Archives of Mining Sciences 51, Issue 4 (2006) 503–528

Archives of Mining Sciences 51, Issue 4 (2006) 503–528
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DANUTA DOMAŃSKA*, ANDRZEJ WICHUR**
THE RESEARCHES ON SLOPE STABILITY EVALUATION
WITH INCLINOMETRIC MEASUREMENTS
BADANIA NAD OCENĄ STATECZNOŚCI SKARP
Z WYKORZYSTANIEM POMIARÓW INKLINOMETRYCZNYCH
Slopes are encountered either as naturally shaped or artificially molded terrain forms (as elements
of various structures, e.g. tanks, trenches, heaps, embankments, earth banks); moreover, they are often
present in surface mining.
Slope stability analysis is usually conducted using the factor (degree) of slope safety or the factor
(coefficient) of slope stability. Classic calculation methods, e.g. Fellenius’, Taylor’s, Bishop’s, Janbu’s have
introduced the notion of slope safety factor in this field, understood as the ratio of the limit value of the
force required to induce displacement of the considered body of earth (stabilizing force) to the value of the
acting sliding force (destabilizing force) (or the ratio of the moment causing its turn to the actual turning
moment). The term of safety factor also functions in contemporary computer programs (e.g. in Z_Soil as
Safety Factor SF), while its estimate is assessed during the generation of the medium’s limit state through
the fictional reduction of its strength parameters, dismissing the arbitrary assumption of the slide surface
shape. The definition also corresponds to the notion of slope stability factor or stability coefficient. The
term of slope stability factor is incorporated in this paper and marked with the symbol SF.
This work presents the concept of ground media stability evaluation drawn on the basis of horizontal
displacement value analysis, measured with the use of inclinometers in the conditions of the „Bełchatów”
brown coal pit. Action in this field was undertaken in two fundamental directions:
• on the basis of the proposed reduction of the soil elasticity modulus value (ch. 2),
• on the basis of the conducted analysis of the state of soil effort in the inclinometer hole axis and
its relation to slope stability (ch. 3).
Both approaches incorporate computer calculations of slope stability factor made with the Z_Soil
program.
In order to evolve the slope stability evaluation method on the basis of inclinometric measurements
with the use of computer calculations (ch. 2), the conditions of placement and the „in situ” results obtained from ten exploratory bore-holes were analyzed. The estimation procedure of allowable horizontal
soil displacements was closely connected with the methodology of numerical determination of the slope
stability factor, defined during the analysis of the stress and strain state in the modeled soil shield.
The stability assessment of the earthen structures was conducted according to the following outline:
• creating the computer models of slopes representing the inclinometers’ working conditions in the
considered moments, which is exemplified by Fig. 1 (in relation to all models notations containing
*
**
D.SC.TECH., MIN.ENG. – MINING MECHANIZATION CENTRE KOMAG, GLIWICE
FULL PROF., D.HAB.SC.TECH., MIN.ENG. – AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY, CRACOW
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•
•
•
•
the time point symbol were implemented, while the symbol served solely an ordinal function
and did not denote any physical dimension, that is it did not reflect any time intervals between
particular measurements),
determining the slope stability factor SF in the aforementioned moments with a computer program,
calculation of horizontal displacements along the lines mapping the location of the inclinometer
hole axes with a computer program,
estimating horizontal displacement critical values upon the reaching of which the slope instability
may occur,
comparing the displacements registered in the inclinometer holes with their critical values and
the evaluation of slope stability on the basis of the formulated criterion.
While determining the slope stability factor SF with the computer method (the Z_Soil program in
particular), it is assumed that the hypothetical slope stability failure will ensue at the reduced values of
soil parameters c/SF and tanΦ/SF, not considering the reduction of the elasticity modulus (researches
show that this value does not influence the SF value). When making this assumption, it is essential to
be aware of the fact that if such a state of slope stability failure did occur, then reducing the cohesion
value and the shearing resistance angle tangent of the soil would inevitably lead to the decrease of the
elasticity modulus value, and the vectors of soil displacements in the moment directly preceding the slope
stability failure would be much larger than those specified for the input values (i.e. for SF = 1.0). Owing
to this, the reduction of the elasticity modulus value based on the correlatives between cohesion, angle of
shearing resistance, soil elasticity modulus and the liquidity index of cohesive soils or the density index
of noncohesive soils was proposed for this task. It was assumed that the introduction of the factor SF,
reducing the values c and tanΦ, matches the simultaneous decrease of the soil elasticity modulus value
E by a certain factor NE, in the consequence of which soil strains, particularly horizontal displacements,
increase. A physical interpretation of this simultaneous reduction could be the stipulated plastification of
the cohesive soil (the change of the liquidity index from the assumed initial value IL0 to the final IL) or the
decrease of the noncohesive soil density index (from the initial value ID0 to the final ID).
In the aim of estimating the value of the factor NE reducing the soil elasticity modulus E (a general
term, related to the modulus of linear deformation, as well as to oedometer modulus), the standard
(PN-81/B-03020) served as a basis. The proposed reductive factor of soil elasticity NE was represented
by the formula (3), and as a result of the analysis of the obtained relations, aiming at safety, horizontal
displacements estimated on the grounds of the relations (4) were acknowledged as critical. For research
purposes, it was assumed that in a given moment of time, characterized by a specific geometry of the
system, the slope is stable when the horizontal displacement recorded by the inclinometer located within
it does not exceed the critical value expressed by the formula (5). The obtained dependences were applied
in the area of an example inclinometer hole of slope geometry at the time of inclinometer placement (t = 1)
and of geotechnical parameters of soils specified in Fig. 1, also assuming that cohesive soils at the initial
state IL0 = 0.5 (Fig. 3) are dealt with.
In ch. 3 a diagram of a slope with an inclinometer hole as in Fig. 4 was considered. The further
assumption was that the slope material has the unit weight γ and is elastic-plastic at the limit condition
described with the dependence (7) (the Coulomb-Mohr criterion). In the first attempt Levy’s solution of the
plane problem of the elasticity theory to the infinite wedge loaded with the dead weight and the hydrostatic
pressure of fluid (Fig. 5) was applied. The solution achieved in this way has a disadvantageous property:
for x → –∞ one obtains σx → –∞; which explains the possibility of its use solely in dam calculations. This
inconvenience could be bypassed by assuming the application of Levy’s solution only to the right side of the
slope (i.e. for x > 0 and α = 0), and for the left side of the slope (i.e. for x ≤ 0) the solution for the primary
stress state in the rock mass could be used (see Fig. 4) (the formulae 18-21). After making appropriate
calculations (with the assumption of the boundary condition: for y = H and x = x0 the horizontal displacement
equals zero ux = 0) the formula (29) was obtained for the horizontal displacement in the hole axis. Zeroing
the horizontal shift value ux stands in contradiction with the results of inclinometric measurements, therefore
this model may not be used in this case as is. In the physical perspective it is clear, since the presence of
the slope edge near the inclinometer hole distorts the primary undisturbed stress state around that hole. It
seems obvious that the nearby location of the slope edge will cause the appearance of shearing stresses
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τxy, which had not occurred in the previously analyzed model. Thus the simplest correction of this model
is the introduction of the shearing stress τxy of a constant value in the vicinity of the inclinometer hole (the
formula 30). At that time the stress state will be given by the formulae (18), (19), (21) and the formula
(30). In this case the horizontal displacements in the hole axis will be represented by the formula (31),
and the horizontal displacement of the inclinometer hole head (x = x0, y = 0) – by the formula (32). The
linear dependence ux along with the depth corresponds to relations occurring in reality (see Fig. 3), and
the quantity t found in the formulae (31) and (32) facilitates their „calibration” in real conditions. Indeed,
the measurement results of horizontal displacements in the inclinometer hole axis could be presented in
the form of a linear function, for instance using the linear regression apparatus, then deriving the u0 value
from this equation, and calculating the value t from the converted formula (32). By marking this value
as t0 the formula (33) is obtained. The value determined in this manner could be implemented in further
calculations, mainly those aiming at the evaluation of the rock mass effort in the inclinometer hole axis,
which will be used for slope monitoring. The expression (34) was employed in the further considerations
as an effort measure. After substituting the stress tensor components, one arrives at the function of effort
in the inclinometer hole axis – the formula (38), and its value on the surface of the ground (i.e. for y = 0)
– the formula (39). After making appropriate calculations (the formulae 40-45) one ultimately arrives at
the formula (46) for the critical displacement value of the inclinometer hole head in relation to the hole
bottom, corresponding to the slope stability failure. Its application is explained on the example of an
inclinometer located in the slope area (acc. to Fig. 1), the stability of which was estimated in ch. 2 on the
basis of the method of reducing the soil elasticity modulus value (Fig. 3). The horizontal displacement
critical values obtained in this way will be reliable in slope stability evaluation on the basis of inclinometric
measurements. In the practical application of the method, it is recommended on the grounds of safety to
reduce that value – in the case of the lack of other premises, the principles stated in the technical rules
(Rozporządzenie 1996) or applied abroad (Gunaratne et al. 2006) should be assumed.
Keywords: slopes’ stability, inclinometric measurements, stability evaluation
Skarpy są spotykane w postaci naturalnej formy ukształtowania terenu, bądź mogą być formowane
w sposób sztuczny (jako elementy różnych budowli np. zbiorników, wykopów, hałd, nasypów, wałów
ziemnych); często występują również w górnictwie odkrywkowym.
Oceny stateczności skarp dokonuje się zwykle przy użyciu współczynnika (stopnia) pewności (bezpieczeństwa) skarpy lub współczynnika (wskaźnika) stateczności skarpy. Klasyczne metody obliczeniowe,
np. Felleniusa, Taylora, Bishopa, Janbu wprowadziły w tym zakresie pojęcie współczynnika pewności
(bezpieczeństwa), rozumianego jako stosunek wartości granicznej siły potrzebnej do wywołania przesuwu rozpatrywanej bryły gruntu do wartości działającej siły zsuwającej (lub momentu powodującego jej
obrót do rzeczywistego momentu obracającego). Określenie współczynnik bezpieczeństwa funkcjonuje
również obecnie w programach komputerowych (np. w Z_Soil jako Safety Factor SF), przy czym jego
szacowania dokonuje się w trakcie generacji stanu granicznego ośrodka w wyniku fikcyjnego zmniejszenia
jego parametrów wytrzymałościowych, przy rezygnacji z arbitralnego założenia kształtu powierzchni
poślizgu. Przedstawionej definicji odpowiada także pojęcie wskaźnik stateczności zbocza, czy też współczynnik stateczności. W niniejszej pracy posłużono się określeniem współczynnik stateczności skarpy
i oznaczono go symbolem SF.
W pracy przedstawiono koncepcję oceny stateczności ośrodka gruntowego opracowaną na podstawie
analizy wartości przemieszczeń poziomych mierzonych za pośrednictwem inklinometrów w warunkach
kopalni odkrywkowej „Bełchatów”. Działania w tym zakresie zostały poprowadzone w dwóch zasadniczych kierunkach:
• w oparciu o zaproponowaną metodę redukcji wartości współczynnika sprężystości gruntu (p. 2),
• w oparciu o przeprowadzoną analizę stanu wytężenia górotworu w osi otworu inklinometrycznego
i jego związku ze statecznością skarpy (p. 3).
W obydwóch podejściach wykorzystano obliczenia komputerowe współczynnika stateczności skarpy
wykonane z użyciem programu Z_Soil.
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W celu opracowania metody oceny stateczności skarpy na podstawie pomiarów inklinometrycznych
z wykorzystaniem obliczeń komputerowych (p. 2), przeanalizowano warunki zabudowy oraz wyniki
„in situ” uzyskane z dziesięciu otworów badawczych. Sposób szacowania dopuszczalnych przemieszczeń poziomych gruntu został ściśle związany z metodyką numerycznego wyznaczenia współczynnika
stateczności zbocza, określanego w trakcie analizy stanu naprężenia i odkształcenia w zamodelowanej
tarczy górotworu.
Sprawdzenie stateczności rozpatrywanych budowli ziemnych przebiegało według następującego
schematu:
• wykonanie modeli komputerowych skarp odwzorowujących warunki pracy inklinometrów
w rozpatrywanych chwilach czasowych, co przykładowo przedstawia rys. 1 (w odniesieniu do
wszystkich modeli zastosowano oznaczenia zawierające symbol punktu czasowego, przy czym
pełnił on wyłącznie funkcję porządkową i nie posiadał żadnego wymiaru fizycznego, tzn. nie
świadczył o odstępie czasowym pomiędzy poszczególnymi pomiarami),
• wyznaczenie za pośrednictwem programu komputerowego w ww. chwilach czasowych współczynnika stateczności skarpy SF,
• określenie przy pomocy programu komputerowego przemieszczeń poziomych wzdłuż linii odwzorowujących położenie osi otworów inklinometrycznych,
• oszacowanie krytycznych wartości przemieszczeń poziomych, po osiągnięciu których może dojść
do utraty stateczności zbocza,
• porównanie przemieszczeń rejestrowanych w otworach inklinometrycznych z ich wartościami
krytycznymi oraz ocena stateczności skarpy na podstawie sformułowanego kryterium.
Przy wyznaczaniu współczynnika stateczności skarpy SF metodą komputerową (w szczególności
przy użyciu programu Z_Soil) zakłada się, że hipotetyczna utrata stateczności skarpy zaistnieje przy
zredukowanych wartościach parametrów gruntu c/SF oraz tgΦ/SF, nie uwzględniając przy tym redukcji
wartości współczynnika sprężystości (badania wykazują, że wartość ta nie wpływa na kształtowanie się
wartości SF). Przyjmując to założenie, należy zdawać sobie sprawę z faktu, że gdyby rzeczywiście taki
stan utraty stateczności skarpy zaistniał, to obniżenie wartości spójności i tangensa kąta tarcia wewnętrznego gruntu musiałoby pociągnąć za sobą obniżenie wartości współczynnika sprężystości tego gruntu,
a wektory przemieszczeń gruntu w chwili bezpośrednio przed utratą stateczności skarpy byłyby znacznie
większe od tychże określonych dla wartości wejściowych (tj. dla SF = 1,0). W związku z tym dla potrzeb rozwiązywanego zadania zaproponowano redukcję wartości współczynnika sprężystości w oparciu
o związki korelacyjne spójności, kąta tarcia wewnętrznego i współczynnika sprężystości gruntów ze
stopniem plastyczności gruntów spoistych lub stopniem zagęszczenia gruntów niespoistych. Przyjęto, że
wprowadzenie współczynnika SF, redukującego wartości c i tgΦ, odpowiada równoczesnemu zmniejszeniu wartości modułu sprężystości gruntu E o pewien współczynnik NE, czego konsekwencją jest wzrost
odkształceń gruntu, w szczególności przemieszczeń poziomych. Interpretacją fizyczną tej jednoczesnej
redukcji może być umowne uplastycznienie gruntu spoistego (zmiana stopnia plastyczności od przyjętej
wartości początkowej IL0 do końcowej IL) lub zmniejszenie stopnia zagęszczenia gruntu niespoistego (od
wartości początkowej ID0 do końcowej ID).
W celu oszacowania wartości współczynnika NE zmniejszającego moduł sprężystości gruntu E
(określenie ogólne, odniesione zarówno do modułu pierwotnego i wtórnego odkształcenia gruntu jak
i do edometrycznych modułów ściśliwości) oparto się na normie (PN-81/B-03020). Zaproponowany
współczynnik redukcyjny sprężystości gruntu NE wyrażono wzorem (3), a w wyniku analizy uzyskanych
zależności, idąc w kierunku bezpieczeństwa, za krytyczne uznano przemieszczenia poziome oszacowane
na podstawie zależności (4). Przyjęto dalej dla celów badawczych, że w danej chwili czasowej, charakteryzującej się konkretną geometrią układu, skarpa jest stateczna, gdy przemieszczenie poziome rejestrowane
przez zabudowany tam inklinometr nie przekracza wartości krytycznej określonej wzorem (5). Uzyskane
zależności zastosowano w rejonie przykładowo wybranego otworu inklinometrycznego o geometrii skarpy
w chwili zabudowy inklinometru (t = 1) i parametrach geotechnicznych gruntów określonych na rys. 1
oraz przy założeniu, że występują grunty spoiste o stanie początkowym IL0 = 0,5 (rys. 3).
W p. 3 rozważono schemat skarpy z otworem inklinometrycznym jak na rys. 4. Założono, że materiał
skarpy posiada ciężar objętościowy γ oraz jest sprężysto-plastyczny o warunku granicznym opisanym
zależnością (7) (model Coulomba-Mohra). W pierwszym podejściu zastosowano rozwiązanie Levy’ego
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płaskiego zadania teorii sprężystości dla nieskończonego klina obciążonego ciężarem własnym oraz
ciśnieniem hydrostatycznym cieczy (rys. 5). Uzyskane w ten sposób rozwiązanie posiada niekorzystną
właściwość: dla x → –∞ uzyskuje się σx → –∞; tłumaczy to możliwość zastosowania go jedynie w obliczeniach zapór. Niedogodność tę można ominąć, zakładając wykorzystanie rozwiązania Levy’ego jedynie
dla prawej części skarpy (tzn. dla x > 0 oraz α = 0), natomiast dla lewej części skarpy (tzn. dla x ≤ 0)
można wykorzystać rozwiązanie dla pierwotnego stanu naprężenia w górotworze (por. rys. 4) (wzory
18–21). Po wykonaniu odpowiednich obliczeń (przy założeniu warunku brzegowego: dla y = H oraz x = x0
przemieszczenie poziome jest równe zeru ux = 0) otrzymano wzór (29) na przemieszczenie poziome w osi
otworu. Zerowanie się wartości przemieszczenia poziomego ux stoi w sprzeczności z wynikami pomiarów
inklinometrycznych, a zatem model ten nie może w takiej postaci być w tym przypadku zastosowany.
Z punktu widzenia fizycznego jest to oczywiste, gdyż obecność krawędzi skarpy w pobliżu otworu inklinometrycznego zniekształca pierwotny nienaruszony stan naprężenia wokół tego otworu. Wydaje się
również oczywiste, że pobliska lokalizacja krawędzi skarpy spowoduje powstanie naprężeń stycznych
τxy, które nie występowały w analizowanym uprzednio modelu. Najprostszą korektą tego modelu jest
zatem wprowadzenie naprężenia stycznego τxy o wartości stałej w pobliżu otworu inklinometrycznego
(wzór 30). Wówczas stan naprężenia będzie dany wzorami (18), (19), (21) oraz wzorem (30). W tym
przypadku przemieszczenia poziome w osi otworu wyrażą się wzorem (31), a przemieszczenie poziome
głowicy otworu inklinometrycznego (x = x0, y = 0) – wzorem (32). Liniowa zależność przemieszczenia
ux wraz z głębokością odpowiada występującym w rzeczywistości relacjom (por. rys. 3), a znajdująca się
we wzorach (31) i (32) wielkość t ułatwia ich „kalibrację” w rzeczywistych warunkach. Rzeczywiście,
można wyniki pomiarów przemieszczeń w osi otworu inklinometrycznego przedstawić w formie funkcji
liniowej, stosując np. aparat rachunku regresji liniowej, następnie z równania tego obliczyć wartość u0
i z przekształconego wzoru (32) obliczyć wartość t. Oznaczając tę wartość przez t0 otrzymuje się wzór
(33). Wyznaczona w ten sposób wartość może być użyta w dalszych obliczeniach, przede wszystkim
mających na celu oszacowanie wytężenia górotworu w osi otworu inklinometrycznego, które posłuży do
monitoringu skarpy. Jako miarę wytężenia przyjęto w dalszych rozważaniach wyrażenie (34). Po podstawieniu składowych tensora naprężenia otrzymuje się funkcję wytężenia w osi otworu inklinometrycznego
– wzór (38), a jego wartość na powierzchni terenu (tj. dla y = 0) – wzór (39). Po wykonaniu odpowiednich
przekształceń (wzory 40-45) otrzymuje się ostatecznie wzór (46) na krytyczną wartość przemieszczenia
głowicy otworu inklinometrycznego w stosunku do dna otworu, odpowiadającą utracie stateczności skarpy.
Jego zastosowanie wyjaśniono na przykładzie odniesionym do inklinometru zabudowanego w rejonie
zbocza (wg rys. 1), którego stateczność została oszacowana w p. 2 w oparciu o metodę redukcji wartości
współczynnika sprężystości gruntu (rys. 3). Uzyskane w ten sposób krytyczne wartości przemieszczenia
poziomego będą miarodajne przy ocenie stateczności skarpy (zbocza) w oparciu o wyniki pomiarów
inklinometrycznych. Przy praktycznym zastosowaniu metody zaleca się, ze względów bezpieczeństwa,
zmniejszenie tej wartości – w przypadku braku innych przesłanek należy przyjąć zasady ujęte w warunkach
technicznych (Rozporządzenie, 1996) lub stosowane za granicą (Gunaratne et al., 2006).
Słowa kluczowe: stateczność skarp, pomiary inklinometryczne, ocena stateczności
1. Introduction
Slopes are encountered either as naturally shaped or artificially molded terrain forms
(as elements of various structures, e.g. tanks, trenches, heaps, embankments, earth banks);
moreover, they are often present in surface mining.
Since slope stability failure may lead to tragic consequences for the human population
and cause major financial problems, the need for evaluating its mechanics conditions,
both during the designing process (stability prognosis) and its exploitation (constant
monitoring).
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Slope stability analysis is usually conducted using the factor (degree) of slope safety
or the factor (coefficient) of slope stability. Classic calculation methods, e.g. Fellenius’,
Taylor’s, Bishop’s, Janbu’s (see Wiłun, 1982; Dembicki et Tejchman, 1974; Jeske et al.,
1966; Piętkowski et Czarnota-Bojarski, 1964; Lambe et Whitman, 1978) have introduced the conception of slope safety factor in this field, understood as the ratio of the
limit value of the force required to induce displacement of the considered body of earth
(stabilizing force) to the value of the acting sliding force (destabilizing force) (or the
ratio of the moment causing its turn to the actual turning moment).
The term of safety factor also functions in contemporary computer programs (e.g.
in Z_Soil as Safety Factor SF (Z_Soil. PC 2003), see (Griffiths et Lane, 1999), while
its estimate is assessed during the generation of the medium’s limit state through the
fictional reduction of its strength parameters, dismissing the arbitrary assumption of
the slide surface shape. The definition also corresponds to the notion of slope stability
factor (see Cała et al., 2004a, b), or stability coefficient (see Thiel, 1980; Cała et al.,
2004a, b), while the latter term had appeared in an earlier work (Rossiński et al., 1963).
In the paper (Chudek et al., 2003) the terms of stability factor and safety factor are used
interchangeably.
The term of slope stability factor is incorporated in this paper and marked with the
symbol SF.
The importance of slope safety for the human life and economy ensures the constant
control (monitoring) of these objects, conducted with inclinometric measurements and
other means (see Wolski, 2001). The application of inclinometric measurements to
evaluate slope stability allows for the dismissal of numerous simplifications utilized in
calculations, as well as for taking into consideration the influence on the slope mechanics exerted by all external agents, stemming from the geological and hydrogeological
situation and strip pit mining, among other things.
In order to provide full implementation of the measurement readings, their correct
interpretation is essential, especially as regards defining the critical value of the measured
parameter, which constitutes the basis of slope stability evaluation.
This work presents the concept of ground media stability evaluation drawn on the
basis of horizontal displacement value analysis, measured with the use of inclinometers
in the conditions of the „Bełchatów” brown coal pit.
Action in this field was undertaken in two fundamental directions:
• on the basis of the proposed reduction of the soil elasticity modulus value (ch. 2),
• on the basis of the conducted analysis of the state of soil effort in the inclinometer
hole axis and its relation to slope stability (ch. 3).
Both approaches incorporate computer calculations of slope stability factor made
with the Z_Soil program.
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2. Slope stability evaluation based on the method of reduction
of the soil elasticity modulus value
In order to evolve the slope stability evaluation method on the basis of inclinometric
measurements with the use of computer calculations, the conditions of placement and the
„in situ” results obtained from ten exploratory bore-holes were analyzed. The estimation
procedure of allowable horizontal soil displacements was closely connected with the
methodology of numerical determination of the slope stability factor, defined during the
analysis of the stress and strain state in the modeled soil shield.
The stability assessment of the earthen structures in concern was conducted according
to the following outline:
• creating the computer models of slopes representing the inclinometers’ working
conditions in the considered moments, which is exemplified by Fig. 1 (in relation to
all models notations containing the time point symbol were implemented, while the
0.0
200.0
400.0
600.0
1
2
3
inclinometric hole
200.0
0.0
-200.0
FE MESH
t = 1.0 [day]
Z_SOIL.PC 2003 v.6.24 Professional
License No: OBR04102001V6D2
Fig. 1. The way of slope modeling in the area of a chosen inclinometer hole t = 1 – location conditions
Medium no. 1: an argillaceous-sandy Quaternary complex: E = 150 MPa, ν = 0.3, γ = 20 kN/m3,
cu = 90 kPa, φu = 13°; Medium no. 2: a subcarbonaceous Tertiary complex: E = 250 MPa, ν = 0.3,
γ = 20 kN/m3, cu = 80 kPa, φu = 13°; Medium no. 3: a Jurassic-Cretaceous bed rock: E = 100 MPa,
ν = 0.3, γ = 16.8 kN/m3, cu = 100 kPa, φu = 30°.
Rys. 1. Sposób zamodelowania zbocza w rejonie wybranego otworu inklinomertycznego
t = 1 – warunki zabudowy
ośrodek nr 1: kompleks ilasto-piaszczysty czwartorzędu: E = 150 MPa, ν = 0.3, γ = 20 kN/m3,
cu = 90 kPa, φu = 13°; ośrodek nr 2: kompleks podwęglowy trzeciorzędu: E = 250 MPa, ν = 0.3,
γ = 20 kN/m3, cu = 80 kPa, φu = 13°; ośrodek nr 3: podłoże jurajsko-kredowe: E = 100 MPa,
ν = 0.3, γ = 16.8 kN/m3, cu = 100 kPa, φu = 30°.
510
•
•
•
•
symbol served solely an ordinal function and did not denote any physical dimension,
that is it did not reflect any time intervals between particular measurements),
determining the slope stability factor SF in the aforementioned moments with
a computer program,
defining horizontal displacements along the lines mapping the location of the
inclinometer hole axes with a computer program,
estimating horizontal displacement critical values upon the reaching of which the
slope instability may occur,
comparing the displacements registered in the inclinometer holes with their critical values and the evaluation of slope stability on the basis of the formulated
criterion.
Numerical calculations were conducted with the MES-based computer program
Z_Soil.PC, allowing for the solution of a number of problems occurring at the design
and execution stages of foundations, earthen and concrete geotechnical objects, highway,
bridge and tunnel engineering, as well as projects realized in underground and surface
mining (Truty et al. 2000). The program mentioned enables the assessment of the maximum resistance capacity, the evaluation of the stability of the considered earthen objects
and structures (including trenches, embankments, slopes, mine excavations at various
execution stages), the analysis of the construction’s co-operation with the earthen foundation, taking into account the fluid flow (stationary and nonstationary filtration). It could
also be used to forecast strains and displacements, to analyze rheological phenomena
(soil consolidation and creep) and problems of heat conductivity and thermomechanics
of concrete hardening (hydratation).
In order to construct calculation models of slopes within the area of inclinometric
measurements, the geological state and available situational-topographic charts were
analyzed. It is worth to emphasize the difficulty of the 2D computer mapping of the
geometric variability of slopes in strip mine conditions, since during model generation
a specific section plane needs to be considered, while in reality we encounter the overlaying of exploitation effects running through various places and in different directions. As
a result, while constructing each model, an attempt was made to retain the section planes
running along the direction of the main exploitation front advance. Basic geotechnical
parameters (modulus of elasticity, Poisson’s number, bulk density, cohesion and angle of
shearing resistance – angle of internal friction) as well as the assumed Coulomb-Mohr
calculation model were determined for all ground complexes specified in the placement
area of each inclinometer.
The evaluation of the slope stability factor SF, understood as the multiplier of shearing
active forces and internal forces maintaining the system in the state of limit equilibrium,
was done on the basis of the c-Φ reduction method, which is one of three criteria offered
by the Z_Soil program in this respect. This method consists in generating an increment
process induced by the fictional reduction of resistance parameters of the medium (the
511
cohesion c and tangent of the angle of shearing resistance tanΦ), until reaching the
limit equilibrium (Truty et al., 2002). The progressing plastic strain range leads to the
revelation of the failure mechanism (the failure surface is not predefined, but determined
numerically) and the estimation of its divisor of material parameters, at which stability
failure occurs (interpreted numerically, as an iterative procedure divergence (Truty et
al. 2002). The program chooses a region in which the unbalancing of internal forces
and those maintaining the system’s limit equilibrium takes place, while the failure may
occur both within a single slope and the entire slope area.
Following the computer calculations of the time moments in consideration, the values
of the slope stability factor SF and of horizontal soil displacements along the inclinometer
axes were determined (since in relation to „in situ” measurements, a zero value of
displacement at the bottom of inclinometer holes is assumed; a similar assumption was
introduced for the results returned by the Z_Soil program), the critical values of the
aforementioned displacements were estimated afterwards.
As from the point of view of the displacement analysis the Z_Soil program generates
reliable results solely in the elastic range of the rock mass mechanics, while in the elasticplastic phase it does not perform meaningful quantitative calculations (values of soil
displacements and strains at the moment of reaching the limit equilibrium state portray
only the qualitative, not quantitative character of its destruction), it could not be used
to specify the sought critical value of horizontal displacements.
While determining the slope stability factor SF with the computer method (the Z_Soil
program in particular), it is assumed that the hypothetical slope stability failure will
ensue at the reduced values of soil parameters c/SF and tanΦ/SF, not considering the
reduction of the elasticity modulus (researches show that this value does not influence
the SF value (see Griffiths et Lane 1999). When making this assumption, it is essential to
be aware of the fact that if such a state of slope stability failure did occur, then reducing
the cohesion value and the shearing resistance angle tangent of the soil would inevitably
lead to the decrease of the elasticity modulus value, and the vectors of soil displacements
in the moment directly preceding the slope stability failure would be much larger than
those specified for the input values (i.e. for SF = 1.0). Owing to this, the reduction of the
elasticity modulus value based on the correlatives between cohesion, angle of shearing
resistance, soil elasticity modulus and the liquidity index of cohesive soils or the density
index of noncohesive soils was proposed for this task. It was assumed that the introduction
of the factor SF, reducing the values c and tanΦ, matches the simultaneous decrease of the
soil elasticity modulus value E by a certain factor NE, in the consequence of which soil
strains, particularly horizontal displacements, increase. A physical interpretation of this
simultaneous reduction could be the arbitrary (stipulated) plastification of the cohesive soil
(the change of the liquidity index from the assumed initial value IL0 to the final IL) or the
decrease of the noncohesive soil density index (from the initial value ID0 to the final ID).
In the aim of estimating the value of the factor NE reducing the soil elasticity modulus
E (a general term, related to the modulus of linear deformation, as well as to oedometer
512
modulus), the standard (PN-81/B-03020) served as a basis. For cohesive soils, the
calculations were initiated with analyzing the relation between the design value of the
angle of shearing resistance of soil and its liquidity index, as well as between design
cohesion and liquidity index. Since the reduction of c and tanΦ for individual soil types
does not induce the same plastification effect, the term of the stipulated average soil
liquidity index was introduced and derived from the formula:
ILc,j =
ILc + I Lj
2
(1)
where:
LLc — the stipulated average soil liquidity index obtained after the reduction of c,
LLΦ — the stipulated average soil liquidity index obtained after the reduction of
tanΦ,
ILc,Φ — the stipulated average soil liquidity index obtained after the reduction of c
and tanΦ for the factor SFc = SFΦ = SF1 according to the dependence:
SFc =
c
tg j
= SFj =
= SF1 £ SF
tgj1
c1
(2)
where:
c, c1 — the design value of the initial and the reduced soil cohesion,
Φ, Φ1 — the design value of the initial and the reduced angle of shearing resistance
of soil,
SFc, SFΦ — the factor reducing the design value of cohesion and the tangent of the
design value of the angle of shearing resistance of soil,
SF1 — the factor stipulately decreasing the above geotechnical parameters of soil
at a given stage of reduction: from the value c to c1 and tanΦ to tanΦ1,
SF — the maximum value of the factor reducing the above geotechnical parameters of rock mass determined by the Z_Soil computer program, upon the
reaching of which slope stability failure occurs.
The proposed reductive factor of soil elasticity NE is represented by the formula:
E
= NE1
E1
(3)
where:
E, E1 — the design value of the initial and the reduced modulus of elasticity of
soil,
NE1 — the reductive factor of soil elasticity.
Drawing on the aforementioned correlatives and analogies of soil elasticity moduli,
relations between the reductive factor of soil elasticity NE and the slope stability factor
SF were drawn (Fig. 2 and Table 1) on the basis of the standard (PN-81/B-03020).
513
Reductive factor of soil elasticity NE1
6,0
5,0
B
A
soil A
4,0
soil B
3,0
soil C
D
soil D
C
2,0
1,0
1,0
1,5
2,0
2,5
3,0
3,5
4,0
Reducing factor SF1
Fig. 2. The dependence of the reductive factor of soil elasticity NE1 on the reducing factor SF1,
the initial soil state IL0(n) = 0.5
Rys. 2. Zależność współczynnika NE1 zmniejszającego wartość modułu sprężystości gruntu spoistego
od współczynnika redukcyjnego SF1, stan początkowy gruntu IL0(n) = 0,5
TABLE 1
The dependence between the reductive factor of soil elasticity NE1 and the reducing factor SF1,
the initial soil state IL0 = 0.5
TABLICA 1
Zależność między współczynnikiem zmniejszającym moduł sprężystości gruntu NE1
a współczynnikiem redukcyjnym SF1, stan początkowy gruntu IL0 = 0,5
The factor reducing the cohesion
value and the value of the tangent of
the internal friction angle of the soil
SF1 = SF
The reductive factor of soil elasticity NE1 = NE
Soil A
Soil B
Soil C
Soil D
1.0
1.0
1.1
1.0
1.0
1.2
1.2
1.2
1.1
1.3
1.5
1.7
1.7
1.5
1.6
1.8
2.1
2.1
1.7
1.9
2.0
2.4
2.3
1.8
2.1
2.5
2.9
3.0
2.1
2.3
3.0
3.5
3.7
2.2
2.5
3.5
4.2
4.6
2.3
2.7
4.0
4.8
5.4
2.4
2.8
514
As a result of the analysis of the obtained relations, aiming at safety, horizontal displacements estimated on the grounds of these relations were acknowledged as critical:
ìm = N E for NE £ SF ü
uxd = u x × m í
ý
îm = SF for NE > SF þ
(4)
where:
uxd — the critical value of the horizontal displacement in a given soil spot,
ux — the extreme horizontal displacement at the length of inclinometer determined
by the computer program in a given moment of time in which the „in situ”
measurement is being taken for SF = 1.0,
m — the displacement multiplier,
NE — the reductive factor of soil elasticity corresponding to the slope stability
factor SF.
On the basis of the performed calculation analysis (see Table 1) it was found that the
estimation of the soil displacement critical values on the basis of factors reducing their
elasticity moduli, will take place primarily in relation to nonconsolidated cohesive soils
(with the exception of moraines) – the symbol C (PN-81/B-03020) and for clays – the
symbol D (PN-81/B-03020). On the other hand, for consolidated cohesive morenic soils
– the symbol A (PN-81/B-03020), other consolidated cohesive soils and nonconsolidated
cohesive morenic soils – the symbol B (PN-81/B-03020), the slope stability factors SF
will play a key part.
For research purposes, it was assumed that in a given moment of time, characterized
by a specific geometry of the system, the slope is stable when the horizontal displacement
recorded by the inclinometer placed within it does not exceed the critical value:
u xi (t i ) ≤ u xd (t i )
(5)
where:
uxi (ti) — the extreme resultant horizontal displacement recorded by the inclinometer
in the time moment ti,
uxd (ti) — the horizontal displacement critical value that would be indicated by the
inclinometer in the time moment ti in the conditions of having reached the
limit equilibrium state – according to the formula (4).
Thus, in the area of an example inclinometer hole of slope geometry at the time
moment of inclinometer placement (t = 1) and of geotechnical parameters of soils specified in Fig. 1, also assuming that we are dealing with cohesive soils at the initial state
IL0 = 0.5, the following results were obtained (Fig. 3):
515
Horizontal displacement [cm]
-1,0
0,0
1,0
2,0
3,0
4,0
5,0
0
-5
-10
displacement from measurements t = 2
displacement from measurements t = 3
Depth [mm]
-15
displacement from measurements t = 4
displacement from calculations t = 2
-20
displacement from calculations t = 3
displacement from calculations t = 4
-25
-30
-35
-39,0
-40
Fig. 3. The comparison of resultant horizontal displacements from measurements
and the displacements from computer calculations in the chosen inclinometric hole axis
Rys. 3. Porównanie przemieszczeń poziomych wypadkowych z pomiarów oraz przemieszczeń
z obliczeń komputerowych w osi wybranego otworu inklinometrycznego
• for the moment of time t = 3:
– the extreme displacement from the computer calculations in the model soil profile corresponding to the location of the inclinometer hole axis: ux3 = 3.03 cm,
– the displacement multiplier: m = 1.7,
– the horizontal displacement critical value (the formula 4): uxd3 = 5.16 cm,
– the extreme displacement from the inclinometric measurement: uxi3 = 2.54
cm,
– the slope stability criterion check (5): 2.54 < 5.16 – criterion fulfilled, slope
stable;
• for the moment of time t = 4:
– the extreme displacement from the computer calculations in the model soil
profile corresponding to the location of the inclinometer hole axis: ux4 = 3.03 cm
516
(a minor change of the model’s geometry in relation to the moment of time
t = 3, did not affect the value of the obtained displacement),
the displacement multiplier: m = 1.7,
the horizontal displacement critical value (the formula 4): uxd4 = 5.16 cm,
the extreme displacement from the inclinometric measurement: uxi34 = 3.20 cm,
the slope stability criterion check (5): 3.20 < 5.16 – criterion fulfilled, slope
stable.
–
–
–
–
3. The estimation of the soil displacement critical values based
on the analysis of the state of soil effort in the inclinometer hole axis
and its relation to slope stability
We are considering a diagram of a slope with an inclinometer hole as in Fig. 4. We
enter data into a cartesian coordinate system (x, y, z) with the origin at the highest point
X0
a
x
b
I
sy
H
t xy
sx
II
sx
t xy
sy
y
Fig. 4. The diagram of a slope with an inclinometer hole
Rys. 4. Schemat skarpy z otworem inklinometrycznym
of the slope, the axis x directed to the right, the axis y directed downwards and the axis z
perpendicular to the plane of the figure. In this system, the stress tensor is described
with the following components:
σx — the normal stress in the section perpendicular to the axis x,
σy — the normal stress in the section perpendicular to the axis y,
σz — the normal stress in the section perpendicular to the axis z,
517
τxy — the shear stress in the section perpendicular to the axis x in the direction
consistent with the direction of the axis y,
τyz — the shear stress in the section perpendicular to the axis y in the direction
consistent with the direction of the axis z,
τzx — the shear stress in the section perpendicular to the axis z in the direction
consistent with the direction of the axis x.
There is an analogous definition of the strain tensor, comprising of three unit
elongations (extensional strains) εx, εy, εz and three shearing strains γxy, γyz, γzx. It is
assumed that the tensile stresses and the elongations are positive.
The displacement vector is described with the following components:
ux — the displacement in the direction of the axis x,
uy — the displacement in the direction of the axis y,
uz — the displacement in the direction of the axis z.
It is assumed that displacements with a sense compatible with that of the appropriate
axes of the coordinate system are positive.
A plane strain state is assumed:
εz = γxz = γyz = 0
(6)
The further assumption is that the slope material has the unit weight γ and is elasticplastic at the limit condition described with the dependence (the Coulomb-Mohr
criterion):
q – psinΦ – ccosΦ = 0
(7)
while
p=q=
1
× (s 1 + s 3)
2
1
× (s 1 - s 3)
2
(8)
(9)
and
σ1
σ3
Φ
c
—
—
—
—
the maximum principal stress,
the minimum principal stress,
angle of the shearing resistance (the internal friction) of slope material,
the cohesion of slope material.
The problem of slope stress state, caused by the dead weight of the slope, has not
been solved analytically up to this point. The solutions applied in designing gravity dams
518
(Balcerski et al., 1969) seem to be closest, as they assume the linear course of stresses
in the body of the dam. This course could be derived from Levy’s solution of the plane
problem of the elasticity theory to the infinite wedge loaded with the dead weight and
the hydrostatic pressure of fluid (Fig. 5) (Rekač, 1977). This solution assumes the stress
function as a polynomial of the third degree:
j = a x3 + b x 2 y + c x y 2 + d y 3
(10)
The stress values are derived from the formulae:
sx =
¶ 2j
= 2 cx + 6d y
¶y 2
(11)
sy =
¶ 2j
= 6 a x + 2by
¶x 2
(12)
¶ 2j
= - (2 b + g ) x - 2 c y
¶x¶ y
(13)
txy =
while:
γ — the unit weight of the wedge material,
a, b, c, d — integration constants specified with the boundary conditions:
– for y = –x cotα (the left part of the wedge)
- s x cosa - txy sin a = g c y cosa
(14)
-txy cosa - sy sin a = g c y sin a
(15)
– for y = x cotβ (the right part of the wedge)
sx cos b - t xy sin b = 0
(16)
t xy cos b - s y sin b = 0
(17)
in which:
γc — the unit weight of the fluid imposing load on the wedge,
α, β — the angles of inclination of the wedge point (Fig. 5).
The solution achieved in this way has a disadvantageous property: for x → –∞ one
obtains σx → –∞; which explains the possibility of its use solely in dam calculations.
This inconvenience could be bypassed by assuming the application of Levy’s solution
519
x
a
b
sy
t xy
sx
sx
t xy
sy
y
Fig. 5. The infinite wedge loaded with the dead weight and the hydrostatic pressure of fluid
Rys. 5. Nieskończony klin obciążony ciężarem własnym oraz ciśnieniem hydrostatycznym cieczy
only to the right side of the slope (i.e. for x > 0 and α = 0), and for the left side of the
slope (i.e. for x ≤ 0) the solution for the primary stress state in the rock mass could be
used (see Fig. 4)
σy = –γy
(18)
σx = –K0γy
(19)
τxy = 0
(20)
K0 =
n
1 -n
(21)
while:
K0 — the coefficient of lateral earth pressure at rest of the rock mass (soil),
ν — the Poisson’s number of the rock mass (soil).
Owing to the location of the inclinometer hole, the stress state in the right side of
the slope will not be considered; the strain and displacement state resulting from the
stress state given by the formulae (18)÷(21) will be examined instead. Upon utilizing
the relations of the elasticity theory, one may record the following dependences for the
components of the strain tensor and the displacement vector:
520
ex =
1
× [s x -n (sy + sz )]
E
(22)
ey =
1
× [s y - n (s z + s x )]
E
(23)
1
× [sz - n (s x + s y )] = 0
E
(24)
2 (1 + n )
× txy
E
(25)
¶ ux
¶x
(26)
ez =
gxy =
ex =
ey =
gxy =
¶uy
¶y
¶ ux ¶ u y
+
¶y
¶x
(27)
(28)
After substituting (18)÷(21) to the equations (22)÷(28) and making appropriate
calculations (with the assumption of the boundary condition: for y = H and x = x0 the
horizontal displacement equals zero ux = 0) one obtains the formula for the horizontal
displacement in the hole axis:
ux =
g x0 (1 +n ) [ K 0 (1 -n ) - n ]
× (H - y ) º 0
E
(29)
while:
H — the inclinometer hole length, m,
x0 — the distance of the inclinometer hole from the edge of slope (x0 < 0), m.
Zeroing the horizontal displacement value ux stands in contradiction with the results
of inclinometric measurements, therefore this model may not be used in this case as
is. In the physical perspective it is clear, since the presence of the slope edge near the
inclinometer hole distorts the primary undisturbed stress state around that hole. It seems
obvious that the nearby location of the slope edge will cause the appearance of shearing
stresses τxy, which had not occurred in the previously analyzed model. Thus the simplest
correction of this model will be the introduction of the shearing stress τxy of a constant
value in the vicinity of the inclinometer hole:
521
(30)
τxy = const = t
At that time the stress state will be given by the formulae (18), (19), (21) and the
formula (30). In this case the horizontal displacements in the hole axis will be represented by the formula:
ux =
2 (1 + n ) t
( y- H )
E
(31)
and the horizontal displacement of the inclinometer hole head (x = x0, y = 0) – by
the formula:
u0 = -
2 (1 +n ) t H
E
(32)
The linear dependence ux along with the depth corresponds to relations occurring in
reality (see Fig. 3), and the quantity t found in the formulae (31) and (32) facilitates their
„calibration” in real conditions. Indeed, the measurement results of horizontal displacements in the inclinometer hole axis could be presented in the form of a linear function,
for instance using the linear regression apparatus, then deriving the u0 value from this
equation, and calculating the value t from the converted formula (32). By marking this
value as t0 the following formula is obtained:
t0 = -
u0 E
2 H (1 + n )
(33)
The value determined in this manner could be implemented in further calculations,
mainly those aiming at the evaluation of the rock mass effort in the inclinometer hole
axis, which will be used for slope monitoring.
The following expression was employed in the further considerations as an effort
measure:
W=
(s y - s x ) 2 + 4t xy2
2 c cos F - (s y + sx ) sin F
£1
(34)
As it is seen, this expression is derived directly from the limit state condition for the
Coulomb-Mohr model. Indeed, this condition has the form (for negative compressive
stresses) (see Wiłun 1982):
(s y - sx ) 2 + 4t xy2
(s y + s x - 2 c ctg F) 2
= sin2 F
(35)
522
It appears from the equation (35), that the safe state of rock mass is characterized
by the inequality:
(s y - s x ) 2 + 4 t xy2
(s y + s x - 2 c ctg F ) 2
£ sin 2F
(36)
Taking into consideration the fact that the internal friction angle is a positive quantity
and assuming that solely normal (negative) compressive stresses occur within the rock
mass, the safe state of rock mass can be characterized by the inequality:
(s y - s x ) 2 + 4 t xy2
2 c cos F - (s y + s x ) sinF
(37)
£1
It follows from the inequality (37) that the quantity defined by the formula (34) could
be accepted as the measure of effort. After substituting the stress tensor components,
one arrives at the function of effort in the inclinometer hole axis:
æ 1 - 2n æ
çç
(1 -n )
+ g y çç
è 1 -n è
W=
2c (1 - n ) cos F + g y sin F
4 t 02
2
2
2
£1
(38)
As it is apparent, the function is generally dependent on the geotechnical parameters
in the inclinometer hole placement area ( γ, c, Φ, ν), the depth y and the quantity t0
determined on the basis of measurement results.
The form of the formula (38) indicates that the effort in the inclinometer hole axis
(for every value y) rises along with the increase of displacement, to which the value t0
is bound. For practical reasons, one can assume the effort factor in the hole axis to be
its value on the surface of the ground (i.e. for y = 0). From the formula (38) one immediately obtains:
W0 =
t0
c cos F
(39)
while:
W0 — the rock mass effort in the inclinometer hole axis on the surface of the
ground (i.e. for x = x0; y = 0),
t0 — the shearing stress value in the inclinometer hole zone (the formula 33),
MPa,
Φ — the internal friction angle of the slope material,
c — the cohesion of the slope material, MPa.
523
As a result of slope stability analysis (e.g. by the parameter reduction method c – tanΦ
with the use of the Z_Soil program), the value of the slope stability factor SF is obtained,
i.e. the slope stability fails at the parameter values (see formula (2) ch. 2):
cf =
c
SF
tan Ff =
while:
Φ
Φf
c
cf
tanF
SF
(40)
(41)
—
—
—
—
the internal friction angle of the slope material,
the internal friction angle in the moment of slope stability failure,
the cohesion of the slope material, MPa,
the cohesion of the slope material in the moment of slope stability failure,
MPa,
SF — the slope stability factor value.
After substituting (40) and (41) to the formula (39), one can calculate the rock mass
(soil) effort value in the inclinometer hole head point (i.e. for x = x0; y = 0) at the time
of slope stability failure. Following appropriate calculations, one obtains:
W0 cr =
t0
SF 2 + tan 2F
c
(42)
while:
W0cr — the rock mass effort in the inclinometer hole axis on the surface of the
ground (i.e. for x = x0; y = 0) in the moment of slope stability failure.
The ratio of efforts equals:
W0cr
= cos F SF 2 + tan 2 F
W0
(43)
Marking the shearing stress value in the inclinometer hole zone for W0 = W0cr by t0cr
we will derive from the formula (39):
t0 c r = W0 c r c cos F = t 0 cos F SF 2 + tan 2F
(44)
And then, after substituting to the formula (32):
u0 c r =
2H (1 +n ) t0 cos F SF 2 + tan2F
E
(45)
524
while:
u0cr — the critical displacement value of the inclinometer hole head in relation to
the hole bottom, m.
Upon taking into account (33), one may state that:
u0 cr = u 0 cos F SF 2 + tan2F
(46)
while:
u0 — the equalized horizontal displacement value of the inclinometer hole head in
relation to the hole bottom in the moment of performing the slope stability
analysis (determining the SF factor value), m.
The formula (46) might be utilized to evaluate the critical displacement value of the
inclinometer hole head (in relation to the hole bottom), corresponding to the slope stability failure. Its application is explained on the example of an inclinometer located in
the slope area (acc. to Fig. 1), the stability of which was estimated in ch. 2 on the basis
of the method of reducing the soil elasticity factor value (Fig. 3).
For the conditions enumerated above, the following values were obtained:
• for the moment of time t = 3:
– the dependence of the hole depth y (in meters) on the horizontal displacement
x (in cm) of the inclinometer hole axis:
y = 12.9074348x – 37.8379159
(47)
– the slope stability factor
SF = 1.7,
– the internal friction angle
Φ = 13o,
– the extreme displacement from
the inclinometric measurement
uxi3 = 2.54 cm,
– the equalized horizontal displacement
value of the inclinometer hole head in relation
to the hole bottom (the formula 47 for y = 0)
u0 = 2.931 cm,
– the critical displacement value of the inclinometer
hole head in relation to the hole bottom
(the formula 46)
u0cr = 4.900 cm,
– the slope stability criterion check: 2.54 cm < 4.900 cm; slope stable;
• for the moment of time t = 4:
– the dependence of the hole depth y (in meters) on the horizontal displacement
x (in cm) of the inclinometer hole axis:
y = 11.2762607x – 36.3616531
(48)
525
– the slope stability factor
SF = 1.7,
– the internal friction angle
Φ = 13o,
– the extreme displacement from the inclinometric
measurement
uxi3 = 3.20 cm,
– the equalized horizontal displacement value
of the inclinometer hole head in relation to the hole
bottom (the formula 48 for y = 0)
u0 = 3.225 cm,
– the critical displacement value of the inclinometer
hole head in relation to the hole bottom
(the formula 46)
u0cr = 5.390 cm,
– the slope stability criterion check: 3.20 cm < 5.390 cm; slope stable.
The calculations were also performed for several other examples of inclinometric
measurements in slopes; their results are presented in Table 2.
TABLE 2
The comparison of the critical displacement values of the inclinometer hole axes
TABLICA 2
Porównanie obliczonych krytycznych wartości przemieszczeń osi otworu inklinometrycznego
cm
cm
5
6
7
1/3
1.1
12
2.54
1
2.95
2.95
1.16
3.11
3.41
1.10
2/3
1.2
8
0.95
1.2
1.25
1.5
1.58
1.19
1.42
1.19
3/3
1.7
13
2.54
1.7
3.03
5.16
2.03
2.93
4.9
1.67
3/4
1.7
13
3.2
1.7
3.03
5.16
1.61
3.22
5.39
1.67
4/3
2.5
13
3.53
2.5
2.84
7.1
2.01
3.91
9.57
2.45
5/3
1.7
11
1.13
1.7
1.59
2.7
2.39
1.2
2
1.67
5/4
1.7
11
1.55
1.7
1.59
2.7
1.74
1.72
2.88
1.67
6/3
2
13
3.16
2
4.22
8.43
2.67
3.3
6.47
1.96
7/3
1.6
8
9.35
1.6
10
16.01
1.71
10.02
15.94
1.59
8/3
1.8
13
3.55
1.8
4.99
8.99
2.53
1.72
3.04
1.77
9/3
1.8
8
1.34
1.8
3.76
6.77
5.05
1.23
2.2
1.79
10/3
1.9
11
1.22
1.9
0.94
1.79
1.47
1.19
2.23
1.87
cm
cm
Fulfilling the stability
condition: col. 10/col. 9
The critical
displacement value
from calculations
–
4
The critical value
of hole head
displacement
The max. displacement value from
measurements
cm
3
The equalized
value of hole head
displacement
The displacement
multiplier
...°
2
Fulfilling the stability
condition: col. 7/col. 4
The extreme displacement
from the inclinometric
measurement
–
1
Hole number/
Moment of time
The internal friction angle
Φ
The analysis method of the effort
in the inclinometer hole axis
The slope stability factor
SF
The reduction method of the elasticity
modulus value
8
9
10
11
526
4. Summary
4.1. Due to the growing rate of inclinometers’ use in the monitoring of slope mechanics, a method of stability evaluation of the aforementioned earthen structures was
developed on the basis of actual horizontal displacement values obtained from “in situ”
measurements.
4.2. In order to specify the slope stability criterion, the proposed method of reduction of
soil elasticity modulus value (ch. 2), as well as the method of analysis of rock mass effort
in the inclinometer hole axis and its relation with slope stability (ch. 3) were used.
4.3. In accordance with the first method, the horizontal displacement critical value
of soil (5) was determined, taking into account computer calculation results for appropriately modeled working conditions of the considered inclinometers, by way of the
simultaneous reduction of cohesion, the tangent of the internal friction angle and the
soil elasticity modulus (a general term, related to the modulus of linear deformation, as
well as to oedometer modulus), based on the standard (PN-81/B-03020). To achieve the
stated aim, work in this area was realized in a few fundamental stages:
• creating the computer models of slopes,
• determining the slope stability factor SF with a computer program,
• calculating horizontal displacements along the lines mapping the location of the
inclinometer hole axes with a computer program,
• estimating horizontal displacement critical values upon the reaching of which the
failure of slope stability may occur,
• comparing the displacements registered in the inclinometer holes with their critical
values and the evaluation of slope stability on the basis of the formulated criterion (5).
4.4. In accordance with the second method, the horizontal displacement critical value
(the formula 46) was determined on the basis of the relation between slope stability and
soil effort in the inclinometer hole axis. The basic realization phases of this task were:
• creating the computer models of slopes (as in 4.3),
• determining the slope stability factor value SF with a computer program (as in 4.3),
• analyzing the state of stress and strain of slope in the area of the inclinometer hole
(on the elastic model) and its correction by introducing a constant value of shearing
stress in the vicinity of the inclinometer hole (the formula 30),
• analyzing the effort in the inclinometer hole axis in the case of the corrected stress
state with the use of the effort function in the form of (34),
• determining the soil effort in the inclinometer hole head point and its relation to
the slope stability factor value (the formula 42),
• determining the horizontal displacement critical value (the formula 46).
527
4.5. In the light of the conducted analyses, the developed method of slope stability
assessment by way of reducing the soil elasticity factor and evaluating the rock mass
effort in the inclinometer hole axis is as follows:
• determining the dependence of the horizontal displacement of the hole axis on its
depth and its alignment (equalizing) with the linear regression apparatus,
• determining the slope stability factor SF with a computer program,
• estimating the horizontal displacement critical value according to the method I
(the formula 4),
• determining the value of the parameter t0 from the linear regression equality (the
formula 33),
• evaluating the horizontal displacement critical value according to the method II
(the formula 46),
• the choice of the horizontal displacement critical value (the smaller value from
those obtained from the formulae 4 and 46).
4.6. The horizontal displacement critical value obtained in this way will be reliable
in slope stability evaluation on the basis of inclinometric measurements. In the practical
application of the method, it is recommended on the grounds of safety to reduce that
value – in the case of the lack of other premises, the principles stated in the technical rules
(Rozporządzenie,1996) or applied abroad (Gunaratne et al., 2006) should be assumed.
Research work no. 4 T12A 014 27 financed from the resources of the Ministry of Sciences
and Higher Education in the years 2004÷2006.
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REVIEW BY: PROF. DR HAB. INŻ. ANTONI TAJDUŚ, KRAKÓW
Received: 28 July 2006