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Influence Of Subsidence Fluctuation On The Determination Of Mining Area Curvatures

Influence Of Subsidence Fluctuation On The Determination Of Mining Area Curvatures Arch. Min. Sci., Vol. 60 (2015), No 2, p. 487­505 Electronic version (in color) of this paper is available: http://mining.archives.pl DOI 10.1515/amsc-2015-0032 ANDRZEJ KOWALSKI*, ELIGIUSZ JDRZEJEC* WPLYW FLUKTUACJI OBNIE NA OKRELANIE KRZYWIZN TERENU GÓRNICZEGO The article concerns the random dispersion of deformation indicators, especially the influence of subsidence fluctuation on the distribution of inclinations and curvatures. Surface curvatures have significant influence on building objects. The article includes the probability studies of displacement fluctuation for two arbitrarily close but different points. It was determined, if the probability is dependent on each other or not. Therefore, the separate deformation indicators can be considered to damage hazard assessment of building objects, if their standard variation of fluctuation is well determined (dependent on the fluctuation of vertical and horizontal displacements). Consequently, it is possible to determine the confidence intervals of fluctuation for all separate deformation indicators. Even in a case of low values of predicted separate curvatures, their values can be significant higher when considering their natural dispersion. Keywords: mining, land deformation, curvatures, fluctuation, forecast, measurement Artykul dotyczy rozproszenia losowego wskaników deformacji, w szczególnoci wplywu fluktuacji obnie na ksztaltowanie si fluktuacji nachyle i krzywizn. W znacznym stopniu dotyczy krzywizn terenu i ich wplywu na obiekty budowlane. Wskanika, do którego panuj dwa pogldy. Jeden o malej jego przydatnoci do oceny szkodliwoci wplywów eksploatacji górniczej na obiekty budowlane, gdy w wyniku pomiarów terenowych stwierdza si duy rozrzut ­ fluktuacje. Drugi, e wskanik ten ma istotne znaczenie, decyduje o zmianie rozkladu pionowych oddzialywa midzy obiektem a podloem. Zaznaczy naley, e wskanik ten jest trudno sprawdzalny geodezyjnymi pomiarami. Wystpowanie fluktuacji ­ naturalnych rozprosze ­ okrelanych pomiarowo wskaników deformacji tlumaczy si przypadkowym spkaniem górotworu, jego przypowierzchniowej warstwy. Deformacje odcinkowe wyznaczane z wzorów (1), (2), (3) na podstawie pomiarów przemieszcze w, u nie s dokladnymi odpowiednikami wskaników deformacji, które s wynikiem prognozy. Prognozowane wskaniki deformacji T, K, e, popularnie zwane wskanikami punktowymi, liczone s w prognozie na podstawie wzorów na pochodne obnie i przemieszcze poziomych w pewnych punktach obliczeniowych. Teoretycznie, oba sposoby bylyby równowane, gdyby byly wyliczane graniczne ich wartoci przy dlugoci boku l 0. CENTRAL MINING INSTITUTE, PLAC GWARKÓW 1, 40-166, KATOWICE, POLAND W artykule przeanalizowano prawdopodobiestwo fluktuacji przemieszcze dwóch dowolnie bliskich, lecz rónych punktów, czy jest od siebie zalene, czy te nie. Najprostsze teoretyczne modele, jakie analizowano s nastpujce: · Model iglowy: fluktuacje w dwóch dowolnie bliskich, lecz rónych punktach s od siebie niezalene. · Model ziarnisty: orodek ma struktur ziarnist (o rónych rozmiarach ziaren, tak rónorodnie rozmieszczonych, e dowolny punkt (x, y, z) naley zawsze do jakiego ziarna lub ley na granicy ziaren ssiednich. · Model falisty: mona go utworzy z modelu ziarnistego, co daje obraz podobny do lekko sfalowanego morza. W takim modelu mona byloby rozwaa te fluktuacje, jako cigle. Z analizy tej wynika, e najprostszym i poprawnym w sensie matematycznym jest model falisty, w którym wszystkie pochodne typu (26) s okrelone z wyjtkiem by moe pewnych punktów lub krzywych, gdzie mog one by niecigle. W obszarach, w których s one skoczone, podstawow funkcj losow jest fluktuacja obnie w. Fluktuacje nachyle i krzywizn s pochodnymi fluktuacji obnie i s jednoznacznie okrelone przez w. Podobnie jest z poziomymi przemieszczeniami i odksztalceniami, gdzie podstawowymi funkcjami losowymi s skladowe poziomego przemieszczenia, a fluktuacje odksztalce s wyznaczone przez ich pochodne. W rozdziale 4, przyjmujc róne dlugoci l oraz lczne odchylenie standardowe obnienia punktu (31 mm) wynikajce z bldu pomiaru (1 mm) oraz wynikajce z naturalnego rozproszenia (30 mm), obliczono wartoci odchyle standardowych T, K rozproszenia wplywów dla nachyle i krzywizn. Obliczono je dla wartoci ekstremalnych nachyle ±Tmax oraz krzywizn ±Kmax, bdcych skutkiem przykladowej eksploatacji w postaci pólplaszczyzny, na rónych glbokociach, od 0 do 1000 m, oraz wartoci wmax = 1 m i parametru r rozproszenia wplywów r = 300 m. Obliczenia wartoci odchyle standardowych wykonano przyjmujc poziom ufnoci = 0,05. Wykresy zalenoci ksztaltowania si maksymalnego nachylenia, krzywizny i promienia krzywizny przedstawiono odpowiednio na rysunkach 1-3. Nastpnie przy zaloeniu, e warto odchylenia standardowego w jest niezalena od poloenia obiektu wzgldem eksploatacji, obliczono rozklady obnie, nachyle i krzywizn w calym obszarze wplywów eksploatacji od 1,5r do ­1,5r, co przedstawiono odpowiednio na rysunkach 4-6. Z rysunków tych wynika, jak w znacznym zakresie mog fluktuowa (z prawdopodobiestwem 95%) nachylenia, a zwlaszcza krzywizny w stosunku do wartoci rednich, które prognozujemy. W konkluzji stwierdzono, e o rozproszeniu naturalnym wskaników nie wiadomo wszystkiego i moliwe s róne podejcia do opisu tego rozproszenia. Z powodu braku wiarygodnego modelu nie jest moliwe okrelenie odchyle standardowych w przypadku tzw. punktowych deformacji prognozowanych. Dlatego do oceny zagroenia obiektów budowlanych mona rozpatrywa odcinkowe wskaniki deformacji, dla których istniej dobrze okrelone oszacowania (5), (6), (7) odchyle standardowych ich fluktuacji wynikajce z ich uzalenienia od fluktuacji obnie i przemieszcze poziomych. W konsekwencji mona okreli przedzialy ufnoci dla tych fluktuacji dla wszystkich odcinkowych wskaników deformacji. Nawet w przypadku, gdy prognozowane odcinkowe krzywizny maj bardzo male wartoci, to w wyniku uwzgldnienia rozproszenia naturalnego ich wartoci mog by istotnie due. Slowa kluczowe: eksploatacja górnicza, deformacja powierzchni, krzywizny, fluktuacja, prognoza, pomiar 1. Introduction Issues concerning observed random dispersion of deformation indicators have been researched by a number of authors, in chronological order: Batkiewicz, Popiolek, Milewski, Ostrowski, Kwiatek, Stoch and Kowalski. Batkiewicz (1971) reviewed various formulas of centre which caused random dispersion of deformation indicators. These assumptions and formulas were later continued by: (Popiolek 1976, 1996; Popiolek et al., 1997a,b; Ostrowski, 2006; Stoch, 2005; Popiolek & Stoch, 2005). They recognized quantities of the dispersion occurring in practice and suggested coefficients of variation as its measure. Works developed by Kowalski focused on research concerning this phenomenon (Kowalski, 2007) and he included it in the forecast (Kowalski, 2014). Research conducted by Kwiatek considered methods that would include these phenomena in methodology of building protection against mining influences. Summary of this research is included in monographs (Kwiatek et al., 1997) and (Kwiatek, 2007), and publications (Kwiatek, 2006, 2008). They mainly relate to land curvatures and the influence they have on buildings. On one hand, it is thought that such indicator is of a small value while estimating a harmfulness level of mining operations on buildings, because field measurements show a great level of dispersion ­ fluctuations ­ of obtained curvatures (Popiolek et al., 1995). On the other hand, such indicator has a significant value and decides on a change in distribution of vertical impacts between a building and foundation. In the case of buildings that are not resistant enough, the influence of curvatures causes typical damages (Kwiatek, 2007). It is difficult to accurately check the indicator, however, it is indispensable for building construction on mining areas (Kwiatek, 2006; Florkowska, 2011, 2012). In addition to curvatures, this paper discusses all other deformation indicators. Therefore, other deformation indicators have been included in this paper with an emphasize on the issues relating to curvatures. 2. Measured deformation indicators In practice, only differences between measuring point's height and distance Li of subsequent points of measuring line i ­ 1, i1 are directly measured. Sometimes the length, angles and differences in height are measured for a closed network of measuring points that allow leveling their coordinates. In both cases such measurements enable to determine the level of point subsidence wi inclinations Ti of measure bases and curvatures Ki (as relative changes of inclinations of two adjacent bases) at specific times. Additionally, length measurements of measured bases allow determining relative changes of such distances, which is a horizontal deformation. When the points coordinates are determines, it is possible to additionally calculate components of horizontal displacement. The paper analyses indicators that are dependent on subsidence (inclination and curvatures) and are calculated as: Ti Ki wi wi 1 Li Ti Ti 1 Li Li 1 2 (1) (2) and horizontal deformation most often calculated as relative continuation of a section at a time of measurement Li' in relation to its initial length Li (1.3). Li' Li Li (3) or less often, when components of horizontal point displacements ux, uy are counted as: such sections are also referred to as measured bases. xi yi u x i u x i 1 xi if xi 0 (4) u u x i 1 xi yi if yi 0 Determined subsidence and horizontal displacement of measured line points, apart from (minor) errors resulting from a measure technique, include also so-called natural dispersion. Accidental cracked rock mass is considered to be the cause of such dispersion. The following issues have not been researched yet: · whether the dispersion occurs independently of frequency of measurements or whether there are specific areas where the dispersion takes places in a regular (smooth) manner2; · what is the probability distributions of natural dispersion of separate deformation indicators, · whether the parameters of such distributions are the same in different locations of subsidence trough. Values of inclinations, curvatures and horizontal deformations are calculated by the formulas (1)...(4) and are characterised also by natural dispersion of vertical and horizontal displacement. The work (Kowalski 2007) presents detailed introduction of relations between fluctuation standard deviation T, K, of discussed deformation indicators depending on standard deviations w of vertical and horizontal displacements (including measured errors). In case when Li = l (for curvatures Li = Li ­ 1 = l) standard deviation amounts to: T K (5) and (6) show that: 2 w l 6 l2 (5) (6) 2 u l (7) and not 3 T l (8) 2 T as it could be calculated for the formula (2), because in this case random varil ables Ti, Ti ­ 1 are not independent (both are dependent on subsidence wi ­ 1). 2 smooth dispersion would indicate an existence of certain center block that are moved as a whole, and potentially deformed in a regular manner If the coefficient of variation is defined acc. to Popiolek (1976) as: Mw Mu MT MK M is calculated by (9): M T T wmax M w w Tmax M K K wmax M w w K max M umax M u u max Budryk-Knothe theory shows that: wmax r Tmax wmax e r2 K max 2 (11) (10) wmax umax Tmax (9) K max wmax max umax e 2 Therefore, after considering relations (5)...(7) and (11) in (10), the result amounts to: MT 2 Mw MK 1 2 Mw M 1 Mu e 3e (12) where are determined: l r (13) An average outcome of research by Popiolek (1976), Kowalski (2007) and Stoch (2005) shows that: M K 0.46 15.33 M w 0.03 M T 0.11 3.67 M w 0.03 M 0.25 2.78 M u 0.09 and they are not dependent on . It should be noted that section deformations calculated by the formulas (1)...(3) on the basis of displacement measurements w, u are not accurate equivalents of deformation indicators resulting from forecast. Deformation indicators T, K, so called point indicators, are calculated on the basis of formulas for derivatives of vertical and horizontal displacements in certain calculation points. Theoretically, both methods would be equivalent if the formulas (1)...(3) contained subsidence derived from a theoretical M model, and instead of the right sides of formulas, their limit values were calculated as the side length approaches to zero l 0. Then, obtained values would be equal to T, K, values calculated from M model. When it is assumed, however, that the forecast corresponds exactly to smooth (without any fluctuations) measurement, the estimated point deformation indicators are not much different than section indicators in the case of correspondingly small distances l 25 m3. 3. Forecast deformation indicators and their natural dispersion Estimated deformation indicators of continuous rock mass are calculated by formulas such as Budryk-Knothe theory (Prusek & Jdrzejec, 2008) treated as constant component of actual rock mass deformation indicators4, to which statistical fluctuations should be added due to the lack of certainty concerning the theory parameters and statistical fluctuations resulting from natural dispersion of corresponding deformation indicators. When a systematic error of theoretical model is neglected, the following aspects can be stated for indicators dependent on the field of vertical and horizontal displacement. wrz ( x, y, z , t ) w( x, y, z , t , P pr ) w ( x, y, z , t ) w ( x, y, z , t ) rz u x ( x, y, z , t ) u x ( x, y, z , t , P pr ) u x ( x, y, z , t ) u x ( x, y, z , t ) (14) u rz ( x, y, z , t ) u y ( x, y, z , t , P pr ) u y ( x, y, z , t ) u y ( x, y, z , t ) y (15) the greatest differences of point and section inclinations for half-planes when r = 500 m, l = 25 m are 3,8% Tmax, and differences of individual curvatures are 20,6% Kmax 4 which center is complex, partially cracked, partially constant. Txrz ( x, y, z , t ) Tx ( x, y, z , t , P pr ) Tx ( x, y, z , t ) Tx ( x, y, z , t ) Tyrz ( x, y, z , t ) Ty ( x, y, z , t , P pr ) Ty ( x, y, z , t ) Ty ( x, y, z , t ) rz z ( x, y, z , t ) z ( x, y, z , t , P pr ) z ( x, y, z , t ) z ( x, y, z, t ) rz xx ( x, y, z , t ) xx ( x, y, z , t , P pr ) xx ( x, y, z , t ) xx ( x, y, z, t ) rz yy ( x, y, z , t ) yy ( x, y, z , t , P pr ) yy ( x, y, z , t ) yy ( x, y, z , t ) (16) (17) rz K xx ( x, y, z , t ) K xx ( x, y, z , t , P pr ) K xx ( x, y, z , t ) K xx ( x, y, z , t ) rz K xy ( x, y, z , t ) K xy ( x, y, z , t , P pr ) K xy ( x, y, z , t ) K xy ( x, y, z , t ) rz K yy ( x, y, z , t ) K yy ( x, y, z , t , P pr ) K yy ( x, y, z , t ) K yy ( x, y, z , t ) (18) where: wrz, Trz, Krz, rz P = (P1, P2, ... Pm) p pr w, T..., K... actual rock mass deformation indicators, m set of theory parameters, a set of these parameters with values used for forecast, constant rock mass deformation indicator calculated for values of Ppr ­ theory parameters adopted for the forecast, ...(x, y, z, t) -- random functions showing the impact of uncertain theory parameters ­ on individual deformation indicators, ...(x, y, z, t) -- random functions showing the impact of natural dispersion of individual deformation indicators. -- -- -- -- Theory parameters adopted for the forecast are never entirely certain, unless the values are verified by deformation indicators measurement taken after mining operations. Approximation of this parameter conducted acc. to the theory results in parameters values P dok, referred to as accurate. Their values are unknown at the time of the forecast. Shall: Pk Pkdok Pkpr , k 1.2, m (19) (20) P P , P2 , Pm 1 It can be said that ...(x, y, z, t) functions: · would be equal to zero if the theory parameters were certain, · are continuous and their values are ... = (­, +) real numbers, · it can be assumed that the distribution of their probability are constant and their density ... is even-numbered ­ they are symmetrical ... (­...) = ... (...)5, · it can be assumed that distributions of difference probability Pk are normal with zero average values, · their values equal zero. what results from assumption that the probability of deviation is independent from its sign In fact, for instance, for subsidence occurs: w x, y, z , t w x, y, z , t P dok w x, y, z , t P pr Considering (2.5), it can be described as: w x, y, z , t w x, y, z , t P pr P w x, y, z , t P pr (21) (22) For minor difference, Pk the formula (2.8) can be approximated by linear formula: w x, y , z , t k 1 m w x, y , z , t , P Pk P pr Pk 2 2 P k (23) 2 Assuming that errors Pk are not dependent, the variation w can be calculated as: 2 w w x, y , z , t , P x, y , z , t Pk k 1 (24) P pr Generally, the differences Pk can not be recognized as small enough and as a result, the issue remains non-linear: 2 w x, y , z , t E ( 2 ) w P1 Pm [ w( x, y, z , t P pr P) w( x, y, z , t P pr )] P d ( P1 ) d ( Pm ) (25) where Pk is the error range Pk, and P (w) is the distribution density of deviation w probability. If the errors Pk are independent, then: P Pk k 1 where Pk are densities of error distribution Pk. Equalities analogical to (21)..(25) occur for the other deformation indicators. It is noteworthy, that standard deviation is a function of the position ... = ...(x, y, z,t). For established realisation of random variables P functions ... are certain continuous surfaces in the whole range of exploitation influences, whereas it has not been decided whether their fluctuation realisation ... is of a continuous character. Due to the fact that the measurements are taken at locations where the points are located in certain distances, it is unknown how fluctuation occurs in two randomly located points. In particular, it has not been determined if the probability of fluctuation occurring at a point is depended on another randomly located point. The simplest theoretical models that seem to be up-to-date can be characterised by: · Needle Model: fluctuations in two randomly located, but different, points. It means that they have an image of independent `needles' in each measuring point in the whole area of exploitation influences (an image similar to `signal noise'). If the measuring points were located in every point of the location impacted by mining operations (in a form of dense-in-itself mosaic), the image of the functions would represent dense-in-itself `bush of needles' with random amplitudes. · Granular Model: the centre has granular structure (with varied granularity and randomly located so that any point (x, y, z) always belongs to a grain or is located at the border of neighbouring grains, because, if a local empty space was created as a result of such centre deformation, it may be recognized as a separate grain), and fluctuations have forms locally continuous (in the area of grains) surfaces ­ prisms with random heights, with bases as curvilinear outlines of grains and tips in a form of certain continuous surfaces. · Corrugated Model: it may be created from the Granular Model by applying certain continuous surface on continuous surface of fluctuation ... (resulting in lightly wavy sea). The functions in such model could be considered as continuous. Discussion often concerning the problem includes an idea that fluctuations ... are independent of a location in the case of fluctuations probability that would be identical in sides of a subsidence trough as well as its centre. It has not, however, been proved by any research. Other theories would also seem justified, such as that the probability is proportional to the size or maximum size of an adequate deformation indicator. Generally, it is safe to assume that the probability is dependent on the location and such dependency can be determined by appropriate research. The state (set) of fluctuation ... at the t moment must have a form of function dependent on the location in the whole area of exploitation influences, otherwise, they would all have the same value which is contrary to observations. It can be said that ...(x, y, z,t) functions: · the values at t moment belong to a continuous and cohesive set of real numbers ...6, · it can be assumed that the distribution of their probability is constant and their density ... is even-numbered, as a result ...(­...) = ...(...)7, · their average values equal zero. In the case of the Granular Model, the set of function characteristics ...(x, y, z,t) can be extended by a feature ­ during the realisation t in the area of every grain, they are continuous functions of the location and time, and on the grain borders they may be non-continuous. Such feature results from the fact that components of inclination are derivatives of subsidence, components of curvatures are inclination derivatives and the granular centre can be calculated by adequately small .8 Tx x, y, z , t lim w x , y , z , t w x, y , z , t 0 (26) There are certain possible locations of two points A = (x, y, z) and B = (x + , y, z), with small enough: 1. A and B point at inside of the same grain: w (x + ,y,z,t) = w (x + ,y,z,t) + · a(x + ,y,z,t) can be added to the linear approximation what gives Tx = a. there is a probability that it is a set (­, +) what results from assumption that the probability of deviation is independent from its sign so small that the section (x, x + ) would be located near the grain 2. One of the points shows the inside of a grain, and the other points at its border: the value near the grain or infinitely near to the grain may be assigned to the border function and this still can be connected with the first case. 3. Both points show (always, with random small value of ) the borders of neighbouring grains (two or more): the value of counter quotient is undetermined, and a value Tx in that point ­ non-continuous. Similarly, it can be determined that the other derivatives Ty, Kxx, Kxy, Kyy in the area of every grain, also in the granular centre are continuous functions of the location and time, and on the grain borders they may be non-continuous. Let in the area of a random grain located at z height, occur at t moment function w, which, if the granularity is small enough, is formed by following equation (accurate to the decimal place): w x, y q0 q1 x q2 y q3 x 2 q4 xy q5 y 2 (27) where the coordinate system xy moved in such a way, that coordinates x = 0, y = 0 point at the centre of a grain. Coefficients q0, q1, ... q5 are random variables and their realisation at t moment is determined by the continuous function (27). Because a grain is subject to forces moving from neighbouring grains, some or all coefficients qi may be dependent on each other9. Therefore: w q1 2q3 x q4 y x Ty x, y w q2 q4 x 2q5 y y Tx x, y (28) K xx x, y K xy x, y K yy x, y Tx x Tx y Ty y 2q3 Ty x q4 (29) 2q5 Similar relations can be calculated for displacement components and horizontal deformations. If a grain is relatively big in relation to the size of measuring base, but small enough to allow measuring point to belong to another grains, local (in the area of the grain) inclinations and curvatures may significantly differ from the ones calculated by formulas (1) and (2). It seems, however, that cracked sections of the centre are moving and turning, but their local curvatures can be omitted. As a result, it is justified to expect that the effects would be registered by properly frequent measurement. Contrary to this, results of measurements conducted at Kazimierz-Juliusz Mine are presented by Kowalski (2007) indicate that for section l = 5 m the curvatures experience is noticeably greater for instance, if the grain is stiff enough, so that it cannot bend, it may be assumed that coefficients q3,q4,q5 dissaper. dispersion than in the case longer measurement bases. This would indicate that the grains are smaller than 5 m (for example few centimetres) or actual rock mass (at least in the surface layer) cannot be determined by the Granular Model. The Needle Model does not include spatial fluctuation derivatives as functions of (26), and fluctuations of inclinations and curvatures should be considered as entirely independent random functions. The easiest mathematical model is the Corrugated Model, in which all derivatives (26) are determined; in exception of some point or curvatures which may be non-continuous. In the areas where they are non-continuous, the main random function is subsidence fluctuation w. Inclination and curvatures fluctuations are derivatives of subsidence fluctuations and are explicitly determined by w. It is the same in the case of horizontal displacement and deformations, where the basic random functions are components of horizontal displacement, and deformation fluctuations are determined by their derivatives. 4. Probabilistic forecast of deformation indicators Formulas (14)...(18) can be formulated as: D rz x, y, z , t D x, y, z , t P pr D x, y, z , t D x, y, z , t (30) where D is any deformation indicator from among subsidence, components of horizontal displacement, inclinations and curvatures, as well as vertical deformations and components of horizontal deformations. A forecasts error of D indicator is marked by: D D rz x, y, z , t D x, y, z , t P pr and its variation is marked by: 2 2 E D D (31) (32) If ranges of random variables D, D amount to D, D and distribution densities are D(D), D(D) and because of random variables D, D are independent, the density of probability that variables take certain values of realization is: D D D (33) As a result of (30) and (31), due to the fact that random variables D, D are independent, the following occurs: 2 2 2 D D D (34) it results for the following calculation: 2 E (D ) 2 ( 2 2 D D D ) D D d D d D D 2 D d D D D d D 2 D D d D D 0 D D d D (35) D 0 D d D 2 2 2 D D d D D D Range of confidence for D deformation indicator at confidence level of , is determined as follows: Shall Z occurrence be based on: q1D q2 D D rz D (36) where: q2 D ( ) P( Z ) q1 D ( ) D (D )d D 1 (37) q1 D ( ) additionally P(D q1D ( )) D (D )d D When the distribution of probability D (D) is an even function sD (­D) = D (D), which occurs when densities D, D are even functions (which was the subject of assumptions), then as a result: q1D q2 D qD and P( D rz D qD ) q D (38) q D D D d D 1 (39) Assuming, that distributions D, D are normal, the distribution D will be normal with an 2 average equal to zero and variations D determined by the function (34). As a result, 1 ­ ­ fractiles are: q D D where () are independent of deformation indicator; their values presented in Table 1. TABLE 1 The value of indicator (), depending on quantiles and 1 ­ 1­ 0.999 0.997300 0.99 0.98 0.975 0.954500 0.95 0.9 0.8 0.75 0.682689 0.5 () 3.290528 3 2.575830 2.326348 2.241402 2 1.959964 1.644854 1.281552 1.150350 1 0.674490 5. Inclination and curvatures of building foundation ­ forecast and measurement An important indicator of deformation for building designers is the curvature of building foundation. In the case of deeper mining excavations, particularly parameters r of influences dispersion, the values of forecast curvatures are small. Consequently, does it mean that measured curvatures are in such cases also small? Authors do not have any measurement of curvatures of such large parameter r that forecast values of maximum curvatures would, for instance, almost equal to values for the zero category of mining areas (in Poland), while the observed ones ­ would be significantly greater. However, as a result of an analysis of a number of mining operations, Kowalski (2014) stated that generally forecasts of extreme curvatures are systematically too small (50% on average). Natural, random subsidence fluctuations may be the cause. The following test has been conducted in order to demonstrate possible influence on observed curvatures. Taking various lengths l of building's foundation (excluding dilatation) and total statistical deviation of point subsidence (31 mm) resulting from a measurement error (1 mm) and natural dispersion (30 mm), calculated the value of standard deviations T, K of influences dispersion for inclination and curvatures. The results are presented in Table 2, which contains emphasized values of typical length l = 25 m. TABLE 2 Calculated values of standard deviations for inclinations and curvatures, depending on the length of measured base l, m T, mm/m K, km­1 At the confidence level = 0.05 or the typical length l confidence ranges were calculated, assuming that parameters wmax, r are correct and free of any errors: 1. for inclinations 1.959964 · 1.75 mm/m = 3.43 mm/m, 2. for curvatures 1.959964 · 0.12 km­1 = 0.2352 km­1. Extreme values of inclinations and curvatures will occur for model exploitation in a form of a half-plane located in known areas: ±Tmax and ±Kmax. Scheme of relation10 for such exploitation carried out at different depths characterised by r parameter of influences dispersion and with assumed value wmax = 1 m. · for inclinations Tmax(r), Tmax(r) + 3.43 mm/m and Tmax(r) ­ 3.43 mm/m (Fig. 1), · for curvatures Kmax(r), Kmax(r) + 0.2352 km­1 and Kmax(r) ­ 0.2352 km­1 (Fig. 2). The charts distinct an example value r = 300 m. Maximum curvatures of such value r are minor (curvature radius = 59.20 km ­ the zero category of mining area), however, limitations of confidence range become more significant for buildings protection (curvature radius = ­4.52 km, 3.92 km ­ 4th and 5th category of mining terrain). Tmax(r) Tmax(r)+1,96 Tmax(r) -1,96 6,77 mm/m 3,33 mm/m -0,10 mm/m T T T T(r), mm/m =1,75 mm/m -2 Fig. 1. Chart of the relation of maximum inclination from the parameter r of influences dispersion. Marked confidence range includes actual value with probability 1 ­ = 0.95 and an example section for r = 300 m Positive and negative signs for inclination and curvatures were assumed according to rules using in building industry Kmax(r) Kmax(r)+ Kmax(r)=0,12 km-1 K(r), km 0.5 0.4 0.3 0.2 0.1 (59,20 km)-1 0 -0.1 -0.2 0 100 200 (-4,52 km)-1 (3,92 km)-1 Fig. 2. Chart of the relation of maximum curvatures from the parameter r of influences dispersion. Marked confidence range includes actual value with probability 1 ­ = 0.95 and an example section for r = 300 m 70 60 50 40 30 20 10 3,92 km 0 -4,52 km -10 -20 -30 0 100 200 300 400 59,20 km 1/Kmax(r) 1/[Kmax(r)+1,96 1/K(r), km ] 1/[Kmax(r) -1,96 K] =0,12 km-1 Fig. 3. Chart of the relation of minimum radius for curvatures from the parameter r of influences dispersion. Marked confidence range includes actual value with probability 1 ­ = 0.95 and an example section for r = 300 m -100 0 100 200 300 400 500 600 700 800 900 1000 1100 w()+1,96 w() w()- 1,96 w(), mm =31 mm deposit zo e -1.5 -1 -0.5 0 0.5 1 1.5 Fig. 4. Chart of subsidence depending on . Marked confidence range includes actual value with probability 1 ­ = 0.95 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 T(), mm/m T()+1,96 =1,75 mm/m deposit zo e T() T()- 1,96 -1.5 -1 -0.5 Fig. 5. Chart of inclination depending on . Marked confidence range includes actual value with probability 1 ­ = 0.95 0.3 (4,52 km)-1 0.2 0.1 (59,20 km) 0 -0.1 -0.2 (-3,92 km)-1 -0.3 zo e deposit -1.5 -1 -0.5 0 0.5 1 1.5 (59,20 km)-1 -1 (3,92 km)-1 Kxx()+1,96 Kxx() Kxx()- 1,96 Kxx(), km-1 =0,12 km-1 (-4,52 km)-1 Fig. 6. Chart of curvatures depending on . Marked confidence range includes actual value with probability 1 ­ = 0.95 Left borders of confidence ranges become negative values, what can be explained with an idea that their value reverses in places of extreme values ­ minimum value appears instead of maximum and inversely. Figures 2 and 3 show the relation between curvatures (curvatures radius) and parameter of influences dispersion. When assuming, that standard deviation value w is independent of the building location in relation to mining workings, T, K ill also be independent of such location. Therefore, calculated confidence range applies to the whole exploitation influences area and it can be marked on charts of deformation indicators at T, K (Fig. 4, 5, 6). 6. Conclusion In order to prepare probabilistic forecast of deformation indicators, especially for curvatures of building's foundations, it is necessary to determine standard, random fluctuation deviations that define the uncertainty of theory parameters for these indicators and standard, random fluctuation deviations resulting from natural dispersion of such indicators. The knowledge of natural indicators dispersion is limited. This paper points out various possible methods towards the dispersion phenomenon. However, due to the lack of its credible model, it is currently not possible to determine such standard deviations in the case of so-called forecast point deformations. Therefore, risk assessment of buildings may include section deformation indicators for which the elements of foundation constructions should serve as the length of sections. There are properly determined calculations (5), (6), (7) for standard deviations of their fluctuation result- ing from their dependence on fluctuation of vertical and horizontal displacements. They can be successfully used for section indicators. As a result, confidence ranges can be determined for such fluctuation for all section deformation indicators. Even if the forecast point curvatures are small, their values may be greater because of natural dispersion. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archives of Mining Sciences de Gruyter

Influence Of Subsidence Fluctuation On The Determination Of Mining Area Curvatures

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Abstract

Arch. Min. Sci., Vol. 60 (2015), No 2, p. 487­505 Electronic version (in color) of this paper is available: http://mining.archives.pl DOI 10.1515/amsc-2015-0032 ANDRZEJ KOWALSKI*, ELIGIUSZ JDRZEJEC* WPLYW FLUKTUACJI OBNIE NA OKRELANIE KRZYWIZN TERENU GÓRNICZEGO The article concerns the random dispersion of deformation indicators, especially the influence of subsidence fluctuation on the distribution of inclinations and curvatures. Surface curvatures have significant influence on building objects. The article includes the probability studies of displacement fluctuation for two arbitrarily close but different points. It was determined, if the probability is dependent on each other or not. Therefore, the separate deformation indicators can be considered to damage hazard assessment of building objects, if their standard variation of fluctuation is well determined (dependent on the fluctuation of vertical and horizontal displacements). Consequently, it is possible to determine the confidence intervals of fluctuation for all separate deformation indicators. Even in a case of low values of predicted separate curvatures, their values can be significant higher when considering their natural dispersion. Keywords: mining, land deformation, curvatures, fluctuation, forecast, measurement Artykul dotyczy rozproszenia losowego wskaników deformacji, w szczególnoci wplywu fluktuacji obnie na ksztaltowanie si fluktuacji nachyle i krzywizn. W znacznym stopniu dotyczy krzywizn terenu i ich wplywu na obiekty budowlane. Wskanika, do którego panuj dwa pogldy. Jeden o malej jego przydatnoci do oceny szkodliwoci wplywów eksploatacji górniczej na obiekty budowlane, gdy w wyniku pomiarów terenowych stwierdza si duy rozrzut ­ fluktuacje. Drugi, e wskanik ten ma istotne znaczenie, decyduje o zmianie rozkladu pionowych oddzialywa midzy obiektem a podloem. Zaznaczy naley, e wskanik ten jest trudno sprawdzalny geodezyjnymi pomiarami. Wystpowanie fluktuacji ­ naturalnych rozprosze ­ okrelanych pomiarowo wskaników deformacji tlumaczy si przypadkowym spkaniem górotworu, jego przypowierzchniowej warstwy. Deformacje odcinkowe wyznaczane z wzorów (1), (2), (3) na podstawie pomiarów przemieszcze w, u nie s dokladnymi odpowiednikami wskaników deformacji, które s wynikiem prognozy. Prognozowane wskaniki deformacji T, K, e, popularnie zwane wskanikami punktowymi, liczone s w prognozie na podstawie wzorów na pochodne obnie i przemieszcze poziomych w pewnych punktach obliczeniowych. Teoretycznie, oba sposoby bylyby równowane, gdyby byly wyliczane graniczne ich wartoci przy dlugoci boku l 0. CENTRAL MINING INSTITUTE, PLAC GWARKÓW 1, 40-166, KATOWICE, POLAND W artykule przeanalizowano prawdopodobiestwo fluktuacji przemieszcze dwóch dowolnie bliskich, lecz rónych punktów, czy jest od siebie zalene, czy te nie. Najprostsze teoretyczne modele, jakie analizowano s nastpujce: · Model iglowy: fluktuacje w dwóch dowolnie bliskich, lecz rónych punktach s od siebie niezalene. · Model ziarnisty: orodek ma struktur ziarnist (o rónych rozmiarach ziaren, tak rónorodnie rozmieszczonych, e dowolny punkt (x, y, z) naley zawsze do jakiego ziarna lub ley na granicy ziaren ssiednich. · Model falisty: mona go utworzy z modelu ziarnistego, co daje obraz podobny do lekko sfalowanego morza. W takim modelu mona byloby rozwaa te fluktuacje, jako cigle. Z analizy tej wynika, e najprostszym i poprawnym w sensie matematycznym jest model falisty, w którym wszystkie pochodne typu (26) s okrelone z wyjtkiem by moe pewnych punktów lub krzywych, gdzie mog one by niecigle. W obszarach, w których s one skoczone, podstawow funkcj losow jest fluktuacja obnie w. Fluktuacje nachyle i krzywizn s pochodnymi fluktuacji obnie i s jednoznacznie okrelone przez w. Podobnie jest z poziomymi przemieszczeniami i odksztalceniami, gdzie podstawowymi funkcjami losowymi s skladowe poziomego przemieszczenia, a fluktuacje odksztalce s wyznaczone przez ich pochodne. W rozdziale 4, przyjmujc róne dlugoci l oraz lczne odchylenie standardowe obnienia punktu (31 mm) wynikajce z bldu pomiaru (1 mm) oraz wynikajce z naturalnego rozproszenia (30 mm), obliczono wartoci odchyle standardowych T, K rozproszenia wplywów dla nachyle i krzywizn. Obliczono je dla wartoci ekstremalnych nachyle ±Tmax oraz krzywizn ±Kmax, bdcych skutkiem przykladowej eksploatacji w postaci pólplaszczyzny, na rónych glbokociach, od 0 do 1000 m, oraz wartoci wmax = 1 m i parametru r rozproszenia wplywów r = 300 m. Obliczenia wartoci odchyle standardowych wykonano przyjmujc poziom ufnoci = 0,05. Wykresy zalenoci ksztaltowania si maksymalnego nachylenia, krzywizny i promienia krzywizny przedstawiono odpowiednio na rysunkach 1-3. Nastpnie przy zaloeniu, e warto odchylenia standardowego w jest niezalena od poloenia obiektu wzgldem eksploatacji, obliczono rozklady obnie, nachyle i krzywizn w calym obszarze wplywów eksploatacji od 1,5r do ­1,5r, co przedstawiono odpowiednio na rysunkach 4-6. Z rysunków tych wynika, jak w znacznym zakresie mog fluktuowa (z prawdopodobiestwem 95%) nachylenia, a zwlaszcza krzywizny w stosunku do wartoci rednich, które prognozujemy. W konkluzji stwierdzono, e o rozproszeniu naturalnym wskaników nie wiadomo wszystkiego i moliwe s róne podejcia do opisu tego rozproszenia. Z powodu braku wiarygodnego modelu nie jest moliwe okrelenie odchyle standardowych w przypadku tzw. punktowych deformacji prognozowanych. Dlatego do oceny zagroenia obiektów budowlanych mona rozpatrywa odcinkowe wskaniki deformacji, dla których istniej dobrze okrelone oszacowania (5), (6), (7) odchyle standardowych ich fluktuacji wynikajce z ich uzalenienia od fluktuacji obnie i przemieszcze poziomych. W konsekwencji mona okreli przedzialy ufnoci dla tych fluktuacji dla wszystkich odcinkowych wskaników deformacji. Nawet w przypadku, gdy prognozowane odcinkowe krzywizny maj bardzo male wartoci, to w wyniku uwzgldnienia rozproszenia naturalnego ich wartoci mog by istotnie due. Slowa kluczowe: eksploatacja górnicza, deformacja powierzchni, krzywizny, fluktuacja, prognoza, pomiar 1. Introduction Issues concerning observed random dispersion of deformation indicators have been researched by a number of authors, in chronological order: Batkiewicz, Popiolek, Milewski, Ostrowski, Kwiatek, Stoch and Kowalski. Batkiewicz (1971) reviewed various formulas of centre which caused random dispersion of deformation indicators. These assumptions and formulas were later continued by: (Popiolek 1976, 1996; Popiolek et al., 1997a,b; Ostrowski, 2006; Stoch, 2005; Popiolek & Stoch, 2005). They recognized quantities of the dispersion occurring in practice and suggested coefficients of variation as its measure. Works developed by Kowalski focused on research concerning this phenomenon (Kowalski, 2007) and he included it in the forecast (Kowalski, 2014). Research conducted by Kwiatek considered methods that would include these phenomena in methodology of building protection against mining influences. Summary of this research is included in monographs (Kwiatek et al., 1997) and (Kwiatek, 2007), and publications (Kwiatek, 2006, 2008). They mainly relate to land curvatures and the influence they have on buildings. On one hand, it is thought that such indicator is of a small value while estimating a harmfulness level of mining operations on buildings, because field measurements show a great level of dispersion ­ fluctuations ­ of obtained curvatures (Popiolek et al., 1995). On the other hand, such indicator has a significant value and decides on a change in distribution of vertical impacts between a building and foundation. In the case of buildings that are not resistant enough, the influence of curvatures causes typical damages (Kwiatek, 2007). It is difficult to accurately check the indicator, however, it is indispensable for building construction on mining areas (Kwiatek, 2006; Florkowska, 2011, 2012). In addition to curvatures, this paper discusses all other deformation indicators. Therefore, other deformation indicators have been included in this paper with an emphasize on the issues relating to curvatures. 2. Measured deformation indicators In practice, only differences between measuring point's height and distance Li of subsequent points of measuring line i ­ 1, i1 are directly measured. Sometimes the length, angles and differences in height are measured for a closed network of measuring points that allow leveling their coordinates. In both cases such measurements enable to determine the level of point subsidence wi inclinations Ti of measure bases and curvatures Ki (as relative changes of inclinations of two adjacent bases) at specific times. Additionally, length measurements of measured bases allow determining relative changes of such distances, which is a horizontal deformation. When the points coordinates are determines, it is possible to additionally calculate components of horizontal displacement. The paper analyses indicators that are dependent on subsidence (inclination and curvatures) and are calculated as: Ti Ki wi wi 1 Li Ti Ti 1 Li Li 1 2 (1) (2) and horizontal deformation most often calculated as relative continuation of a section at a time of measurement Li' in relation to its initial length Li (1.3). Li' Li Li (3) or less often, when components of horizontal point displacements ux, uy are counted as: such sections are also referred to as measured bases. xi yi u x i u x i 1 xi if xi 0 (4) u u x i 1 xi yi if yi 0 Determined subsidence and horizontal displacement of measured line points, apart from (minor) errors resulting from a measure technique, include also so-called natural dispersion. Accidental cracked rock mass is considered to be the cause of such dispersion. The following issues have not been researched yet: · whether the dispersion occurs independently of frequency of measurements or whether there are specific areas where the dispersion takes places in a regular (smooth) manner2; · what is the probability distributions of natural dispersion of separate deformation indicators, · whether the parameters of such distributions are the same in different locations of subsidence trough. Values of inclinations, curvatures and horizontal deformations are calculated by the formulas (1)...(4) and are characterised also by natural dispersion of vertical and horizontal displacement. The work (Kowalski 2007) presents detailed introduction of relations between fluctuation standard deviation T, K, of discussed deformation indicators depending on standard deviations w of vertical and horizontal displacements (including measured errors). In case when Li = l (for curvatures Li = Li ­ 1 = l) standard deviation amounts to: T K (5) and (6) show that: 2 w l 6 l2 (5) (6) 2 u l (7) and not 3 T l (8) 2 T as it could be calculated for the formula (2), because in this case random varil ables Ti, Ti ­ 1 are not independent (both are dependent on subsidence wi ­ 1). 2 smooth dispersion would indicate an existence of certain center block that are moved as a whole, and potentially deformed in a regular manner If the coefficient of variation is defined acc. to Popiolek (1976) as: Mw Mu MT MK M is calculated by (9): M T T wmax M w w Tmax M K K wmax M w w K max M umax M u u max Budryk-Knothe theory shows that: wmax r Tmax wmax e r2 K max 2 (11) (10) wmax umax Tmax (9) K max wmax max umax e 2 Therefore, after considering relations (5)...(7) and (11) in (10), the result amounts to: MT 2 Mw MK 1 2 Mw M 1 Mu e 3e (12) where are determined: l r (13) An average outcome of research by Popiolek (1976), Kowalski (2007) and Stoch (2005) shows that: M K 0.46 15.33 M w 0.03 M T 0.11 3.67 M w 0.03 M 0.25 2.78 M u 0.09 and they are not dependent on . It should be noted that section deformations calculated by the formulas (1)...(3) on the basis of displacement measurements w, u are not accurate equivalents of deformation indicators resulting from forecast. Deformation indicators T, K, so called point indicators, are calculated on the basis of formulas for derivatives of vertical and horizontal displacements in certain calculation points. Theoretically, both methods would be equivalent if the formulas (1)...(3) contained subsidence derived from a theoretical M model, and instead of the right sides of formulas, their limit values were calculated as the side length approaches to zero l 0. Then, obtained values would be equal to T, K, values calculated from M model. When it is assumed, however, that the forecast corresponds exactly to smooth (without any fluctuations) measurement, the estimated point deformation indicators are not much different than section indicators in the case of correspondingly small distances l 25 m3. 3. Forecast deformation indicators and their natural dispersion Estimated deformation indicators of continuous rock mass are calculated by formulas such as Budryk-Knothe theory (Prusek & Jdrzejec, 2008) treated as constant component of actual rock mass deformation indicators4, to which statistical fluctuations should be added due to the lack of certainty concerning the theory parameters and statistical fluctuations resulting from natural dispersion of corresponding deformation indicators. When a systematic error of theoretical model is neglected, the following aspects can be stated for indicators dependent on the field of vertical and horizontal displacement. wrz ( x, y, z , t ) w( x, y, z , t , P pr ) w ( x, y, z , t ) w ( x, y, z , t ) rz u x ( x, y, z , t ) u x ( x, y, z , t , P pr ) u x ( x, y, z , t ) u x ( x, y, z , t ) (14) u rz ( x, y, z , t ) u y ( x, y, z , t , P pr ) u y ( x, y, z , t ) u y ( x, y, z , t ) y (15) the greatest differences of point and section inclinations for half-planes when r = 500 m, l = 25 m are 3,8% Tmax, and differences of individual curvatures are 20,6% Kmax 4 which center is complex, partially cracked, partially constant. Txrz ( x, y, z , t ) Tx ( x, y, z , t , P pr ) Tx ( x, y, z , t ) Tx ( x, y, z , t ) Tyrz ( x, y, z , t ) Ty ( x, y, z , t , P pr ) Ty ( x, y, z , t ) Ty ( x, y, z , t ) rz z ( x, y, z , t ) z ( x, y, z , t , P pr ) z ( x, y, z , t ) z ( x, y, z, t ) rz xx ( x, y, z , t ) xx ( x, y, z , t , P pr ) xx ( x, y, z , t ) xx ( x, y, z, t ) rz yy ( x, y, z , t ) yy ( x, y, z , t , P pr ) yy ( x, y, z , t ) yy ( x, y, z , t ) (16) (17) rz K xx ( x, y, z , t ) K xx ( x, y, z , t , P pr ) K xx ( x, y, z , t ) K xx ( x, y, z , t ) rz K xy ( x, y, z , t ) K xy ( x, y, z , t , P pr ) K xy ( x, y, z , t ) K xy ( x, y, z , t ) rz K yy ( x, y, z , t ) K yy ( x, y, z , t , P pr ) K yy ( x, y, z , t ) K yy ( x, y, z , t ) (18) where: wrz, Trz, Krz, rz P = (P1, P2, ... Pm) p pr w, T..., K... actual rock mass deformation indicators, m set of theory parameters, a set of these parameters with values used for forecast, constant rock mass deformation indicator calculated for values of Ppr ­ theory parameters adopted for the forecast, ...(x, y, z, t) -- random functions showing the impact of uncertain theory parameters ­ on individual deformation indicators, ...(x, y, z, t) -- random functions showing the impact of natural dispersion of individual deformation indicators. -- -- -- -- Theory parameters adopted for the forecast are never entirely certain, unless the values are verified by deformation indicators measurement taken after mining operations. Approximation of this parameter conducted acc. to the theory results in parameters values P dok, referred to as accurate. Their values are unknown at the time of the forecast. Shall: Pk Pkdok Pkpr , k 1.2, m (19) (20) P P , P2 , Pm 1 It can be said that ...(x, y, z, t) functions: · would be equal to zero if the theory parameters were certain, · are continuous and their values are ... = (­, +) real numbers, · it can be assumed that the distribution of their probability are constant and their density ... is even-numbered ­ they are symmetrical ... (­...) = ... (...)5, · it can be assumed that distributions of difference probability Pk are normal with zero average values, · their values equal zero. what results from assumption that the probability of deviation is independent from its sign In fact, for instance, for subsidence occurs: w x, y, z , t w x, y, z , t P dok w x, y, z , t P pr Considering (2.5), it can be described as: w x, y, z , t w x, y, z , t P pr P w x, y, z , t P pr (21) (22) For minor difference, Pk the formula (2.8) can be approximated by linear formula: w x, y , z , t k 1 m w x, y , z , t , P Pk P pr Pk 2 2 P k (23) 2 Assuming that errors Pk are not dependent, the variation w can be calculated as: 2 w w x, y , z , t , P x, y , z , t Pk k 1 (24) P pr Generally, the differences Pk can not be recognized as small enough and as a result, the issue remains non-linear: 2 w x, y , z , t E ( 2 ) w P1 Pm [ w( x, y, z , t P pr P) w( x, y, z , t P pr )] P d ( P1 ) d ( Pm ) (25) where Pk is the error range Pk, and P (w) is the distribution density of deviation w probability. If the errors Pk are independent, then: P Pk k 1 where Pk are densities of error distribution Pk. Equalities analogical to (21)..(25) occur for the other deformation indicators. It is noteworthy, that standard deviation is a function of the position ... = ...(x, y, z,t). For established realisation of random variables P functions ... are certain continuous surfaces in the whole range of exploitation influences, whereas it has not been decided whether their fluctuation realisation ... is of a continuous character. Due to the fact that the measurements are taken at locations where the points are located in certain distances, it is unknown how fluctuation occurs in two randomly located points. In particular, it has not been determined if the probability of fluctuation occurring at a point is depended on another randomly located point. The simplest theoretical models that seem to be up-to-date can be characterised by: · Needle Model: fluctuations in two randomly located, but different, points. It means that they have an image of independent `needles' in each measuring point in the whole area of exploitation influences (an image similar to `signal noise'). If the measuring points were located in every point of the location impacted by mining operations (in a form of dense-in-itself mosaic), the image of the functions would represent dense-in-itself `bush of needles' with random amplitudes. · Granular Model: the centre has granular structure (with varied granularity and randomly located so that any point (x, y, z) always belongs to a grain or is located at the border of neighbouring grains, because, if a local empty space was created as a result of such centre deformation, it may be recognized as a separate grain), and fluctuations have forms locally continuous (in the area of grains) surfaces ­ prisms with random heights, with bases as curvilinear outlines of grains and tips in a form of certain continuous surfaces. · Corrugated Model: it may be created from the Granular Model by applying certain continuous surface on continuous surface of fluctuation ... (resulting in lightly wavy sea). The functions in such model could be considered as continuous. Discussion often concerning the problem includes an idea that fluctuations ... are independent of a location in the case of fluctuations probability that would be identical in sides of a subsidence trough as well as its centre. It has not, however, been proved by any research. Other theories would also seem justified, such as that the probability is proportional to the size or maximum size of an adequate deformation indicator. Generally, it is safe to assume that the probability is dependent on the location and such dependency can be determined by appropriate research. The state (set) of fluctuation ... at the t moment must have a form of function dependent on the location in the whole area of exploitation influences, otherwise, they would all have the same value which is contrary to observations. It can be said that ...(x, y, z,t) functions: · the values at t moment belong to a continuous and cohesive set of real numbers ...6, · it can be assumed that the distribution of their probability is constant and their density ... is even-numbered, as a result ...(­...) = ...(...)7, · their average values equal zero. In the case of the Granular Model, the set of function characteristics ...(x, y, z,t) can be extended by a feature ­ during the realisation t in the area of every grain, they are continuous functions of the location and time, and on the grain borders they may be non-continuous. Such feature results from the fact that components of inclination are derivatives of subsidence, components of curvatures are inclination derivatives and the granular centre can be calculated by adequately small .8 Tx x, y, z , t lim w x , y , z , t w x, y , z , t 0 (26) There are certain possible locations of two points A = (x, y, z) and B = (x + , y, z), with small enough: 1. A and B point at inside of the same grain: w (x + ,y,z,t) = w (x + ,y,z,t) + · a(x + ,y,z,t) can be added to the linear approximation what gives Tx = a. there is a probability that it is a set (­, +) what results from assumption that the probability of deviation is independent from its sign so small that the section (x, x + ) would be located near the grain 2. One of the points shows the inside of a grain, and the other points at its border: the value near the grain or infinitely near to the grain may be assigned to the border function and this still can be connected with the first case. 3. Both points show (always, with random small value of ) the borders of neighbouring grains (two or more): the value of counter quotient is undetermined, and a value Tx in that point ­ non-continuous. Similarly, it can be determined that the other derivatives Ty, Kxx, Kxy, Kyy in the area of every grain, also in the granular centre are continuous functions of the location and time, and on the grain borders they may be non-continuous. Let in the area of a random grain located at z height, occur at t moment function w, which, if the granularity is small enough, is formed by following equation (accurate to the decimal place): w x, y q0 q1 x q2 y q3 x 2 q4 xy q5 y 2 (27) where the coordinate system xy moved in such a way, that coordinates x = 0, y = 0 point at the centre of a grain. Coefficients q0, q1, ... q5 are random variables and their realisation at t moment is determined by the continuous function (27). Because a grain is subject to forces moving from neighbouring grains, some or all coefficients qi may be dependent on each other9. Therefore: w q1 2q3 x q4 y x Ty x, y w q2 q4 x 2q5 y y Tx x, y (28) K xx x, y K xy x, y K yy x, y Tx x Tx y Ty y 2q3 Ty x q4 (29) 2q5 Similar relations can be calculated for displacement components and horizontal deformations. If a grain is relatively big in relation to the size of measuring base, but small enough to allow measuring point to belong to another grains, local (in the area of the grain) inclinations and curvatures may significantly differ from the ones calculated by formulas (1) and (2). It seems, however, that cracked sections of the centre are moving and turning, but their local curvatures can be omitted. As a result, it is justified to expect that the effects would be registered by properly frequent measurement. Contrary to this, results of measurements conducted at Kazimierz-Juliusz Mine are presented by Kowalski (2007) indicate that for section l = 5 m the curvatures experience is noticeably greater for instance, if the grain is stiff enough, so that it cannot bend, it may be assumed that coefficients q3,q4,q5 dissaper. dispersion than in the case longer measurement bases. This would indicate that the grains are smaller than 5 m (for example few centimetres) or actual rock mass (at least in the surface layer) cannot be determined by the Granular Model. The Needle Model does not include spatial fluctuation derivatives as functions of (26), and fluctuations of inclinations and curvatures should be considered as entirely independent random functions. The easiest mathematical model is the Corrugated Model, in which all derivatives (26) are determined; in exception of some point or curvatures which may be non-continuous. In the areas where they are non-continuous, the main random function is subsidence fluctuation w. Inclination and curvatures fluctuations are derivatives of subsidence fluctuations and are explicitly determined by w. It is the same in the case of horizontal displacement and deformations, where the basic random functions are components of horizontal displacement, and deformation fluctuations are determined by their derivatives. 4. Probabilistic forecast of deformation indicators Formulas (14)...(18) can be formulated as: D rz x, y, z , t D x, y, z , t P pr D x, y, z , t D x, y, z , t (30) where D is any deformation indicator from among subsidence, components of horizontal displacement, inclinations and curvatures, as well as vertical deformations and components of horizontal deformations. A forecasts error of D indicator is marked by: D D rz x, y, z , t D x, y, z , t P pr and its variation is marked by: 2 2 E D D (31) (32) If ranges of random variables D, D amount to D, D and distribution densities are D(D), D(D) and because of random variables D, D are independent, the density of probability that variables take certain values of realization is: D D D (33) As a result of (30) and (31), due to the fact that random variables D, D are independent, the following occurs: 2 2 2 D D D (34) it results for the following calculation: 2 E (D ) 2 ( 2 2 D D D ) D D d D d D D 2 D d D D D d D 2 D D d D D 0 D D d D (35) D 0 D d D 2 2 2 D D d D D D Range of confidence for D deformation indicator at confidence level of , is determined as follows: Shall Z occurrence be based on: q1D q2 D D rz D (36) where: q2 D ( ) P( Z ) q1 D ( ) D (D )d D 1 (37) q1 D ( ) additionally P(D q1D ( )) D (D )d D When the distribution of probability D (D) is an even function sD (­D) = D (D), which occurs when densities D, D are even functions (which was the subject of assumptions), then as a result: q1D q2 D qD and P( D rz D qD ) q D (38) q D D D d D 1 (39) Assuming, that distributions D, D are normal, the distribution D will be normal with an 2 average equal to zero and variations D determined by the function (34). As a result, 1 ­ ­ fractiles are: q D D where () are independent of deformation indicator; their values presented in Table 1. TABLE 1 The value of indicator (), depending on quantiles and 1 ­ 1­ 0.999 0.997300 0.99 0.98 0.975 0.954500 0.95 0.9 0.8 0.75 0.682689 0.5 () 3.290528 3 2.575830 2.326348 2.241402 2 1.959964 1.644854 1.281552 1.150350 1 0.674490 5. Inclination and curvatures of building foundation ­ forecast and measurement An important indicator of deformation for building designers is the curvature of building foundation. In the case of deeper mining excavations, particularly parameters r of influences dispersion, the values of forecast curvatures are small. Consequently, does it mean that measured curvatures are in such cases also small? Authors do not have any measurement of curvatures of such large parameter r that forecast values of maximum curvatures would, for instance, almost equal to values for the zero category of mining areas (in Poland), while the observed ones ­ would be significantly greater. However, as a result of an analysis of a number of mining operations, Kowalski (2014) stated that generally forecasts of extreme curvatures are systematically too small (50% on average). Natural, random subsidence fluctuations may be the cause. The following test has been conducted in order to demonstrate possible influence on observed curvatures. Taking various lengths l of building's foundation (excluding dilatation) and total statistical deviation of point subsidence (31 mm) resulting from a measurement error (1 mm) and natural dispersion (30 mm), calculated the value of standard deviations T, K of influences dispersion for inclination and curvatures. The results are presented in Table 2, which contains emphasized values of typical length l = 25 m. TABLE 2 Calculated values of standard deviations for inclinations and curvatures, depending on the length of measured base l, m T, mm/m K, km­1 At the confidence level = 0.05 or the typical length l confidence ranges were calculated, assuming that parameters wmax, r are correct and free of any errors: 1. for inclinations 1.959964 · 1.75 mm/m = 3.43 mm/m, 2. for curvatures 1.959964 · 0.12 km­1 = 0.2352 km­1. Extreme values of inclinations and curvatures will occur for model exploitation in a form of a half-plane located in known areas: ±Tmax and ±Kmax. Scheme of relation10 for such exploitation carried out at different depths characterised by r parameter of influences dispersion and with assumed value wmax = 1 m. · for inclinations Tmax(r), Tmax(r) + 3.43 mm/m and Tmax(r) ­ 3.43 mm/m (Fig. 1), · for curvatures Kmax(r), Kmax(r) + 0.2352 km­1 and Kmax(r) ­ 0.2352 km­1 (Fig. 2). The charts distinct an example value r = 300 m. Maximum curvatures of such value r are minor (curvature radius = 59.20 km ­ the zero category of mining area), however, limitations of confidence range become more significant for buildings protection (curvature radius = ­4.52 km, 3.92 km ­ 4th and 5th category of mining terrain). Tmax(r) Tmax(r)+1,96 Tmax(r) -1,96 6,77 mm/m 3,33 mm/m -0,10 mm/m T T T T(r), mm/m =1,75 mm/m -2 Fig. 1. Chart of the relation of maximum inclination from the parameter r of influences dispersion. Marked confidence range includes actual value with probability 1 ­ = 0.95 and an example section for r = 300 m Positive and negative signs for inclination and curvatures were assumed according to rules using in building industry Kmax(r) Kmax(r)+ Kmax(r)=0,12 km-1 K(r), km 0.5 0.4 0.3 0.2 0.1 (59,20 km)-1 0 -0.1 -0.2 0 100 200 (-4,52 km)-1 (3,92 km)-1 Fig. 2. Chart of the relation of maximum curvatures from the parameter r of influences dispersion. Marked confidence range includes actual value with probability 1 ­ = 0.95 and an example section for r = 300 m 70 60 50 40 30 20 10 3,92 km 0 -4,52 km -10 -20 -30 0 100 200 300 400 59,20 km 1/Kmax(r) 1/[Kmax(r)+1,96 1/K(r), km ] 1/[Kmax(r) -1,96 K] =0,12 km-1 Fig. 3. Chart of the relation of minimum radius for curvatures from the parameter r of influences dispersion. Marked confidence range includes actual value with probability 1 ­ = 0.95 and an example section for r = 300 m -100 0 100 200 300 400 500 600 700 800 900 1000 1100 w()+1,96 w() w()- 1,96 w(), mm =31 mm deposit zo e -1.5 -1 -0.5 0 0.5 1 1.5 Fig. 4. Chart of subsidence depending on . Marked confidence range includes actual value with probability 1 ­ = 0.95 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 T(), mm/m T()+1,96 =1,75 mm/m deposit zo e T() T()- 1,96 -1.5 -1 -0.5 Fig. 5. Chart of inclination depending on . Marked confidence range includes actual value with probability 1 ­ = 0.95 0.3 (4,52 km)-1 0.2 0.1 (59,20 km) 0 -0.1 -0.2 (-3,92 km)-1 -0.3 zo e deposit -1.5 -1 -0.5 0 0.5 1 1.5 (59,20 km)-1 -1 (3,92 km)-1 Kxx()+1,96 Kxx() Kxx()- 1,96 Kxx(), km-1 =0,12 km-1 (-4,52 km)-1 Fig. 6. Chart of curvatures depending on . Marked confidence range includes actual value with probability 1 ­ = 0.95 Left borders of confidence ranges become negative values, what can be explained with an idea that their value reverses in places of extreme values ­ minimum value appears instead of maximum and inversely. Figures 2 and 3 show the relation between curvatures (curvatures radius) and parameter of influences dispersion. When assuming, that standard deviation value w is independent of the building location in relation to mining workings, T, K ill also be independent of such location. Therefore, calculated confidence range applies to the whole exploitation influences area and it can be marked on charts of deformation indicators at T, K (Fig. 4, 5, 6). 6. Conclusion In order to prepare probabilistic forecast of deformation indicators, especially for curvatures of building's foundations, it is necessary to determine standard, random fluctuation deviations that define the uncertainty of theory parameters for these indicators and standard, random fluctuation deviations resulting from natural dispersion of such indicators. The knowledge of natural indicators dispersion is limited. This paper points out various possible methods towards the dispersion phenomenon. However, due to the lack of its credible model, it is currently not possible to determine such standard deviations in the case of so-called forecast point deformations. Therefore, risk assessment of buildings may include section deformation indicators for which the elements of foundation constructions should serve as the length of sections. There are properly determined calculations (5), (6), (7) for standard deviations of their fluctuation result- ing from their dependence on fluctuation of vertical and horizontal displacements. They can be successfully used for section indicators. As a result, confidence ranges can be determined for such fluctuation for all section deformation indicators. Even if the forecast point curvatures are small, their values may be greater because of natural dispersion.

Journal

Archives of Mining Sciencesde Gruyter

Published: Jun 1, 2015

References