A method of sand liquefaction probabilistic estimation based on RBF neural network model.

Guoxing, Chen ; Fangming, Li

Based on the 344 liquefaction site data during 25 strong earthquakes in the world, through training and testing the neural network model of the Radial Basis Function, the nonlinear relation between corrected blow count of standard penetration test N1 and cyclic resistance ratio of saturated sand soils CRR is analyzed, and also the empirical equation CR[R.sub.cri] of the liquefaction limit state curve or critical cyclic resistance ratio curve of saturated sand soils is constructed. By the statistic analysis, the probability density functions of liquefaction and non-liquefaction as well as empirical equation between the safety factor and liquefaction probability of saturated sand soils are given, then the empirical equation of the cyclic resistance ratio of saturated sand CRR under the different probability is deduced. The method used in this paper makes the sand liquefaction probabilistic estimation on engineering sites is as easy as the traditional deterministic method of sand liquefaction estimation.

INTRODUCTION

Earthquake-induced saturated sand liquefaction is one of important causes for making ground failure and building damage. Thus, many methods of soil liquefaction estimation are developed. The SPT method is comparatively perfect and universally accepted by the engineering domain in the world. In consistent with the reliability theory of superstructure design, the evaluation of soil liquefaction should also adopt the probability method, which gives the soil liquefaction estimation result under the different probability. Based on the 344 liquefaction site data during 25 strong earthquakes, with Radial Basis Function (RBF) neural network model, the cyclic resistance ratio CRR curve of saturated sand under the different probability is presented.

SAND LIQUEFACTION LIMIT STATE FUNCTION BASED ON THE RBF NEURAL NETWORK MODEL

Design of the Neural Network Model

RBF neural network is a multi-nonlinear dynamic system with good self-adaptive, self-systematical and finer ability of learning, association, compatibility and anti-jamming. It can expediently construct the model for complicated and unknown systems, thereby realizing the self-estimation of sand liquefaction potential under the different influencing factors.

RBF network model comprises three layers, and its construction is shown in Figure 1. A input layer node passing the input signal to a hidden layer is usually a simple linear function while the hidden layer node usually consists of the fundamental function. The fundamental function in hidden layer nodes will be affected in local once the signal is inputted.

[FIGURE 1 OMITTED]

The hidden layer function usually expressed as Gaussian function is

[u.sub.j] = exp [- [(X - [C.sub.j]).sup.T] (X - [C.sub.j]) / 2 [[sigma].sup.2.sub.j]] j = 1,2, ... , [N.sub.h] (1)

Where [u.sub.j] = the output value of the hidden layer node of No. j, X = input samples, [C.sub.j] = the central value of the Gaussian function, [[sigma].sub.j] = the standardization constant, [N.sub.h] = the quantity of hidden layer nodes.

The output of RFB network model is a linear combination of the output of hidden layer nodes, and is expressed in the equation (2).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Where [W.sub.i] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The learning of the RBF network model classifies into two processes. The first one, the central value of the Gaussian function and standardization constant [[sigma].sub.j] of each hidden layer node are determined by all the input samples. The second one, after the hidden layer parameters determined, the weights of the output layer can be gained by applying with the least square theory.

Sand Liquefaction Limit State Function

Among the 344 liquefaction site data of 25 strong earthquakes in the world (Xie junfei 1984), there are 206 liquefaction sites and 138 non-liquefaction sites. According to H.B. Seed's (2001) equation, the cyclic stress ratio of saturated sand layer caused by earthquake ground motion is

CSR = 0.65 [[sigma].sub.v] / [[sigma]'.sub.v] [a.sub.max] / g [r.sub.d] MS[F.sup.-1] (3)

Where [[sigma].sub.v] and [[sigma]'.sub.v] are the total and effective vertical overburden stress, respectively; [a.sub.max] is the peak value of the horizontal ground motion acceleration; g is the acceleration of gravity; [r.sub.d] is the stress reduction coefficient; and is the magnitude scaling factor. NCEER suggests using the equation (4) to determine the value of MSF:

MSF = [(M / 7.5)/sup.-2.56] (4)

The cyclic resistance ratio (CRR) of saturated sand is mainly determined by soil density (adopting SPT-N value representation), vertical overburden stress ([[sigma].sub.v] or [[sigma]'.sub.v]), the peak value of horizontal ground motion acceleration, seismic scale and so on. So the liquefaction potential function of saturated sand can be shown as:

L = f(N, [[sigma].sub.v], [[sigma].sub.v], a, M) (5)

In order to calculate conveniently, a reduced acceleration is adopted which considers the influence of horizontal ground motion acceleration and seismic scale defined as 7.5:

[A.sub.M] = [a.sub.max] / g MS[F.sup.-1] (6)

The input layer of the neural network model have 4 neurons, namely [N.sub.1], [[sigma].sub.v], [[sigma]'.sub.v] and [A.sub.M]. The determined method of the liquefaction limit state equation of saturated sand proposed by JUANG et al (2000) is that any variable determining liquefaction potential such as reduced acceleration [A.sub.M], with the trained artificial neural network method, is used to check their state whether from one state convert to another state. Then, through the increase or decrease of the variable, the critical point of converting state is resulted (Fig.2).For example, the liquefaction point A, if the variable [A.sub.M] is decreased (equals to reduce the seismic stress), becomes non-liquefaction, or non-liquefaction point B, and if the variable [A.sub.M] is increased, it gets into liquefaction. Thus the critical value of CSR can be found. Then curve fitting of all the critical values of CSR and [N.sub.1] can make the liquefaction limit state curve and the critical cyclic resistance ratio obtained:

CR[R.sub.cri] = 0.0002 [N.sub.1.sup.2] + 0.005 [N.sub.1] + 0.03 (7)

[FIGURE 2 OMITTED]

The limit state curve of saturated sand of the 344 site data is marked in Fig 3.

[FIGURE 3 OMITTED]

When the equivalent cyclic stress ratio of soil layers caused by earthquake ground motion is great than CR[R.sub.cri], the critical cyclic resistance ratio determined in the equation (7), the saturated sand layer will be a liquefaction case, otherwise a non-liquefied case.

PROBABILITY EVALUATION METHOD OF SAND LIQUEFACTION POTENTIAL

Probability Density Function of Sand Liquefaction Potential

The cyclic resistance safety factor of sand liquefaction can be defined as:

[F.sub.s] = CR[R.sub.cri] / CSR = (8)

Where CSR is a calculated cyclic stress ratio generated by the earthquake ground motion, calculated by the equation (3); CR[R.sub.cri] is a critical cyclic resistance ratio of sand liquefaction, calculated by the equation (7).

With the above 344 site data, the equation (8) is used to calculate the cyclic resistance safety factor of sand liquefaction for every sample. Fig.4 is a histogram describing the safety factor of liquefaction and non-liquefaction cases. By statistic, the probability density function of liquefaction [f.sub.L] ([F.sub.s]) and non-liquefaction [f.sub.NL] ([f.sub.s]) is shown as follows:

[f.sub.L] ([F.sub.s] = 1 / [F.sub.s] [square root of 2 [pi][[sigma].sup.2.sub.L] exp[ - (ln ([F.sub.s] - [micro].sub.L]).sup.2] / 2 [sigma].sup.2.sub.L]] (9a)

[f.sub.NL] ([F.sub.s] = 1 / [F.sub.s] [square root of 2[pi][[sigma].sup.2.sub.NL] exp[ - (ln ([F.sub.s] - [micro].sub.NL]).sup.2] / 2 [sigma].sup.2.sub.NL]] (9b)

where [[micro].sub.L] = -0.4627, [[sigma].sub.L] = 0.443, [[micro].sub.NL] = 0.4507, [[micro].sub.NL] = 0.4753.

[FIGURE 4 OMITTED]

According to the basic concept of probability, if the stylebook is large enough, then

P(L / [F.sub/s]) = [f.sub.L] ([F.sub.s]) / [[f.sub.L]([F.sub.s]) + [f.sub.NL] ([F.sub.s])] (10)

Where P(L / [F.sub/s]) = the liquefaction probability of saturated sand for a given safety factor. With the equation (9) and (10), the liquefaction probability of the 344 site data can be calculated and a Scatters diagram can be drawn accordingly which is shown in Fig.5, and the fitting curve can be presented as:

[P.sub.L] = 1/(1 + [F.sup.4.297.sub.s]) (11)

[FIGURE 5 OMITTED]

If the safety factor equals to 1, the probability of liquefaction or non-liquefaction is both 50%. Juang (2000) pointed out that if the safety factor [F.sub.s] = 1, 30% of the liquefaction probability by H. B. Seed's empirical formula. So it is shown that the cyclic resistance stress curve proposed by H. B. Seed's empirical formula is not the same as the limit state curve of sand liquefaction. By shifting equation (7) and (11), the sand liquefaction resistance stress curve under the different probability can be shown as:

CRR = [[[P.sub.L] / (1 - [P.sub.L])].sup.0.233]. CR[R.sub.cri] (12a)

CRR = [[[P.sub.L] / (1 - [P.sub.L])].sup.0.233]. (0.0002 [N.sup.2.sub.1] + 0.005[N.sub.1] + 0.03) (12b)

The Evaluation Criteria of Sand Liquefaction Potential

In order to be practical and convenient in engineering projects, it suggests that the liquefaction potential of saturated sand is classified into 3 grades under the different liquefaction probability level, and the criteria suggested is shown in table1. According to the importance of engineering projects, an acceptable liquefaction probability level is defined and the estimation criteria of sand liquefaction with different probability can be given by the equation (12). The probability evaluation of sand liquefaction is different to the estimation of sand liquefaction probability. the former is to estimate whether the sand liquefaction will happen under the given probability level. while The latter is given the liquefaction probability of the engineering site so as to make relative decisions,

CONCLUSION

Based on the RBF neural network model, this paper constructs the empirical equations of the limit state curve and critical cyclic resistance ratio curve of saturated sand as well as the empirical equation between the safety factor and liquefaction probability of saturated sand, then deduces the empirical equation of the cyclic resistance ratio of saturated sand under the different probability. The equation is simple and practical, which makes the sand liquefaction probabilistic estimation on engineering sites as easy and convenient as the traditional deterministic method of sand liquefaction estimation. So it is possible that the method of sand liquefaction probability estimation is applied in the engineering practice and adopted in codes for seismic design.

REFERENCES

Juang, C.H., Chen J., Tao J., and Andrus, R.D. (2000). "Risk-based liquefaction potential evaluation using standard penetration tests". J. Can. Geotech. 37:6, 1195-1208.

Xie Junfei. (1984). "Some Comments on the Formula for Estimation the Liquefaction of Sand in Revised Seismic Design Code". Earthquake Engineering and Engineering Vibration 4(2), 95-126.

Youd T.L, and Idriss I. M. et al. (2001). "Liquefaction resistance of soils: summary report from the 1996 NCEER and 1998 NCEER/NSF workshops on evaluation of liquefaction resistance of soils". Geotechnical and Geoenvironmental Engineering. ASCE, 127:10, 297-313.

CHEN GUOXING

Institute of Geotechnical Engineering, Nanjing University of Technology, Nanjing, China

LI FANGMING

Institute of Geotechnical Engineering, Nanjing University of Technology, Nanjing, China

Table 1. Standard for probability evaluation of saturated sand
liquefaction

Liquefaction Sand liquefaction Liquefaction action
probability level safety factor potential evaluation

0.0 [less than or [F.sub.s][greater non-liquefaction
equal to] [P.sub.L] than or equal to]1.2
<0.30

0.30 [less than or 0.81 <[F.sub.s]<1.2 possible liquefaction
equal to] [P.sub.L]
<0.70

0.70 [less than or [F.sub.s][greater liquefaction
equal to] [P.sub.L] than or equal to]0.81
<1.0