Multi-Segment Linear Function Approximation to Nonlinear Shear Strength Envelope

Multi-Segment Linear Function Approximation to Nonlinear Shear Strength Envelope

При подготовке инженерных проектов большинство геотехнических специалистов используют линейный критерий прочности Мора Кулона вместо точного упругопластического решения Хоека Брауна. Это обусловлено тем, что изменение наклона нелинейной огибающей линии сдвиговой прочности согласно решению Хоека Брау...

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Дата:2016
Автори: Lei, X.Z., Fu, W.X., Liao, Y.
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Опубліковано: Інститут проблем міцності ім. Г.С. Писаренко НАН України 2016
Назва видання:Проблемы прочности
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Цитувати:Multi-Segment Linear Function Approximation to Nonlinear Shear Strength Envelope / X.Z. Lei, W.X. Fu, Y. Liao // Проблемы прочности. — 2016. — № 3. — С. 147-157. — Бібліогр.: 12 назв. — англ.

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spelling irk-123456789-1734842020-12-06T01:26:31Z Multi-Segment Linear Function Approximation to Nonlinear Shear Strength Envelope Lei, X.Z. Fu, W.X. Liao, Y. Научно-технический раздел При подготовке инженерных проектов большинство геотехнических специалистов используют линейный критерий прочности Мора Кулона вместо точного упругопластического решения Хоека Брауна. Это обусловлено тем, что изменение наклона нелинейной огибающей линии сдвиговой прочности согласно решению Хоека Брауна не может быть описано каким-либо одним линейным уравнением. Для аппроксимации этой линии используются мультисегментные линейные уравнения. Данная методика позволяет уменьшить погрешности при расчете в случае условного разбиения каждого материала на несколько зон, соответствующих истинному изменению градиента напряжений в материале. Для верификации предложенной методики выполнены расчеты погрешностей и сходимости результатов расчетов на конкретных примерах. При підготовці інженерних проектів більшість геотехнічних фахівців використовують лінійний критерій міцності Мора Кулона замість пружно-пластичного розв язку Хоека Брауна. Це зумовлено тим, що зміна нахилу нелінійної обвідної лінії зсувної міцності згідно з розв язком Хоека Брауна не може бути описана будь-яким одним лінійним рівнянням. Для апроксимації цієї лінії використовуються мультисегментні лінійні рівняння. Дана методика дозволяє зменшити похибки при розрахунках у випадку умовного розбиття кожного матеріалу на декілька зон, що відповідають істинній зміні градієнта напружень у матеріалі. Для верифікації запропонованої методики оцінено похибки і збіжність результатів розрахунків на конкретних прикладах. 2016 Article Multi-Segment Linear Function Approximation to Nonlinear Shear Strength Envelope / X.Z. Lei, W.X. Fu, Y. Liao // Проблемы прочности. — 2016. — № 3. — С. 147-157. — Бібліогр.: 12 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/173484 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
topic Научно-технический раздел
Научно-технический раздел
spellingShingle Научно-технический раздел
Научно-технический раздел
Lei, X.Z.
Fu, W.X.
Liao, Y.
Multi-Segment Linear Function Approximation to Nonlinear Shear Strength Envelope
Проблемы прочности
description При подготовке инженерных проектов большинство геотехнических специалистов используют линейный критерий прочности Мора Кулона вместо точного упругопластического решения Хоека Брауна. Это обусловлено тем, что изменение наклона нелинейной огибающей линии сдвиговой прочности согласно решению Хоека Брауна не может быть описано каким-либо одним линейным уравнением. Для аппроксимации этой линии используются мультисегментные линейные уравнения. Данная методика позволяет уменьшить погрешности при расчете в случае условного разбиения каждого материала на несколько зон, соответствующих истинному изменению градиента напряжений в материале. Для верификации предложенной методики выполнены расчеты погрешностей и сходимости результатов расчетов на конкретных примерах.
format Article
author Lei, X.Z.
Fu, W.X.
Liao, Y.
author_facet Lei, X.Z.
Fu, W.X.
Liao, Y.
author_sort Lei, X.Z.
title Multi-Segment Linear Function Approximation to Nonlinear Shear Strength Envelope
title_short Multi-Segment Linear Function Approximation to Nonlinear Shear Strength Envelope
title_full Multi-Segment Linear Function Approximation to Nonlinear Shear Strength Envelope
title_fullStr Multi-Segment Linear Function Approximation to Nonlinear Shear Strength Envelope
title_full_unstemmed Multi-Segment Linear Function Approximation to Nonlinear Shear Strength Envelope
title_sort multi-segment linear function approximation to nonlinear shear strength envelope
publisher Інститут проблем міцності ім. Г.С. Писаренко НАН України
publishDate 2016
topic_facet Научно-технический раздел
url http://dspace.nbuv.gov.ua/handle/123456789/173484
citation_txt Multi-Segment Linear Function Approximation to Nonlinear Shear Strength Envelope / X.Z. Lei, W.X. Fu, Y. Liao // Проблемы прочности. — 2016. — № 3. — С. 147-157. — Бібліогр.: 12 назв. — англ.
series Проблемы прочности
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AT fuwx multisegmentlinearfunctionapproximationtononlinearshearstrengthenvelope
AT liaoy multisegmentlinearfunctionapproximationtononlinearshearstrengthenvelope
first_indexed 2025-07-15T10:08:56Z
last_indexed 2025-07-15T10:08:56Z
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fulltext UDC 539.4 Multi-Segment Linear Function Approximation to Nonlinear Shear Strength Envelope X. Z. Lei, W. X. Fu1, and Y. Liao State Key Laboratory of Hydraulic and Mountain River Engineering, Sichuan University, Chengdu, China 1 wxf_lee@scu.edu.cn ÓÄÊ 539.4 Àïïðîêñèìàöèÿ íåëèíåéíîé îãèáàþùåé ëèíèè ñäâèãîâîé ïðî÷íîñòè ñ ïîìîùüþ ìóëüòèñåãìåíòíîé ëèíåéíîé ôóíêöèè Êñ. Ç. Ëåè, Â. Êñ. Ôó, É. Ëÿî Ñè÷óàíñêèé óíèâåðñèòåò, ×åíãäó, Êèòàé Ïðè ïîäãîòîâêå èíæåíåðíûõ ïðîåêòîâ áîëüøèíñòâî ãåîòåõíè÷åñêèõ ñïåöèàëèñòîâ èñïîëü- çóþò ëèíåéíûé êðèòåðèé ïðî÷íîñòè Ìîðà–Êóëîíà âìåñòî òî÷íîãî óïðóãîïëàñòè÷åñêîãî ðåøåíèÿ Õîåêà–Áðàóíà. Ýòî îáóñëîâëåíî òåì, ÷òî èçìåíåíèå íàêëîíà íåëèíåéíîé îãèáàþùåé ëèíèè ñäâèãîâîé ïðî÷íîñòè ñîãëàñíî ðåøåíèþ Õîåêà–Áðàóíà íå ìîæåò áûòü îïèñàíî êàêèì- ëèáî îäíèì ëèíåéíûì óðàâíåíèåì. Äëÿ àïïðîêñèìàöèè ýòîé ëèíèè èñïîëüçóþòñÿ ìóëüòè- ñåãìåíòíûå ëèíåéíûå óðàâíåíèÿ. Äàííàÿ ìåòîäèêà ïîçâîëÿåò óìåíüøèòü ïîãðåøíîñòè ïðè ðàñ÷åòå â ñëó÷àå óñëîâíîãî ðàçáèåíèÿ êàæäîãî ìàòåðèàëà íà íåñêîëüêî çîí, ñîîòâåòñòâó- þùèõ èñòèííîìó èçìåíåíèþ ãðàäèåíòà íàïðÿæåíèé â ìàòåðèàëå. Äëÿ âåðèôèêàöèè ïðåäëî- æåííîé ìåòîäèêè âûïîëíåíû ðàñ÷åòû ïîãðåøíîñòåé è ñõîäèìîñòè ðåçóëüòàòîâ ðàñ÷åòîâ íà êîíêðåòíûõ ïðèìåðàõ. Êëþ÷åâûå ñëîâà: îãèáàþùàÿ ëèíèÿ Õîåêà–Áðàóíà, àïïðîêñèìàöèÿ ôóíêöèè, ïîãðåø- íîñòü àïïðîêñèìàöèè, ãðàäèåíò íàïðÿæåíèé. Introduction. Practice of rock engineering projects has revealed that the shear strength of the in-situ rock mass has a non-linear feature. This feature is very significant under low stress levels. The Hoek–Brown (HB) criterion proposed by Hoek and Brown [1] can account for this observation. This criterion has been updated many times – see the paper by Hoek and Marinos [2]. Its most recent version is named the general HB criterion [3]. In spite of precise implementation made for the HB elasto-plastic analysis, e.g., Priest [4], Zhang [5], Fu and Liao [6], most geotechnical engineers and technicians usually choose the linear Mohr–Coulomb (MC) criterion for designs of engineering projects. In the HB elasto-plastic calculation, moreover, much iteration time needs being consumed when a failure stress state returns to the HB yield surface. The generalized HB criterion can evaluate the equivalent MC parameters [3]. Some researchers, e.g., Priest [7], Yang and Yin [8], Brown [9], also presented determining the MC parameters from the HB envelope. However, a particular linear MC equation cannot well match the curvature of a non-linear HB envelope. Moreover, when directly using the existing methods, there exist some difficulties in some situations, e.g., where there are two or more HB materials in the analytical model. © X. Z. LEI, W. X. FU, Y. LIAO, 2016 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 147 In order to match the curvature variation of the non-linear HB envelope as closely as possible, this study employs multi-segment linear functions for its approximation. This paper gives the expressions for estimating the fitting error and for evaluating the equivalent MC strength parameters over an arbitrary stress interval. The fitting error analysis for three cases of equivalent MC fittings of the two materials in the slope example validates the present approximate description for the non-linear HB envelope. Moreover, the stability analysis for the slope shows that, when finely dividing the actual HB curve of each material, the results based on the multi-segment linear MC equations well agree with those obtained from the HB model. 1. Best-Fitting MC Equation for Non-Linear HB Envelope. The generalized HB criterion [6] in the � �1 3� plane is expressed in � � � � �1 3 3� � �ci b ci am s( / ) , (1) where m m eb i GSI D� � �( )/ ( ) ,100 28 14 s e GSI D� � �( )/ ( ) ,100 9 3 a e eGSI� � �� �1 2 615 20 3/ ( ) / ,/ / � ci is the uniaxial compressive strength of the intact rock specimen, mi is the material constant of the intact rock determined by the tri-axial compression testing, GSI is the abbreviation of the geological strength index, D is the disturbance factor, and �1 and � 3 are the major and minor principal stresses at failure, respectively (compressive stress is considered to be positive). The � �1 3� relation of Eq. (1) is plotted in Fig. 1a. In this figure the two intersection points between the fitting MC line and HB curve are denoted by B and C. The linear MC equation over the arbitrary stress interval of [� �3 3L U, ] in Fig. 1a is expressed in � �� � �b k 3 , (2) where k e e� � �( sin ) / ( sin ),1 1� � b c ke� 2 1 2/ , and �e and ce are the equivalent friction angle and cohesive force, respectively. Using the best-fitting linear relation for approximating the non-linear HB envelope, i.e., the least-squares method, we derive the equations for estimating the fitting error and for evaluating the MC strength parameters over [� �3 3L U, ] – see Appendix. The expressions of �e and ce are � � � � � � e ci b U L U L a a m u l a � � � �� � arcsin ( ) ( )( ) ( 12 2 2 3 3 3 3 3 1 1 � � � � � � � � � � � 1 1 2 2 12 2 2 2 3 ) ( ) ( )( ) ( � � � ci a a b ci b U u l m a a m � � � � � �� � � � � � � � 3 3 3 3 1 1 2 2 2 1 L U L a a ci a a b u l a u l m) ( )( ) ( ) ( ) (a a� � � � � � � � � � � � � � � � � � �1 2)( ) , (3) c u l m a e ci a a b U L ci U L� � � � � �� �2 1 122 1 1 3 3 2 3 3� � � � � �( ) ( )( ) ( ) ( )mb U L� �3 3 3� � � � �� � � � � � � � � � � � �( )( ) ( ) ( ) ( )( � � �3 3 1 1 2 2 2 1 1 U L a a ci a a b u l a u l m a a� � � � � � � �� � 2) X. Z. Lei, W. X. Fu, and Y. Liao 148 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 � � � � � � � � � 4 1 12 2 1 2 3 3 3 3 3 1 1� � � � � �ci b U L U L a a m u l a( ) ( )( ) ( ) ci a a b u l m a a ( ) ( )( ) /� �� � � � � � � � � �� � � � �� � � � �� �2 2 1 2 1 2 � � �� �1 , (4) where u m sb U ci� �� �3 / and l m sb L ci� �� �3 / . Multi-Segment Linear Function Approximation ... ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 149 a b Fig. 1. Linear fitting for the HB curve over [� �3 3L U, ] (a) and definition of the safety factor on the arbitrary section of a microunit (b). 2. Multi-Segment Approximation to Non-Linear HB Envelope. It should be noted that the values of �e and ce under different stress intervals, calculated by Eqs. (3) and (4), have obvious differences. If the specified stress range is too large, the shear strength under the low and high stress levels will be overestimated and that under the intermediate stress level will be underestimated. In fact, the stress levels at different positions in each material (or even in a particular material) of the analytical model are different. Therefore, it is necessary to divide each material into different zones appropriate to the actual stress gradient variation of that material. Accordingly, the stress intervals chosen for different zones will yield a better agreement with the curvature variation of the HB envelope. 2.1. Determination of a Potential Failure Stress State. The minor principal stress employed for calculating �e and ce in Eqs. (3) and (4) must be consistent with the stress state at failure. However, the stresses first calculated in the numerical analysis are the elastic solutions. In the elasto-plastic analysis, the stress path of any point in the numerical simulation model is very complicated and its possible position at the HB curve is unknown. Accordingly, the minor principal stress interval determined according to the elastic analysis cannot be directly used in Eqs. (3) and (4). Clearly, the interval of the minor principal stress at failure needs to be first determined. As illustrated in Fig. 1b, any point in the numerical analytical model is treated as a microunit with one unit thickness. The principal stresses of the micro-unit, denoted by � e1 and � e3 , respectively, can be determined by the elastic analysis. The corresponding stress circle and HB envelope are plotted in Fig. 1b. The principal stresses at failure, i.e., �1 and � 3 , will be determined in the following – see the massive stress group in Fig. 1b. In the � �1 3� plane the slope of the tangent equation corresponding to any position on the HB curve can be easily derived: 1 1 1 3 1 � � � � � � � �� � � �� � sin sin , � � � � i i b b ci a am m s (5) where �i is the instantaneous friction angle as illustrated in Fig. 1b. Performing a simple transformation for Eq. (5) we can lead to � � � � � 3 1 1 2 1 � � � � � � � ci b i b i a ci bm am s m sin ( sin ) . / ( ) (6) Substituting Eq. (6) into Eq. (1) we attain � � � � � �1 1 1 2 1 � � � � � � � � ci b i b i a ci b ci m am s m sin ( sin ) / ( ) 2 1 1 sin ( sin ) . / ( ) � � i b i a a am � � � � � (7) According to the relation between the massive stress circle and HB envelope in Fig. 1b, we can write the expressions of the normal and shear stresses denoted by � n and � s , respectively: � � � � � �n i� � � �� � � 1 3 1 3 2 2 sin , (8) � � � �s i� �� � � 1 3 2 cos . (9) 150 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 X. Z. Lei, W. X. Fu, and Y. Liao Substituting Eqs. (6) and (7) into Eqs. (8) and (9) we obtain � � � � � n ci b i b i a i m am a � � � � � � � � � � 2 1 1 1 1 sin ( sin ) sin / ( ) � � �� s m ci b � , (10) � � � � �s ci i i b i a a am � � � � � � cos sin ( sin ) . / ( ) 2 2 1 1 (11) Observation for Fig. 1b shows that both of � n and � s are monotonically-decreasing functions with respect to �i . The radius of the large stress circle in Fig. 1b, i.e., the maximum shear strength � s max , equals ( ) / .� �1 3 2� Substituting Eqs. (6) and (7) into � � �s max ( ) /� �1 3 2 we attain � � � �s ci i b i a a am max / ( ) sin ( sin ) .� � � � � � 2 2 1 1 (12) The radius of the small stress circle in Fig. 1b, i.e., the maximum shear stress � max , is � � � max .� �e e1 3 2 (13) Here we define the ratio of � s max to � max as the safety factor Fs of the microunit in Fig. 1b. Substituting Eqs. (12) and (13) into Fs s� � �max max/ we obtain F am s ci e e i b i a a � � � � � � � � � � � �( ) sin ( sin ) . / ( ) 1 3 1 2 1 (14) Equation (14) is a function with respect to sin �i . When controlling the center position of the large stress circle in Fig. 1b to be within the interval of [� �e e3 1, ], we can find the value of sin �i corresponding to the minimum of Fs by numerical iteration techniques, e.g., the Newton–Raphson method. When determining �i , we can calculate � n and � s by Eqs. (10) and (11). Then we can determine � 3 by Eq. (15), which is derived through a simple transformation for Eqs. (8) and (9) � � � � �3 1 � � � n i i s sin cos . (15) When determining �i , � n , and � s , we can also calculate the instantaneous cohesive force ci by the � �n s� relation of the MC criterion, i.e., the equation as follows: ci n i s� �� � �tan . (16) 2.2. Brief Description of the Application to the Analytical Model. A brief description for the application of the present method is given here. It is assumed that necessary material parameters, e.g., the Young modulus E, Poisson’s ratio �, and bulk weight �, have been input into the numerical analytical model and that all the boundary conditions have been correctly set. The initial estimate for the elastic stresses of each element can be first solved. ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 151 Multi-Segment Linear Function Approximation ... The minor principal stress of each element at the potential failure state can be calculated by the above method. The interval of the minor principal stress at failure can be found by searching all elements of each material. To agree with the curvature variation of each HB envelope under different stress levels as closely as possible, each material in the analytical model is divided into several zones. These zones can well match the actual stress variations at different positions of that material. When determining the interval of the minor principal stress at failure for each zone, one can calculate the corresponding MC parameters by Eqs. (3) and (4) and employ these parameters in the elasto-plastic analysis. 3. Validity of the Approximate Description. 3.1. Comparison of Fitting Error. To validate the present approximate description for the non-linear HB envelope, we first compare the fitting error for a simple slope with two materials. Figure 2a illustrates the sandstone and mudstone distributions and the slope geometrical dimensions. The basic material parameters are � ci � 30 MPa, mi � 4, GSI � 5, D� 0, E� 410 MPa, �� 0.3, and �� 25 kN/m3 for the sandstone, and � ci �15 MPa, mi � 2, GSI � 5, D� 0, E� 290 MPa, �� 0.35, and �� 23 kN/m3 for the mudstone. The elastic stress analysis was first performed for the slope model. Only the gravitational stress field was considered. The horizontal displacements with zero values were made for all nodes at the left- and right-hand side boundaries. The horizontal and 152 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 X. Z. Lei, W. X. Fu, and Y. Liao a b Fig. 2. Slope geometrical dimensions (a) and multi-segment approximations to the HB envelopes (b). vertical displacements were fixed to be zero values for all nodes at the bottom boundary. The actual � �n s� curves of the sandstone and mudstone are plotted in Fig. 2b. Three cases of equivalent MC fittings were analyzed, referred to below as the cases I, II, and III. For the case I, the equivalent MC equations of each material were evaluated over the actual stress interval of that material; for the case II, each HB envelope was divided into two segments; and for the case III, each HB envelope was divided into four segments. Here we only give the case III for example. As illustrated in Fig. 2b, the increment of the minor principal stress is unequal, while the absolute difference between the two instantaneous friction angles at the two end points of each segment is equal. Figure 2b shows that the curvature variation of the HB curve can be well matched by use of an increasing step length for the increment of the minor principal stress. The fitting error, i.e., the absolute error � total , over the actual stress interval of each material was also calculated for the three cases I, II, and III. The actual interval of the minor principal stress is [3.70, 199.09] in kPa for the sandstone and [5.03, 232.85] in kPa for the mudstone. For the case I, � total equal 0.027691 for the sandstone and 0.009875 for the mudstone; for the case II, � total equal 0.023599 for the sandstone and 0.008562 for the mudstone; and for the case III, � total equal 0.021029 for the sandstone and 0.007600 for the mudstone. The results clearly indicate that the present method can reduce the fitting error when finely dividing each HB envelope. Comparatively, since the curvature of the mudstone envelope varies more obviously, the corresponding error decreases more significantly. 3.2. Further Discussion. To further validate the present method in the application to the stability calculation of slope, we implemented the elasto-plastic finite element method (FEM) analysis for the slope example by the shear strength reduction technique proposed by Zienkiewicz et al. [10], and the rigid body limit equilibrium analysis by the simplified Bishop method [11] as well. We also performed the HB elasto-plastic analysis referred to below as the case of iv for comparison. In the HB elasto-plastic analysis, the HB envelope was described as a series of instantaneous equations and each element in the slope was assigned an MC equation [5]. In the MC elasto-plastic analysis, the corresponding parameters were evaluated by Eqs. (3) and (4). The shear strength was reduced by letting tan tan /� �trial e trialF� and ctrial � � c Fe trial/ , where Ftrial is a reduction factor. The failure analysis was repeatedly performed by the bracketing approach, e.g., Dawson et al. [12], until the elasto-plastic solutions no longer converged. In the limit equilibrium analysis by the simplified Bishop method, the HB model was applied to the slope as well. Accordingly, the instantaneous friction angle �i at the base of a slice were determined by Eq. (10) in terms of the normal stress � n acting on the base of each slice. Due to difficulties to write the explicit expression of �i , the Newton–Raphson method was employed for determining �i . Then the shear stress � s acting on the base of each slice was calculated by Eq. (11). In the limit equilibrium analysis by the simplified Bishop method, the MC model was also applied to the slope for comparison. Accordingly, the shear strength at the base of the slice was calculated by the � �n s� relation of the MC criterion. The Fs-values under different cases are given in Table 1. Observation for Table 1 shows that the Fs-value of each case obtained from the elasto-plastic FEM analysis is slightly less than that calculated by the simplified Bishop method. Also, when more finely dividing each HB envelope – as in the case III for example – the corresponding Fs-value more closely agrees with that determined in terms of the HB elasto-plastic analysis. Table 1 shows the same variations for the limit equilibrium analysis results by the simplified Bishop method. Moreover for the case of iii, the failure trend and slip zone based on the equivalent MC model closely agree with those obtained from the HB model – see Fig. 3. ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 153 Multi-Segment Linear Function Approximation ... Conclusions. In this paper we propose the multi-segment linear equations for approximately describing the non-linear HB envelope. When dividing each material of the analytical mode into several zones appropriate to the actual stress gradient variation of that material, the present method can significantly reduce the error. Moreover, the comparison and analysis for the slope example have validated that, when more finely dividing the actual HB strength curve of each material, the safety factor and slip surface determined from the equivalent MC model more closely agree with those obtained from the HB model. Acknowledgments. The present authors thank the National Nature Science Foundation of China (Grant No. 40972190) and the National Basic Science Foundation for the Sichuan University by the Ministry of Education of China (Grant No. 2010SCU23004) for financial supports. They thank the peer referees for good suggestions. They also thank Miss Zhe Li for polishing this paper. Appendix. Derivations for Eqs. (3) and (4). The HB curve is divided into three segments over [ , ]� �3 3L U as illustrated in Fig. 1a. The three parts of areas between the MC line and HB curve, i.e., A1, A2, and A3 in Fig. 1a, can be found by integration. Using the state to equalize the areas above and below the MC line, i.e., A A A1 3 2� � , we can write the expression as follows: 154 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 X. Z. Lei, W. X. Fu, and Y. Liao T a b l e 1 Safety Factor and Relative Error Case FEM | |/ , %F F FmL hL hL� Bishop method | |/ , %F F FmB hB hB� I FmL �1173. 8.210 FmB �1228. 7.155 II FmL �1103. 1.753 FmB �1166. 1.745 III FmL �1098. 1.292 FmB �1162. 1.396 IV FhL �1084. – FhB �1146. – a b Fig. 3 Results for the cases III (a) and IV (b). [ ( ) ] ( / ) .b k d m s dci b ci a L U L U � � � � � 1 03 3 3 3 3 3 3 3 � � � � � � � � � � (A1) The explicit expression of Eq. (A1) is 2 1 3 3b k tU L� � � �( )( ) ,� � (A2) where t u l m a ci a a b U L � � � � � �2 1 2 1 1 3 3 � � � ( ) ( )( ) , (A3) and the definitions of u and l in Eq. (A3) have been given in Eqs. (3) and (4). We can decompose Eq. (A3) as b k t k U U L� � � � � � ( ) ( )( ) ,1 1 2 3 3 3� � � (A4) b k t k L U L� � � � � � ( ) ( )( ) .1 1 2 3 3 3� � � (A5) For any � 3 within the interval of [� �3 3L U, ], the squared error between the MC line and HB curve is defined by � � � � � � �( ) [( ) ( / ) ] .3 2 3 3 3 2� � � � �b k m sci b ci a (A6) The total squared error over [� �3 3L U, ] is � � � � � � � � � � total d k b d L U L U � � � � � ( ) [( ) ]3 2 3 3 2 3 3 3 3 3 1 � � � � � 2 1 3 3 3 3 3 3 � � � � � � � � � ! ci b ci a ci bk b m s d m L U [( ) ]( / ) ( / � � � � ci as d L U � ) .2 3 3 3 (A7) The explicit expression of � total is � � � � � total U L U L t t k � � � � � �� � � �� � � � �� ( ) ( ) ( )3 3 2 2 2 3 3 2 3 1 4 � � � � � � � � � �� � � ��2 1 2 1 12 3 3 1� � �ci b U L a m t k u a t k[ ( )( )] ( ) [ ( )( )] ( ) � �3 3 1 2 1 U L al a � � � � � � � � � � � � �� � �� � �� �ci a a b ci ak u l m a a u l( )( ) ( )( ) (1 1 2 2 2 3 2 1 2 1 2 1 a bm a � � ) ( ) . (A8) ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 155 Multi-Segment Linear Function Approximation ... When considering Eq. (A8) as a function with respect to k , we can write the first-order derivative of � total with respect to k: " � � � �� � � total U L k( ) ( )3 3 3 1 6 � � � � � �� � � �2 2 1 2 3 3 1 1 2 2� � � �ci b U L a a ci a a m u l a u l m ( )( ) ( ) ( ) b a a( )( ) . � � � � � 1 2 (A9) According to the condition of minimizing � total , i.e., " �� total 0, we lead to k m u l a ci b U L U L a a � � � � � � � � � 1 12 2 1 2 3 3 3 3 3 1 1� � � � � � ( ) ( )( ) ( ) ci a a b u l m a a ( ) ( )( ) . � �� � � � � � 2 2 1 2 (A10) Substituting Eq. (A10) into k e e� � �( sin ) / ( sin )1 1� � we obtain Eq. (3), and substituting b of Eq. (A2) and k of Eq. (A10) into b c ke� 2 1 2/ we obtain Eq. (4). Ð å ç þ ì å Ïðè ï³äãîòîâö³ ³íæåíåðíèõ ïðîåêò³â á³ëüø³ñòü ãåîòåõí³÷íèõ ôàõ³âö³â âèêîðèñòîâó- þòü ë³í³éíèé êðèòåð³é ì³öíîñò³ Ìîðà–Êóëîíà çàì³ñòü ïðóæíî-ïëàñòè÷íîãî ðîçâ’ÿçêó Õîåêà–Áðàóíà. Öå çóìîâëåíî òèì, ùî çì³íà íàõèëó íåë³í³éíî¿ îáâ³äíî¿ ë³í³¿ çñóâíî¿ ì³öíîñò³ çã³äíî ç ðîçâ’ÿçêîì Õîåêà–Áðàóíà íå ìîæå áóòè îïèñàíà áóäü-ÿêèì îäíèì ë³í³éíèì ð³âíÿííÿì. Äëÿ àïðîêñèìàö³¿ ö³º¿ ë³í³¿ âèêîðèñòîâóþòüñÿ ìóëüòèñåãìåíòí³ ë³í³éí³ ð³âíÿííÿ. Äàíà ìåòîäèêà äîçâîëÿº çìåíøèòè ïîõèáêè ïðè ðîçðàõóíêàõ ó âèïàäêó óìîâíîãî ðîçáèòòÿ êîæíîãî ìàòåð³àëó íà äåê³ëüêà çîí, ùî â³äïîâ³äàþòü ³ñòèíí³é çì³í³ ãðà䳺íòà íàïðóæåíü ó ìàòåð³àë³. Äëÿ âåðèô³êàö³¿ çàïðîïîíîâàíî¿ ìåòî- äèêè îö³íåíî ïîõèáêè ³ çá³æí³ñòü ðåçóëüòàò³â ðîçðàõóíê³â íà êîíêðåòíèõ ïðèêëàäàõ. 1. E. Hoek and E. T. Brown, “Empirical strength criterion for rock masses,” J. Geotech. Engng. Div., 106, No. GT9, 1013–1035 (1980). 2. E. Hoek and P. Marinos, “A brief history of the development of the Hoek–Brown failure criterion,” Soils and Rocks, 30, No. 2, 85–92 (2007). 3. E. Hoek, C. Carranza-Torres, and B. Corkum, “Hoek–Brown failure criterion – 2002 edition,” in: R. Hammah (Ed.), NARMS-TAC 2002: Mining and Tunnelling Innovation and Opportunity (Proc. of the 5th North American Rock Mechanics Symposium and the 17th Tunnelling Association of Canada Conference: NARMS-TAC 2002, July 7–10, 2002, Toronto, Ontario, Canada), Vol. 1, University of Toronto Press (2002), pp. 267–273. 4. S. D. Priest, “Comparisons between selected three-dimensional yield criteria applied to rock,” Rock Mech. Rock Eng., 43, No. 4, 379–389 (2010). 5. L. Zhang, “A generalized three-dimensional Hoek–Brown strength criterion,” Rock Mech. Rock Eng., 41, No. 6, 893–915 (2008). 6. W. X. Fu and Y. Liao, “Non-linear shear strength reduction technique in slope stability calculation,” Comput. Geotech., 37, 288–298 (2010). 7. S. D. Priest, “Determination of shear strength and three-dimensional yield strength for the Hoek–Brown criterion,” Rock Mech. Rock Eng., 38, No. 4, 299–327 (2005). 156 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 3 X. Z. Lei, W. X. Fu, and Y. Liao 8. X. L. Yang and J. H. Yin, “Slope equivalent Mohr–Coulomb strength parameters for rock masses satisfying the Hoek–Brown criterion,” Rock Mech. Rock Eng., 43, No. 4, 505–511 (2010). 9. E. T. Brown, “Estimating the mechanical properties of rock masses,” in: Y. Potvin, J. Carter, A. Dyskin, and R. Jeffrey (Eds.), Proceedings of the First Southern Hemisphere International Rock Mechanics Symposium (SHIRMS 2008, Sept. 16–19, 2008, Perth, Western Australia), Vol. 1, Australian Centre for Geomechanics (2008), pp. 3–21. 10. O. C. Zienkiewicz, C. Humpheson, and R. W. Lewis, “Associated and nonassociated viscoplasticity in soil mechanics,” Géotechnique, 25, No. 4, 671–689 (1975). 11. A. W. Bishop, “The use of the slip circle in the stability analysis of slopes,” Géotechnique, 5, No. 1, 7–17 (1955). 12. E. M. Dawson, W. H. Roth, and A. Drescher, “Slope stability analysis by strength reduction,” Géotechnique, 49, No. 6, 835–840 (1999). Received 15. 01. 2015 ISSN 0556-171X. 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<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> /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. 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