Application of R / S Method for Dynamic Analysis of Additional Strain and Fracture Warning in Shaft Lining

In the past few decades, enormous losses have been induced by hundreds of vertical shafts collapse. This study is based on longterm in situ monitoring data of several mines with overburden soil layers in East China in the past six years. R/Smodeling based on fractal theory was used to analyze the development of additional strains in shafts by using Hurst exponent. It can be found that from the monitoring results the Hurst effect is significant ranging from 0.5 to 1.0. Its trend also is in good agreement with forward-biased random distribution. Hence, R/Smethod can be used to predict the additional strain along the shaft lining. In this paper, Hurst exponent shows an irregular phenomenon before cracking. It is proved that Hurst exponent can be used to predicate the progressive failure of shaft lining from abnormal state to normal state.This paper presents the prediction of shaft lining failures using strains measured by embedded strain gauges in the thick overburden soil layers in east China.


Introduction
The disasters of shaft lining fracture have frequently occurred in China since 1987, which greatly threatened the production and safety of collieries.According to the data, there are more than 100 shaft linings with similar fracture, which has caused a large amount of property loss and a great deal of potential disaster [1].So the present research focused on the causes, mechanism, and solving methods for these kinds of geotechnical calamities and the engineering projects have always been the hot topics in recent 20 years in the fields of shaft building, geology, and mining.The vertical additional stress theory on the shaft fracture put forward by China University of Mining and Technology (CUMT) can well describe the main characters of the shaft fracture and are gradually accepted by the experts and scholars both here and abroad because it has been supported by many results from theories, experiments, and practical researches [2,3].Most of the mines were completed in 1970s.The designers had not known the mechanism of shaft fracture.Now, in view of the broken shaft, researchers have developed several kinds of monitoring system for prediction of the fracture [4][5][6][7][8][9].We still cannot predict the fracture of shaft lining for the stress history and current stress value having not been known.
The rescaled range is a statistical measure of the variability of a time series introduced by the British hydrologist Harold Edwin Hurst.Its purpose is to provide an assessment of how the apparent variability of a series changes with the length of the time-period being considered.The rescaled range is calculated from dividing the range of the values exhibited in a portion of the time series by the standard deviation of the values over the same portion of the time series.For example, consider a time series {2, 5, 3, 7, 8, 12, 4, 2} which has a range, , of 12 − 2 = 10.Its standard deviation, , is 3.46, so the rescaled range is / = 2.71.If we consider the same time series, but increase the number of observations of it, the rescaled range will generally also increase.The increase of the rescaled range can be characterized by making a plot of the logarithm of / versus the logarithm of .The slope of this line gives the Hurst exponent, .If the time series is generated by a random walk (or a Brownian motion process) it has the value of  = 1/2.Many physical phenomena that have a long time series suitable for analysis exhibit a Hurst exponent greater than 1/2.For example, observations of the height of the Nile River measured annually over many years give a value of  = 0.77 [12].
In the light of self-similarity of the fractal characteristics appearing in shaft lining fracture, rescaled range analysis method is applied to study the early warning of shaft lining fracture [10].As strain is one of the most critical structural response parameters and plays an important role in structural health monitoring [19][20][21][22][23], an additional strain monitoring system was established [5].A large amount of time series data are obtained using this system, which makes it possible that we study the evolution of additional strain and forecasting of shaft lining fracture.The additional strain during a shaft lining production run is studied in [13], using rescaled range analysis method.The value of Hurst exponent indicates that the method is feasible and effective.In this paper, additional strain data before fracture, obtained from a representative mine embedded in deep alluvium of east China, are studied using rescaled range analysis method.The result shows that Hurst exponent becomes abnormal before fracture.The evolution of Hurst exponent shows that it has anomalous change from 2 to 5 months before shaft lining fracture.

𝑅/𝑆 Method [13]
The dynamic analyses process for additional strain is shown as follows: the corresponding time series is  1 ,  2 ,  3 , . . .,   at time of  1 ,  2 ,  3 , . . .,   , and the time span of the mentioned time series is The average value of the time series within time span  is where  is the number of time series.The cumulative departure of  relating to their mean values at   can be written as The expression of (, ) is not only related to  but also related to .The difference between maximum () and minimum () at the similar  value is denoted as Hurst adopted the following standard departure: Then introducing the dimensionless ratio / and rescaled , Hurst's relationship can be expressed as where  is Hurst index, / is the rescaled range value of sequence   , and  is a constant.Hurst index can be obtained by Hurst obtained the explanation to  based on a series of studies: (1) when  = 0.5, the sequence is Brownian motion, variables are independent, the corresponding coefficient is 0, variables will not affect the future, and, therefore, the time series is random.(2) When 0 ≤  < 0.5, it indicates that time series presents long-term correlation, but the future overall trend is contrary to the past; the process is antisustainability.
(3) When 0.5 <  < 1, it indicates that the time series present the characteristic of long-term correlation; that is to say, the process is sustainable.
Generally speaking, the shaft additional forces variation is consistent with the previous variation, and the maintenance is stronger as the increase of the .

Dynamic Analysis of Additional Strain before the Fracture of Shaft Lining
The stress of shaft lining will change accompanied by the initiation of the shaft fracture.Therefore, the variation of additional strain is of great importance to the prediction of shaft lining fracture.Taking a mine as an example, dynamic analysis of the additional strain is presented before the fracture of shaft lining, which is used to provide scientific basis for the disaster forecasting of shaft lining fracture.The thickness of alluvium over the shaft is 153.22 m.The shaft lining is built in 1977 with 400 mm thickness and 6000 mm diameter.It fractured on December 16, 2004, and was cured by cement grouting in around stratum from July 20, 2015, to June 26, 2006.
Figure 1 shows the variation of the additional strain at the depth of 145 m from January 1, 2004, to November 1, 2007.During the period of production and operation in a mine, the additional strain is increasing steadily.It presents a different phenomenon before fracture of shaft lining.The increasing rate becomes bigger than before.We calculate the average additional strain every 3 days as data of time series,  1 ,  2 ,  3 , . . .,   .100 pieces of information are obtained.Figure 2 shows the relationship between additional strain and time at different depths of 100 m, 125 m, 145 m, and 165 m.The monitoring record in ten months shows dynamic changes of the additional strains.
According to the above data, a series of lg(/) and lg is obtained in log-log coordinate.It is easily seen that the slope and intercept are obtained through matching the curve, as is shown in Figure 3. Slope of the line is greater than 0.5 along with the increasing of the time scale.Analysis results are as follows.
(1) Because the time series of additional strain before fracture of shaft lining is not a pure random process, the in situ  measurement data are not independent.Dynamic changes of additional strain have long-term memory measurably.
(2) Hurst exponents are 0.9961, 0.9933, 0.8922, and 0.9795 at different depths 100 m, 125 m, 145 m, and 165 m.They are all greater than 0.5, which shows that the current trend of the additional strain exerts a tremendous influence on the next period of time.So we can draw the conclusion that the strain will continue to accumulate.
(3) Hurst exponent is changing with time scale especially at depth of 145 m.
It is easily seen from Figure 3(c) that there is an obviously inflection point when  = 50.In Figures 3(a) and 3(b), there are also inflection points when "" equals 90 nearly though it is not obvious.It is necessary to make a deep analysis on the inflection point.

Hurst Exponent Study of Additional Strain before Fracture
In order to carry out Hurst exponent accumulative experiments, we should take actions as follows.
(1) We select the data from 30 May 2004 to 15 July 2004 as the initial window ( = 35∼50) and then calculate Hurst exponent.
(2) Based on the initial window, we add the data from 16 July to 30 July ( = 5); the length of window data is 35 ( = 35∼55); then we calculate Hurst exponent again.If the value changes, we consider that the change is caused by the new added data for the initial window data does not change.
(3) We can seek the inflection point of Hurst exponent through these accumulative experiments.
We choose the data at depth of 145 m, and the results are shown in Figure 4.
It is easily seen that Hurst exponent significantly reduced with the increase of sample data since 15 July 2004.It began to increase slowly after a period of months.The increasing rate was less than the decreasing rate.
Figure 5 shows the trend of Hurst exponent while adding the sample data every half a month until the fracture moment.The time process of Hurst exponent shows an unnatural variation before fracture (2∼5 months).Before the shaft lining fracture, the abnormal feature of Hurst exponent appears to be a trend where there are 3 stages.In the first stage, the Hurst exponent declines until its value is less than 0.5.It maintains a lower value in the second stage.Some days later, the Hurst exponent begins to arise and it enters the third stage.The fracture takes place during the rising process.Hurst exponent is useful for forecasting of shaft lining fracture.It can be used to predicate the progressive failure of shaft lining from disorder state to order state.When  < 0.5, the trend of additional strain in the future will change into a reversal.This heralds that shaft lining is  likely to fracture without the influence of external load, which causes stress release.As shown in Figure 6.The shaft lining fractured from 16 December to 18 December 2004.In situ measurement data at that time were shown in Figure 6.

Conclusions
The aim of this paper is to propose an effective method for predicting shaft lining fracture.According to the test results, the following conclusions can be drawn as follows.
(a) The / analysis method is applicable for studying the variation of additional strain of shaft lining.
(b) The Hurst exponent of additional strain is between 0.5 and 1.The sequence is positive biased random walk and presents a characteristic of strong positive correlation.
(c) Hurst exponent shows abrupt changes before fracture.It can be used to predict the progressive failure of shaft lining from disorder state to order state.

Figure 1 :Figure 2 :
Figure 1: Monitoring data of shaft lining additional strain at 145 m.

Figure 4 :
Figure 4: Results of accumulated experiments at depth of 145 m.