice, the balanced function is:

<span id='eq0020'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>B(x_1,\ldots ,x_n)=\sum_{j=1}^nx_j^{\alpha }\mbox{ }(0<\alpha \leq 1)</math>
|}
| style="text-align: right;" | ( 4)
|}

simultaneous Eqs.  [[#eq0015|(3)]]  and [[#eq0020|(4)]] , it can get:

<span id='eq0025'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>w_j(x_1,\ldots ,x_n,w_1^{(0)},\ldots ,w_n^{(0)})=\frac{w_j^{(0)}x_j^{\alpha -1}}{\sum _{k=1}^nw_k^{(0)}x_k^{\alpha -1}}</math>
|}
| style="text-align: right;" | ( 5)
|}

so

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>V=\sum_{j=1}^m\frac{100w_j^{(0)}x_j^{\alpha }}{\sum _{k=1}^nw_k^{(0)}x_k^{\alpha -1}}</math>
|}
| style="text-align: right;" | ( 6)
|}

''α'' : balance coefficient, according to experience of engineering practice, ''α''  = 0.2, it can meet most of engineering situation.      

The inspection data of bridge can be divided into three categories: the first kind is non-numeric data with describing the state of bridge components only or dividing grade simply; the second is a single numerical; and the third is a data series. It is needed that applies dimensionless method to the third kind data, compare and analyse the two curves of data series. Introduce grey correlativity analysis and calculate non-uniform variation coefficient to get the assessment result, the formula as follows:

<span id='eq0035'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>V=\sum_{j=1}^m\frac{100w_j^{(0)}x_j^{\alpha }}{\sum _{k=1}^nw_k^{(0)}x_k^{\alpha -1}}r(x_0,x_i)</math>
|}
| style="text-align: right;" | ( 7)
|}

And ''r'' (''x''<sub>0</sub> , ''x''<sub>''i''</sub> ) calculate according to the formula as below  [[#eq0025|(5)]]  and [[#eq0040|(8)]] :

<span id='eq0040'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\begin{array}{l}\displaystyle r(X_0\\\displaystyle X_i)=\frac{1}{n-1}\sum _{k=1}^{n-1}\left[1+\left|\frac{a^{(1)}(x_0(k+1))}{x_0(k+1)}-\right.\right.\\\left.\left.\displaystyle -\frac{a^{(1)}(x_i(k+1))}{x_i(k+1)}\right|\right]^{-1}\end{array}</math>
|}
| style="text-align: right;" | ( 8)
|}

==Advantage of variable weight synthesis method==

Compared with constant weight synthesis method, variable weight synthesis method can make the influence of components with damage badly more prominent. In structure, maybe individual component damage can be lead to the bridge operational be limited, even impact to the safety of the bridge ([[#bib0010|Gao and Wang, 2000]] , [[#bib0015|Lan and Shi, 2001]] , [[#bib0030|Wu and Xiang, 1995]] , [[#bib0035|Xiang and Wu, 2005]]  and [[#bib0040|Zhang and Feng, 2001]] ). Comparison with the result of constant weight synthesis method and variable weight synthesis method is given.      

Using the data in [[#tbl0005|Table 2.1]]  and [[#tbl0010|Table 2.2]] , assessment of the health of bridge with variable weight synthesis method and constant weight synthesis method is shown.
* Constant weight synthesis method:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\begin{array}{l}\displaystyle BCI_2=80\times 0.31+80\times 0.13+75\times 0.14+80\times 0.37+5\times 0.04=\\\displaystyle =75.51>6\end{array}</math>
|}
| style="text-align: right;" | 
|}

The conclusion is “qualified”;
* Variable weight synthesis method (''α''  = 0.2):

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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| 
{| style="text-align: center; margin:auto;" 
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| <math>\begin{array}{l}\displaystyle \sum _{k=1}^mw_k^{(0)}x^{\alpha -1}=\sum _{k=1}^9w_k^{(0)}x_k^{-0.8}=\\\displaystyle =0.31\times 0.80^{-0.8}+0.13\times 0.80^{-0.8}+0.14\times 0.75^{-0.8}+\\\displaystyle +0.37\times 0.80^{-0.8}+0.04\times 0.05^{-0.8}=1.5840\end{array}</math>
|}
| style="text-align: right;" | 
|}

score ''V''<sub>2</sub> :

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>\begin{array}{l}\displaystyle V_2=\frac{100w_j^{(0)}x_j^{\alpha }}{\sum _{k=1}^nw_j^{(0)}x_k^{\alpha -1}}=\frac{100}{1.5840}(0.2965+0.1243+0.1322+\\\displaystyle +0.3539+0.0220)=58.64\end{array}</math>
|}
| style="text-align: right;" | 
|}

The conclusion is “disqualified”.

<span id='tbl0005'></span>

{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+

Table 2.1.

Assess table of a bridge.

|-

! Indicator
! Weight
! Technical status
! Explain
|-

! Pier and abutment
| 0.31
| 80
| Better
|-

! Girder
| 0.13
| 80
| Better
|-

! Pylon
| 0.14
| 75
| Normal
|-

! Stay cable
| 0.37
| 80
| Better
|-

! Subsidiary facilities
| 0.04
| 5
| Failure badly
|}

<span id='tbl0010'></span>

{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+

Table 2.2.

Consultable table of equal coefficient.

|-

! Equal coefficient
! ''α''  = 0                                                    
! ''α''  = 0.2                                                    
! ''α''  = 0.5                                                    
! ''α''  = 0.8                                                    
! ''α''  = 1.0                                                    
|-

! Score of girder
| 49.52
| 58.64
| 68.59
| 74.19
| 75.51
|}

Explain: The value of ''α''  reflects tolerance of local damage, with the increase of ''α'' , tolerance increase.      

By the above examples, it can be got that when the subsidiary facilities damage badly, the result of constant weight synthesis method is “qualified” while variable weight synthesis method is “disqualified”. It is show that variable weight synthesis method can well reflect the local defect and considering the malconformation into assessment. It is more tallies with the actual situation.

==Application example of method==

The inspection data of line shape control about abridge is in [[#tbl0015|Table 3.1]] .

<span id='tbl0015'></span>

{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+

Table 3.1.

Level evaluation of reference point.

|-

! Coordinate (''m'' )                                                    
! Designed elevation (''X''<sub>0</sub> )                                                    
! Measured elevation (''X''<sub>''i''</sub> )                                                    
! Error (''m'' )                                                    
! Assessed value (''D''<sub>''i''</sub> )                                                    
|-

| 0
| 86.550
| 86.538
| 0.012
| 90
|-

| 20
| 86.844
| 86.830
| 0.014
| 90
|-

| 40
| 87.079
| 87.056
| 0.023
| 80
|-

| 60
| 87.247
| 87.235
| 0.012
| 90
|-

| 80
| 87.432
| 87.419
| 0.013
| 90
|-

| 100
| 87.534
| 87.520
| 0.014
| 90
|-

| 120
| 87.541
| 87.517
| 0.024
| 80
|-

| 140
| 87.531
| 87.508
| 0.023
| 80
|-

| 160
| 87.371
| 87.357
| 0.014
| 90
|-

| 180
| 87.168
| 87.153
| 0.015
| 90
|-

| 200
| 86.993
| 86.977
| 0.016
| 90
|-

| 220
| 86.774
| 86.752
| 0.022
| 80
|-

| 240
| 86.537
| 86.526
| 0.011
| 90
|}

According to formula [[#eq0025|(5)]] , calculate variation of weight shown in [[#fig0005|Fig. 3.1]] .

<span id='fig0005'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_200301432-1-s2.0-S2213020915000786-gr1.jpg|center|362px|Variational weight in observation point of girder line.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 3.1.

Variational weight in observation point of girder line.

</span>
|}

According to formula [[#eq0035|(7)]]  and [[#eq0040|(8)]]  calculate as:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
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{| style="text-align: center; margin:auto;" 
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| <math>V=\sum_{j=1}^m\frac{100w_j^{(0)}x_j^{\alpha }}{\sum _{k=1}^nw_k^{(0)}x_k^{\alpha -1}}r(x_0,x_i)=84.14</math>
|}
| style="text-align: right;" | 
|}

''x''<sub>''k''</sub>  = ''D''<sub>''i''</sub> /100, ''D''<sub>''i''</sub> : assessed value. <math display="inline">w^{(0)}=1/n</math>  (''n''  = 13) initial weight value, see [[#tbl0015|Table 3.1]] .

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle r(X_0\\\displaystyle X_i)=\frac{1}{n-1}\sum _{k=1}^{n-1}\left[1+\left|\frac{a^{(1)}(x_0(k+1))}{x_0(k+1)}-\right.\right.\\\left.\left.\displaystyle -\frac{a^{(1)}(x_i(k+1))}{x_i(k+1)}\right|\right]^{-1}=0.968\end{array}</math>
|}
| style="text-align: right;" | 
|}
* ''X''<sub>0</sub>  = [''X''<sub>0</sub> (1), ''X''<sub>0</sub> (2), …, ''X''<sub>0</sub> (13)] designed elevation;                          
* ''X''<sub>''i''</sub>  = [''X''<sub>''i''</sub> (1), ''X''<sub>''i''</sub> (2), …, ''X''<sub>''i''</sub> (13)] measured elevation.                  

Assessment score of girder alignment is 84.14, contrast [[#tbl0020|Table 3.2]] , the bridge can be classified grade 1.

<span id='tbl0020'></span>

{| class="wikitable" style="min-width: 60%;margin-left: auto; margin-right: auto;"
|+

Table 3.2.

The reference standard of bridge technique condition evaluation.

|-

! Score
! Grade
! Explain
|-

! 80 < ''V''  ≤ 100                                                    
| Grade 1
| Well, only daily maintenance
|-

! 60 < ''V''  ≤ 80                                                    
| Grade 2
| Normal, daily maintenance and minor repair
|-

! 40 < ''V''  ≤ 60                                                    
| Grade 3
| Weak, repair and load test
|-

! 20 < ''V''  ≤ 40                                                    
| Grade 4
| Bad, Stop running and strengthen
|-

! 0 < ''V''  ≤ 20                                                    
| Grade 5
| Danger, considering dismantle and rebuild
|}

''Note'' : The evaluation standard is made according to “Evaluation procedure for bearing capacity of highway bridge”.

==Conclusion and prospect==

In the safety process of assessment, the method, by using the variable weight principle to assess actual state of the assessment indexes, extrude the influence of the parts of damaged badly, especially the effect of secondary cause. And on this basis, apply method of grey relational to calculate the correlation degree of sequence data, it make up malconformation and incomplete collection of the test data.

The article explains the variable weight synthesis method, and applies the method into the process of bridge assessment. Though it has the rationality of the method, but it still exist some problem. First, although the assessment applies new method, but most of the index still reference codes and specifications and previous research results, it needs to do a more in-depth discussion. Second, the initial weight value used in the paper, it is still the average weight which has certain blindness, it needs to do a more in-depth discussion, too.

==Conflict of interest==

The author declares that there is no conflict of interest.

==References==

<ol style='list-style-type: none;margin-left: 0px;'><li><span id='bib0005'></span>
[[#bib0005|Bai, 2007]] Y. Bai; The application of variable weight synthesis method in estimate of real estate investment environment; J. Adult Educ. Coll. Hebei Univ., 9 (3) (2007)</li>
<li><span id='bib0010'></span>
[[#bib0010|Gao and Wang, 2000]] H. Gao, J. Wang; A review of research and application of detective and evaluation technology for bridges; World Inf. Earthq. Eng., 16 (2) (2000)</li>
<li><span id='bib0015'></span>
[[#bib0015|Lan and Shi, 2001]] H. Lan, J. Shi; Degree of grey incidence and variable weight synthesizing applied in bridge assessment; J. Tongji Univ., 29 (l) (2001)</li>
<li><span id='bib0025'></span>
[[#bib0025|Liu et al., 2004]] S. Liu, Y. Dang, Z. Fang; The Grey System Theory and Application; Science publishing house, Beijing (2004)</li>
<li><span id='bib0030'></span>
[[#bib0030|Wu and Xiang, 1995]] H. Wu, Y. Xiang; Assessment the tension and the linetype state cable 2 stayed bridge based on variable weight synthesis; China Rail. Sci., 4 (1995), p. 16</li>
<li><span id='bib0035'></span>
[[#bib0035|Xiang and Wu, 2005]] Y. Xiang, H. Wu; Research on condition assessment system of long-span prestressed concrete cable-stayed bridge; J. China Foreign Highw., 25 (5) (2005)</li>
<li><span id='bib0040'></span>
[[#bib0040|Zhang and Feng, 2001]] Y. Zhang, Z. Feng; To evaluate the bridge safety by AHP; J. Xi’an Highway Univ., 21 (3) (2001)</li>
</ol>