FS3-010 Fuzzy Neural Network Calculating Theory for Saturated Sand Liquefaction

The 24th ICTPA Annual Conference & NACGEA International Symposium on Geo-Trans ISBN 978-0-615-42857-4

Paper No. S3-010 P a g e | 1

Fuzzy Neural Network Calculating Theory for Saturated Sand Liquefaction Potential under Action of Earthquake

Wang Xinghua1, Wang Zhenyu2, Wang, Mian-C3 and Zhou, Hailin4

1Professor, College of Civil and Architectural Engineering, Central South University, Shaoshan Road 22#,Changsha,410075, Hunan, P.R.China; [email protected], [email protected] 2Master, Tunnel Engineering Group, China Communication Construction Group Company, Beijing 100011, China 3Professor, Department of Civil and Environment Engineering, College of Engineering, the Penn State University, University Park, PA, 16802, USA 4Doctor, College of Civil and Architectural Engineering, Central South University, Shaoshan Road 22#,Changsha,410075, Hunan, P.R.China ABSTRACT: Based on the fact that saturated sand may become liquefaction under the dynamic loading and the liquefaction potential has fuzzy characteristics, In the paper, the authors set up a system of fuzzy-neural network by combining fuzzy mathematics with neural network. By means of the fuzzy nerve network, the possibility of saturated sand liquefaction were forecasted, some parameters of the sand, such as: relative density (Dr/%), beating counts of standard penetration test (N63.5/beating), overlying effective stress (σv/kPa), and max value of vibration force (I0), were known as calculation parameters. This system can solve the problems of saturated sand liquefaction potential. The simulation results show the effectiveness of this method. INTRODUCTION Saturated sand liquefaction will harm the constructions on the sand very much. Prognostication of the saturated sand liquefaction is a very important work. After saturated sand liquefaction was researched more than dozens years, there are many methods to forecast saturated sand liquefaction, such as: experience formula method, simply Seed’s analysis, probability and statistic method, dynamical reactivity analysis, etc. (Chen 1994, 1996; Chen and Luo 1997; Chen and Liu 2000; Li 1999; Ye et al. 1997; Zhao and Xu 1997). Because there are many factors affecting saturated sand liquefaction that are complexity and multiformity, and linear relationship between the factors is not very evidence, the reliability of the results is still be improved. In recently years, many researchers have forecasted the possibility of saturated sand liquefaction with some no-linearity theories, such as: Fuzzy mathematics and Grey theory, etc (Chen and Luo 1997; Weng 1993). But the method of fuzzy judge need that the different values of weighting should be given to the selected parameters. The selection of the values of weighting has inevitably some subjectivity and randomness, it should cause distortion of researching results. With the grey theory, when there is more fluctuate in the origin data series and the information spread around too, the precision of forecast will be reduced. Moreover fuzzy logic and neural network are two self-complementary



techniques. They have very strong function of non-linear mapping and adaptive learning. Neural network get some information from the trained and controlled system. And the most typical method used in fuzzy technique is that oral and/or language information is obtained from experts. When the two techniques are combined with, the system can posses the characteristic of neural network, such as: study ability, optimization capacity, and link construction, but also have the virtues of fuzzy system, such as; rule concept of apery if-then and inference mechanism. THE FUZZY CONCEPT OF APPLYING TO SAND LIQUEFACTION Zhao and Xu (1997) put forward some exact process and methods that distinguish sand liquefaction with fuzzy colligate judge, put out a statistical average of liquefaction index data in Tianjin someplace that is 8 degree in earthquake. And liquefaction level of three sand samples in the place were carried through distinguished. Because they used the fuzzy judge method that there was a bug that much information was lost, i.e. taken large or little operators, and assertion of the degree of membership was more subjective in principle, it should be that distinguish conclusion of saturated sand liquefaction was absence persuasion in theory. On the reference (Zhao and Xu, 1997, Chen,K., Liu, X.C. 2000), the paper gave a theory and method of fuzzy identification that was used to forecast degree and level of saturated sand liquefaction under vibration force. Because liquefaction is a fuzzy concept, it is impossible that sand liquefaction is divided exactly. Sand liquefaction under vibration force can be divided four degrees, such as: 1) no-liquefaction; 2) venial liquefaction; 3) middle liquefaction; and 4) deeply liquefaction, according to the extent index of liquefaction (such as: deformation quantity; value of pore water pressure and strength loss) (Jin et al 1992; Weng 1993). The criterion of division is shown in Table 1.

Table 1. Division of liquefaction extent of sand

Rank of liquefaction Displacement extent State of sand

No-liquefaction Little displacement, little affecting coverage

There is little pore water pressure. State of sand is stable. Fluid in sand does not flow basically.

Veniality liquefaction

Some displacement, little affecting coverage

There is little pore water pressure, light softening phenomenon, little fluid flowing.

Middle liquefaction

Large displacement, little affecting coverage

There is some pore water pressure, softening phenomenon, some fluid flowing in sand.

Deeply liquefaction

Large displacement, much affecting coverage

Greater pore water pressure produced in sand, sand in liquid form, much fluid flowing in sand.

It is defined that when strain value of the sand arrives at a standard value, the relative degree of membership of state of sand to deeply liquefaction is 1; that when the value of



strain of sand under cycle force is less than or equal to some standard value, the relative degree of membership of state of sand to no-liquefaction is 1. It can also be defined that the extent value of liquefaction of sand to deeply liquefaction is 1, the extent value of liquefaction of sand to no-liquefaction is 0, and middle liquefaction, and veniality liquefaction are between deeply liquefaction and no-liquefaction. Their values of relative extent of liquefaction are in the span 1 and 0. They can be confirmed by linear interpolation. The 4 degrees of membership function sand from no-liquefaction to deeply liquefaction is able to be confirmed. Also it is the same method that the standard index, such as: relative density (Dr/%), beating counts of standard penetration test (N63.5/beating), overlying effective stress (σv/kPa), and max value of vibration force (I0), are able to be divided fuzzify. For beating counts of standard penetration test, at first, a beating counts of standard penetration test MAXN 5.63 that is considered to be maximal is selected, the index of beating counts of standard penetration test imported after normalization is defined as

MAXNNN

5.63

5.635.63 = , and then it will be fuzzified as 5 degrees between 0~1, 1)very

big, 2)larger, 3)middle, 4)less, 5)very little. General Sigmoid nerve cell can be used as usable membership function (Chen 1993). Such as for on a language real axis expressing variable X that has three fuzzy sets, ‘small(S)’, ‘middle (M)’, ‘large(L)’, the network showed in Fig. 1 may be used. In the Fig. 1, Y1, Y2 and Y3 are separately represented membership function )(xSµ , )(xMµ and )(xlµ .

( ) ( ){ }11

1 expsg c

y xx

µω ω

= =⎡ ⎤+ − +⎣ ⎦

(1)

So the central site and width of Sigmoid function are separately decided by right values wc and wg (Chen 1993, 1995). After the right values are initialized properly, the membership functions of the fuzzy set S, M and L can be arranged as Fig.2. The similar trapezoid membership function )(xMµ is made up of two Sigmoid functions. In Fig. 2 membership functions are separately expressed by 1 2 5, ,y y yL .

It is evident that to define parameters of membership function in real engineering is very difficulty. For getting optimum of parameters and structure of fuzzy inference

FIG. 1 Achieved simple membership function by neural network

FIG. 2 Sigmoid membership function



system, it has to make continually alteration of fuzzy inference system according to the new knowledge get from reality. The modified method is that fuzzy inference system is drove through introducing neural network. ARITHMETIC MODELING OF FUZZY NEURAL NETWORK (SPATIAL STRUCTURE) Neural network is made up of neural cell. The model of neural cell is showed in Fig. 3. The model has multi-import and single export. Internal state of the model is defined by weighted sum of input signal. Its output can be expressed as:

FIG. 3 Model chart of neural cell FIG. 4 B-P network

( ) ( )1

n

i ii

y t f x tω θ=

⎛ ⎞= −⎜ ⎟⎝ ⎠∑ (2)

Where: θ is threshold of neural cell; n is number of imported data; t is the time; ω is weighting coefficients; Y is output function that is two-valued function that its value is usual 1 and 0, or is a continuum, nonlinear Sigmoid function. If there is some part of neural cell that will be corresponded as the membership function in the fuzzy inference system, neural network could be introduced into fuzzy reasoning. It is the process that is built through neural network training that the known input and output are all endued to the input and output of neural cell, and then the parameters in neural cell are ascertained through iterative computation. According to the analysis of fuzzy inference system, the fuzzy inference model driven by neural network is made up of two parts network in space. Before the network is trained, at first, initial construction of network is structured. And then, during study, the ultimate construction is formed through deleting or uniting the some nodes and joins in initial construction. Modeling process of inference system is: Step 1: Choosing input-output variable and trained data； Step 2: Clustering the training data; Step 3: Training NNmem corresponding to IF part of fuzzy inference rule; Step 4: Training network, NNs, corresponding to the then part of the sth rule; Step 5: Expurgating the then part with back-elimination method; Step 6: Deciding final output value. Based on the above modeling thought, Fig 4 shows the structural chart of fuzzy



network. The fuzzy neural network is divided into 5 layers as showed in Fig. 5. The nodes in first layer are input nodes, they accept input language variable. The fifthly layer is output layer. Every output variable has two language nodes. The one as training data (expecting input) is sent to the network, the other one as output of network is decision-making signal (actual output). The nodes on the second and fourth layers are term nodes. They work as membership function. They represent the term of corresponding variable. The node of the second layer can be a simple node. It can complete a simple membership function. Also it can be made up of nodes of many layers. The nodes of the third layer are rule nodes what represent a fuzzy logic rule. The connection and its function between the third layer and fourth layer likes as an inference machine of connection mode. The connection of the third layer defines the reason of rule node. The node of the fourth layer defines the conclusion of rule node. The connection between the second and the first layer is full connection between language node and corresponding term node. Connective arrowhead represents flow direction of normal signal during operation after the network is set up and trained. The signal flow will only reverse during study and training. FIG.6 shows the time division of fuzzy inference system of neural network

ALGORITHM ANALYSIS OF FUZZY-NEURAL NETWORK (TIME DIVISION) Training is made up of two independent phases of a learning strategy. It combined with unsupervised learning and learning process of supervisory gradient degression for setting up rule node and training membership function. The mixed learning algorithm trains data classification through the crossover region of acceptance before learning. The first phase: independent learning phase. Before the network is trained, at first

FIG.5 Space division of fuzzy inference system of neural network

FIG.6 Time division of fuzzy inference system of neural network



initial structure of network is constructed, and then during learning some nodes and connections in the initial structure will be deleted or united for forming the final structure. Also the connection between rule nodes and output term nodes is full connection at the first. It is mean that the conclusion of rule nodes is not still decided. After learning every output variable term can only be selected. The second phase: supervised learning phase. After the rule of fuzzy logic is confirmed, the fuzzy rule of network has been confirmed at the first learning phase or provided by experts, so that whole network structure has been established. When the network comes into the second learning phase, it can best adjust the parameters of input and output membership function using feedforward working. The final purpose of supervised learning is that the error between output and desired output will be least. If the error is found is more than specified value, connection weight value of network will be adjusted using inverse-transmission method. The parameters of membership function can be adjusted effectively. Detail derivation is showed in the reference (Li, 1999). CALCULATING EXAMPLE The information comes from the reference (Weng, 1993). They are obtained through sorting. There are 35 training samples, of which 22 samples of liquefaction, 13 no-liquefaction samples, as well as 4 checking samples. There are many factors of affecting liquefaction of saturated sand. They can be divided into three large parts of condition of soil characters, mode of occurrence, vibration load. It should be thought that the best basic confinement factor of sand liquefaction is self characters of sand. And it should be thought that selection of index should be simplicity, facility and representativity, average grain diameter (d50/mm) (when in fact operation the unit choose µm), relative density (Dr/%), beating counts of standard penetration test (N63.5/beating), overlying effective stress (σv/kPa), and earthquake intensity (I0) of sand are chosen as the calculating index. According to given data, liquefaction of sand is divided into two kinds; liquefaction and no-liquefaction. And 1 and 0 are used as their mark values separate. The Fuzzy (i.e. fuzzy logic tool) in Matlab tool box is used to programming calculate. During calculating, the existing data will be divided into training data and checking data at first. They will be inputted in the form of matrix M×N. The every row of the matrix represents a data. Front (N-1) columns represent affecting factors. The last column represents anticipant output value. The model is established using fuzzy inference system based subtraction kind gather in the tool box. After modeling clustering center of data point (i.e. the rule that data is obeyed to) and degree of membership of some data obeying the rule can be obtained. The Fig.7 and 8 show that 21 rules can be obtained from the subtraction clustering. The membership function of relative density to rule is more decentralization. The membership function of earthquake intensity to rule is mainly centralized on the four curves. The survey data of 34 liquefaction yields in 19 areas are showed in Fig. 7 and Fig. 8. The obtained membership functions of affecting factors obeyed each rule are plotted after modelling. And then the model is established using neural fuzzy system based on



Sugeno fuzzy inference in the tool box. The method that uses reverse propagation arithmetic and least square method completes modeling to input-output data. It has the advantages that the calculating is simple and propitious to mathematical analysis. Variety of root meat square error during fuzzy training is showed in Fig. 9.

The comparison between the inference output value of the system and inference value of actual expectation is showed in Fig. 10. It can be found that the inference result of the inference system is fit with the actual value very well. The analysis of calculating results indicate: the error between the output value of training samples and actual value is very little comparing the actual data and forecasting data in Fig. 10. Forecasting results of 4 samples is fit with actuality (i.e. accuracy rating is 100%). The performance of network generalization is very good. And the conclusion is directviewing. But there is a defect that because BP network uses black box learning

FIG.10 Comparison between output value of system inference and inference value of actual expectation

FIG.9 Change of root-mean-square error during training

FIG. 7 Membership degree of relative density to rule

FIG.8 Membership degree of earthquake to rule



pattern, after completing learning, the obtained relationship of input/output can not use the manner that can be accepted by human easily. CONCLUSION 1) Because high capacity training is used, the error induced by accident can be avoided. When using condition is more complex or it is not judged using single factor easily, using the network can obtain obvious effect. 2) That the data information is used the technique of distributivity storage and treating can avoid the mistake that is leaded to by individual cell breakage. So the ability of fault-tolerance of the system that is used to forecast the sand liquefaction is very strong, and its reliability is high. It can become an effect method of forecasting of saturated sand earthquake liquefaction. ACKNOWLEDGMENTS The authors appreciate the finance support of the China National Nature Science Fund (No. 59979001). REFERENCES Chen, S.Y. (1993). “Theory model of fuzzy optimal classify and its application in assess

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FS3-010 Fuzzy Neural Network Calculating Theory for Saturated Sand Liquefaction

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