The Application of the Principle for Synthesizing Measurements of Multiple System Parameters into A Single Parameter toOptimization
Analysis of Bracing Systems for Deep Foundation Pits
Ying Liao1 Qiangguo Pu2 Nikos Mastorakis3
1Department of Civil Engineering,2Computer Center, University of Science and Technology of Suzhou,298 Binhe Road,Suzhou, Jiangsu,215011,China
3Hellenic Naval Academy, Terma Hatzikyriakou, 18539 Piraeus, Greece
Also: WSEAS, Ag.I.Theologou 17-23, 15773, Zografou, Athens, Greece
Abstract: - The optimization analysis of bracing systems for deep foundation pits is a rather complex problem of system engineering which relates to many indexes belonging to safety and feasibility, economy and rationality, environmental protection and convenience of construction. In this paper, the evaluation index system of bracing systems for deep foundation pits is established; every index’s overall sorting weight is determined; the comprehensive evaluation results are obtained to evaluate the degree of superiority and inferiority of bracing schemes by means of the system parameter fusion principle; and then the optimum scheme is concluded.
Keywords: - deep foundation pits, bracing systems, parameter fusion, optimization analysis, system parameter fusion principle
- Introduction
The bracing systems for deep foundation pits is a complex problem of Geo-technical Engineering, the optimization analysis of it relates to four essential criteria, i.e. safety and feasibility, economy and rationality, environmental protection and convenience of construction. Inside every essential criterion, there are both certain indexes such as the project’s general construction cost and uncertain indexes such as the scientific and advanced nature of the scheme, the influence of construction on the environment, etc. The traditional methods of evaluating the bracing systems for deep foundation pits are qualitative methods, e.g. effort inquiry method, weighted average method, etc. Because those methods involve many subjective factors, they can’t scientifically and objectively reflect the degree of superiority and inferiority of the bracing systems for deep foundation pits. This paper selects 12 relevant indexes relative to the four essential criteria to establish the evaluation index system of bracing systems for deep foundation pits. On the basis of the determination of every index’s overall sorting weight, the system parameter fusion principle is applied to synthesize every certain and uncertain index to a simple quantitative result—the comprehensive evaluation result. By means of the simple quantitative result we can judge the degree of superiority and inferiority of the bracing scheme for deep foundation pits and then the optimum scheme is obtained from the bracing schemes that were proposed.
- The evaluation index system and the determination of index’s overall sorting weight of bracing systems for deep foundation pits
2.1The evaluation index system
In order to make the bracing scheme satisfy the four criteria, i.e. safety and feasibility, economy and rationality, environmental protection and convenience of construction, selecting 12 relevant indexes to establish the evaluation index system of bracing systems for deep foundation pits according to Analytical Hierarchy Process (AHP)[1] shown as figure 1.
2.2The determination of index’s overall sorting weight
According to the AHP method, the system analysis of the evaluation index system can be reduced to the problem of determining the overall sorting weight of every index belonging to the lowest hierarchy, i.e. index hierarchy relative to the highest hierarchy, i.e. objective hierarchy. The main calculating steps are as follows.
①．Establishing the judgment matrix
Make the relative importance judgment of each pair of factors belonging to the same hierarchy relative to the relevant element of the upper hierarchy. By introducing
Figure 1. The evaluation index system of bracing systems for deep foundation pits
suitable scales, these judgments can be indicated as values, therefore a judgment matrix is established. In this paper, 1—9 scales method is adopted to establish the judgment matrix for every factor belonging to the same hierarchy.
②．Hierarchy’s single sorting
This paper employs square root method to determine
the single sorting weight of every factor belonging to the same hierarchy relative to the relevant element of the upper hierarchy, this process is called “hierarchy’s single sorting”. The single sorting weight WBk (k=1,2,3,4) of every criterion belonging to criterion hierarchy relative to the upper hierarchy, i.e. objective hierarchy; the single sorting weight WCi (i=1,2,…,12) of every index belonging to index hierarchy relative to the relevant element of the upper hierarchy , i.e. criterion hierarchy is calculated shown in table 1.
③．Hierarchy’s overall sorting
According to the calculating step of hierarchy’s single sorting, after determining the single sorting weight of every factor belonging to the same hierarchy relative to the relevant element of the upper hierarchy, the weight of the relevant element itself is again weighted synthetically, therefore, the overall sorting weight of every index of the lowest hierarchy, i.e. index hierarchy relative to the highest hierarchy, i.e. objective hierarchy is calculated. This overall sorting weight is called “index’s overall sorting weight” and this calculating process from the highest hierarchy to the lowest hierarchy is called “hierarchy’s overall sorting”.
Hierarchy’s overall sorting must pass the consistency check. When the random consistency ratio CR of hierarchy’s overall sorting satisfies formula (1), the results of hierarchy’s overall sorting are said to have satisfied consistency.
(1)
Here, CR is the random consistency ratio, WBk (k=1,2,3,4) is the single sorting weight of every criterion belonging to criterion hierarchy relative to objective hierarchy, CIkandRIk (k=1,2,3,4) expresses respectively the consistencyindex and the average random consistency index of the hierarchy’s single sorting of every index belonging to index hierarchy
relative to the relevant criterion of criterion hierarchy.
The results of hierarchy’s overall sorting and the consistency check are shown in table 1 too. The vector of indexes’ overall sorting weights of the evaluation index system of bracing systems for deep foundation pits can be written as follows.
(2a)
Here, Wi (i=1,2,…,12)is the index’s overall sorting weight of index Ci belonging to the lowest hierarchy, i.e. index hierarchy relative to the highest hierarchy, i.e. objective hierarchy. Then known from table 1,
Table 1. The single sorting weights WBk and WCi, index’s overall sorting weight Wi
and consistency check of hierarchy’s overall sorting
Criterion hierarchyIndex hierarchy / B1
0.351 / B2
0.351 / B3
0.189 / B4
0.109 / Index’s overall
sorting weight
Wi（i=1，…，12） / Consistency check of
hierarchy’s overall sorting
C1 / 0.082 / 0.0288 /
=0.0086 < 0.1
C2 / 0.260 / 0.0913
C3 / 0.260 / 0.0913
C4 / 0.260 / 0.0913
C5 / 0.138 / 0.0484
C6 / 1.000 / 0.351
C7 / 0.297 / 0.0561
C8 / 0.163 / 0.0308
C9 / 0.540 / 0.102
C10 / 0.249 / 0.0271
C11 / 0.157 / 0.0171
C12 / 0.594 / 0.0648
(2b)
- The evaluation and optimization analysis of bracing system for deep foundation pits
3.1Establishing the evaluation matrix
These paper adopts the system parameter fusion principle[2] to build the evaluation matrix R. The method is as follows.
①．Firstly, determine a few experts, e.g. N experts to constitute the expert team for evaluating bracing schemes. Secondly, according to engineering conditions and characteristics of the actual project, the expert team defines the optimum value Oi, the critical values, i.e. the lowest value Li and the highest value Hi of index Ci (i=1,2,…,12)of the evaluation index system of bracing systems for deep foundation pits. Thirdly, in the light of the index’s optimum and critical values, every one of the expert team gives a mark to an uncertain index of No. j bracing scheme’s by adopting centesimal system. This mark can be written as xij, and equation (3) used to treat xij statistically.
(3)
Here, r ij is the evaluating value of an uncertain index Ci of No. j bracing scheme’s given by N experts, p is the average value of all marks to Ci given by N experts, d is Ci’s average value of marks lower than p, g is Ci’s average value of marks higher than p.
The evaluating value r ij calculated by equation (3) not only attaches importance to the average value of all marks given by experts but also gets rid of the extreme values, i.e. the highest value and the lowest value, so it conforms to the actual situation better than that of the general average method. As to the certain index, i.e. C2, C3, C4, C6, C12 of the evaluation index system of bracing systems for deep foundation pits, its evaluating value is the index’s definite value itself of the bracing scheme and doesn’t need the mark given by the expert team.
②．Ascertain the element f(rij) of the evaluation matrix R in accordance with function (4), here f(rij) is called the evaluating function.
(4a)
If Oi–Li = Hi–Oi, then function(4a) becomes the simple style, i.e. the semicircular function shown as function (4b).
(4b)
Figure 2. f(rij) can be the semicircular function of r ij
③．If there are altogether m groups of bracing schemes that were proposed for deep foundation pits, their evaluation matrix R is shown as function (5a).
(5a)
Here, Rj is the evaluation vector of No. j bracing scheme for deep foundation pits (j=1, 2,…,m):
(5b)
Adopting the evaluating function f(rij) as the element of the evaluation matrix R can adequately refer to the index’s optimum value Oi, critical values Li and Hi so as to give a relatively subjective and fair comprehensive evaluation result for the bracing scheme. Furthermore, according to function (4) when rij≤Li or rij≥Hi , the evaluating function f(rij) is given the value 0, that means the unusual circumstances when evaluating value rij equates the lowest value, the highest value or exceeds it’s limits which may result in giving a wrongly comprehensive evaluation result for the bracing scheme can be eliminated , therefore, the actual situation of the foundation pit system can be reflected fully and really.
3.2The comprehensive evaluation results and optimization analysis of bracing systems for deep foundation pits
Comprehensively considering every index’s overall sorting weight and evaluating function, the comprehensive evaluation results for the foundation pit’s bracing schemes are obtained as shown in formula (6).
(6)
Here, Bj (j=1, 2,…,m) is the comprehensive evaluation result for No. j bracing scheme which synthesized 12 indexes. Let every Bj is arranged in dimensions, the greatest Bj is relative to the optimum result which adequately considers every index’s overall sorting weight and the expert team’s evaluating values, so the scheme relates to Bj is the optimum bracing scheme for deep foundation pits.
- Real example
An underground park’s deep foundation pit covers 8100 square meters and its depth is 7.2 meters[3]. To the east, west and south are neighboring streets, to the north is 15 meters from a teaching building. The main technical demands on one hand are controlling the lateral displacements in order to ensure the bracing structures have enough stiffness to resist the deformations arising from ground loads and the pressure of water and soil; strictly controlling stability safety coefficient in order to avoid unstable damages such as drift sand damage, piping damage and so on. On the other hand, it requires advanced and feasible design schemes, cheaper construction, short-term construction and reliable quality of construction.
There are altogether 3 groups of bracing schemes for deep foundation pits comprehensively considering the project’s actual conditions and characteristics. Through consulting and estimating from the expert team (consisting of 8 experts), the feedback information about the optimum value, critical values and evaluation value towards every index of these 3 groups of bracing schemes are obtained, thereby the evaluating functions are calculated adopting function (4). The results of calculations for evaluating value and evaluating function are shown in table 2.
According to function (5b), the evaluating functions f(rij)of the last 3 columns in table 2 constitute respectively the evaluation vector of scheme 1, scheme 2 and scheme 3, i.e. R1, R2, R3. Thus the evaluation matrix R is composed of them in accordance with function (5a):.
Leading functions (2b) and (6) to calculate the comprehensive evaluation result of scheme 1, scheme 2 and scheme 3, the respective result is as follows.
Obviously, B3> B1> B2, therefore, the superior and inferior sorting of the 3 groups of schemes is scheme 3> scheme 1> scheme 2. In the light of the system parameter fusion principle, the chosen scheme, i.e. the optimum scheme should be scheme 3. Combining with the actual evaluation results of the bracing schemes of the project,
Table 2. The evaluating value and evaluating function of the indexes
Criterion hierarchy / Indexes / The optimumvalue
Oi / The critical
value / Evaluation value rij (i=1,2,…,12 ;)
( j=1,2,3) / Evaluation function f(rij) (i=1,2,…,12 )
(j=1,2,3)
The
lowest
value
Li / The
highest
value
Hi / Scheme
1 / Scheme
2 / Scheme
3 / Scheme
1 / Scheme
2 / Scheme
3
B1 / C1 Scientific and advanced nature of the scheme / 0.9 / 0.7 / 1.0 / 0.723 / 0.890 / 0.913 / 0.466 / 0.999 / 0.992
C2 Stability safety coefficient of bracing systems / 2.0 / 1.8 / — / 1.83 / 1.75 / 1.89 / 0.527 / 0.000 / 0.835
C3 The allowable unstable probability of bracing systems meeting the strength requirements / 0.35% / 0.1% / 1% / 0.28% / 0.49% / 0.1% / 0.960 / 0.977 / 0.000
C4 The lateral displacements of bracing systems meeting the deformation requirements（millimeter） / 20 / — / 40 / 26 / 29 / 34 / 0.954 / 0.893 / 0.714
C5Reliability of construction quality / 0.9 / 0.7 / 1.0 / 0.920 / 0.810 / 0.891 / 0.980 / 0.893 / 0.999
B2 / C6 General construction cost（tenthousand RMB yuan） / 200 / — / 300 / 283.7 / 252.3 / 223.9 / 0.547 / 0.852 / 0.971
B3 / C7 Influence of construction on the environment / 0.7 / — / 1.0 / 0.841 / 0.882 / 0.793 / 0.883 / 0.795 / 0.951
C8Influence of project on the environment / 0.7 / — / 1.0 / 0.830 / 0.913 / 0.781 / 0.901 / 0.704 / 0.963
C9 Possibility of generating subsequent disaster by
construction / 0.7 / — / 1.0 / 0.810 / 0.904 / 0.785 / 0.930 / 0.733 / 0.959
B4 / C10 Level of difficulty of construction / 0.7 / — / 1.0 / 0.809 / 0.945 / 0.879 / 0.932 / 0.577 / 0.802
C11 Level of disturbance between the construction and other construction / 0.6 / — / 1.0 / 0.881 / 0.921 / 0.787 / 0.712 / 0.597 / 0.884
C12 Length of time of construction（days） / 35 / — / 70 / 60 / 70 / 40 / 0.700 / 0.000 / 0.990
the actual chosen scheme is scheme 3 too. This result indicates that the optimum scheme analyzed by this paper’s method is the same as the actual project’s chosen scheme.
- Conclusion
This paper establishing the evaluation index system of bracing systems for deep foundation pits by adopting the AHP method can respect all aspects of the actual situation and characteristics of the foundation pit project. Applying the system parameter fusion principle can evaluate bracing schemes subjectively and thoroughly, moreover, it makes possible to make an optimization analysis of bracing schemes for deep foundation pits. The results of this paper show that combining the AHP method with the system parameter fusion principle to evaluate bracing schemes is a good way for scheme optimization and right choice for deep foundation pits.
References
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[2]Qiangguo Pu.The principle forsynthesizing measurement of multiple system parameters into a single parameter. Proceedings of SPIE(The International Society for Optical Engineering), Vol.4731-Sensor Fusion, AeroSense 2002, Orlando,USA,2002, pp.295~301
[3]Xuan Zhang, Peiyin Lv, Determination of supporting and protecting scheme of deep base pit with fuzzy similar precedable ratio. Journal of Liaoning Institute of Technology, Vol. 19(5), Liaoning , China, 1999, pp. 47~52 (in Chinese)