Features of shear failure of brittle materials and concrete structures on rock foundations

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International Journal of Rock Mechanics & Mining Sciences 45 (2008) 976–992 www.elsevier.com/locate/ijrmms

Features of shear failure of brittle materials and concrete structures on rock foundations Yu. A. Fishman Jenkelweg 16, D-22119 Hamburg, Germany Received 24 September 2007; accepted 29 September 2007 Available online 19 November 2007

Abstract The failure of brittle materials under compressive shear loading is described mainly on the basis of experimental investigations. The failure is accompanied by the formation of a combined fracture of a curvilinear or broken line shape, developing along the trajectories of the principal planes. A primary tensile crack is formed along the principal tensile stress planes. A thin streak or tight crack named a compressive crack develops along the principal compressive stress planes. Formation of the compressive crack coincides with a limiting condition, the peak shear strength tp. Reasons for the discordance of conventional shear theories as applied to brittle materials are discussed. The criterion of a material strength parameter, crushing resistance Rcr, is proposed and corrections are introduced into traditional methods of analysis of concrete dam stability. Examples of engineering applications of the failure features obtained are given. r 2007 Elsevier Ltd. All rights reserved. Keywords: Shear resistance; Brittle failure; Dam foundation failure; Crushing resistance; Tensile crack; Compressive crack; Stability factor

1. Introduction The traditional phenomenological strength theories continue to be applied widely in various spheres of science and engineering, mainly where they give satisfactory results and have been verified by long-term practice. However, in some cases, traditional approaches may result in inconsistent, often erroneous, estimation of the deformability and strength of materials and structures. This applies particularly to the shear failure of brittle materials working under compressive shear loading, for example, rock and concrete in retaining structures, dam abutments, rock slopes, mine openings and underground structures. More mechanistic approaches such as Fracture Mechanics give more reliable results than traditional phenomenological strength theories. However, at present, they have not been developed sufficiently for engineering analysis of such structures. Therefore, the traditional strength theories, in particular, shear theories, despite their discrepancies, continue to be used widely in many engineering areas. Tel.: +49 40 736 729 52.

E-mail address: yu.fi[email protected] 1365-1609/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2007.09.011

The traditional shear concepts are based mainly on a Coulomb conception. In engineering practice, the modified Coulomb theory, according to which shear failure is considered as the formation of a shear surface characterized by the shear parameters tan f (the coefficient of friction) and c (cohesion) is widely adopted. The strength criterion accepted in design practice is the achievement by shear stress, t, of the peak value, tp. The shear resistance, tp, depends on the magnitude of the normal compressive stresses, sn, acting along the shear surface, and is characterized by the relationship tp ¼ c+sn tan f. The parameters of this linear relationship, naturally, are the constants for the material. The classical shear theory, to a greater degree, corresponds to the failure of earth media and brittle materials (rock, concrete and others) along weak surfaces. So, numerous experiments of block shear along rock mass discontinuities: joints, interlayers, tectonic zones, etc., demonstrate the clear shear character of failure [1]. Deformation of one part of a rock mass relative to another proceeds as plane-parallel displacement, and shear parameters tan f and c are characterized by stable values. A very different pattern of shear failure is observed in continuous and quasicontinuous brittle materials.

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The trajectories of cracks formed in them do not coincide with the direction of the shear stresses. A complex process of material disintegration takes place. The interpretation of the traditional shear parameters tan f and c as material constants is not necessarily justified in these experiments. The nature of deformation processes differs from traditional plane-parallel shear displacement. For example, the great number of field and laboratory experiments by shear of concrete blocks along a rock foundation has shown that shear failure along the concrete–rock interface, as a rule, has no place [2]. The complicated mode of block– foundation system failure has been noted by many experts [3,4] for some time. Almost all experts observe that at the initial stage of failure the formation of a tensile crack develops from the loaded side of the block deep into the foundation. The final stage of failure is treated differently. Nevertheless, many experts are inclined to the conception that the final shear of the block occurs along the foundation. The author has shown in a number of works [2,5] that at the final stage of failure for brittle materials, a discontinuity is formed in the compression zone that develops under the unloaded side of the block and has stated a hypothesis that this is the result of rock crushing. The initiation and development of cracks has been thoroughly investigated using the principles of Fracture Mechanics by an enormous number of experts, including

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those works devoted to the failure of rock and concrete [6–10]. A simple loading (uniform stress field) of brittle materials always results in the formation of tensile cracks. Under pure tensile stresses, a tensile crack develops in the plane perpendicular to the stress direction. Under pure compressive stresses, longitudinal tensile (splitting) cracks appear parallel to the stress direction. Under pure shear stresses, inclined tensile cracks propagate at an angle to the stress direction. Under a mixed loading mode, the cracking process is much more complex, especially under a compressive shear mode. Such loading is typical for rock, concrete and other brittle materials in most underground and open structures, in particular, in rock slopes, concrete dams, rock foundations, etc. The goal of this work is to interpret the mechanism of brittle material failure in compression-shear conditions through the investigation of failure of concrete retaining structures on rock foundations. 2. Field tests and accompanying investigations The shear failure of concrete blocks along rock foundations was studied in specialized field experiments carried out on sandstone and argillite at the Kurpsay gravity dam site (Kyrgyzstan) (Fig. 1) and on diabasic porphyrite at the Krapivino gravity dam site (Russia) (Fig. 2).

Fig. 1.

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Fig. 2.

Fig. 3.

The experiments were carried out in special chambers the bottom of which was excavated below the block base, which make it possible to observe all stages of the failure process (Fig. 1a and c). The rock wall, along which the concrete blocks were sheared, was 1 m in thickness and did not contain any subhorizontal planes of weakness or any discontinuities that may promote out-of-plane shear. The blocks had in-plan sizes of 1  1 m (Kurpsay) or 1  1.5 m (Krapivino). The tests were carried out in standard momentless loading mode, where the resultant force passes through the plane of the block base through its center of gravity (Fig. 1c). The shear stress t ¼ T cos a/A in the base plane was increased whereas the normal stress sn ¼ (N+T sin a)/A remained constant, where a is the angle of inclination of force T to the block base plane and A is the base plane area.

In both cases, the failure took place almost identically. It was accompanied by a turn of the block and adjoining rock mass (Fig. 3a). The turn took place, despite the momentless loading mode under the shear load T directed obliquely downwards. On the loaded side, a decompression of the rock mass and opening of joints occurred, whereas on the opposite side, a compaction of the rock mass was observed. The joint opening was estimated from indicator data or through the opening of plaster markers attached to the rock wall. In the beginning, a steep tensile crack formed (‘1’ in Fig. 1c), developing about 1.0 m deep into the foundation and opening up to 5–6 mm at the crack mouth. The further increase of shear load led to the formation of another gently dipping tensile crack, which has broken off the concrete and then the rock mass approximately along a direction of shear force (see ‘1’ in Fig. 1b and rightmost diagram of c). Finally, at the end of tensile crack

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development, a compression zone caused a crushing of the rock material under the unloaded side of the block (‘2’ in Fig. 1b and c). This point in time coincided with the limit of bearing capacity, the peak of shear stress, tp (Fig. 3b). Then the pressure in the shearing jacks decreased and any further injection of oil into the jacks resulted in shear of part of the block and the rock along the rather rough surface formed by the tensile crack and crushing zone (Fig. 1b) with a small rise of the unloaded side. The shear occurred at values of shear stresses t ¼ tres lower than tp (Fig. 3b). The concrete–rock interface remained unbroken, the block– foundation system failed as a uniform body. However, at a post-peak state after the formation of the through going fracture (tensile crack plus crushing zone) the system began to work as two interacting bodies. The rock crushing under the toe of loaded block was also observed in field tests, for example, at tests on limestone of the Ingury arch dam (Georgia) and aleurolite of the Tashkumir gravity dam (Kyrgyzstan). The crushing zone was clearly visible in the foundation after the tests. The effect of crushing manifested itself more strongly for weaker the rocks. Also, experiments on weathered granite of the Bureya buttress dams (Russia) have shown that, under shear load, an indentation of the unloaded block side into the rock was occurred with intensive turning of the block. Similar phenomena were noted at a tectonic zone in granite of the Krasnoyarsk gravity dam (Russia) [4]. All field studies demonstrated the three failure stages of the block–foundation system: initiation of tensile cracks (beginning of failure, ti, Fig. 3), formation of crushing zone (peak stress, tp) and continuous displacement of the block along the fractured surface (residual resistance, tres). The second stage (the material crushing at peak strength) is of crucial importance; it determines the bearing capacity of the system. To understand these failure features the investigation of the stress state of the block–foundation system has been carried out on photoelastic models (Fig. 4) simulating field experiments at the Kurpsay dam shown in Fig. 1 [11].

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The stress state was studied both at an elastic stage, and during material failure. Photoelastic investigations were carried out for the momentless loading mode. The photoelastic models were made of epoxymal. After fixing the stress field in an elastic stage, the models were brought to failure. The opportunity to study the stresses in the cracking process was realized in special methodical experiments. It was established during these methodical experiments that the stress–strain diagram of the model material is linear to the moment of its complete fracture. The residual stresses in the material are rather insignificant. The linear dependence of the stresses at the locations of the greatest concentration on the block loading was also established, which allowed to fix the isocline and isochromat for the entire stress field. Lastly, the linear stress–strain diagram from zero stress almost up to fracture indicates that the failure of epoxymal is brittle. The tests were carried out at a constant normal load N and increasing inclined load T. The principal stresses trajectories before the onset of cracking is shown in Fig. 4a. The dotted line shows the location of the primary tensile crack. It initiated at the corner of the interface of the block with the foundation, which characteristically has a high stress concentration. After the development of the primary tensile crack and a change in the stress field, secondary tensile cracks formed (Fig. 4b). These tensile cracks seen in Fig. 4b propagated approximately along the trajectory of the principal tensile planes, following the criteria of global fracture [12]. A further increasing of the shear load up to the peak value, Tp, resulted in formation of a plastic zone (shaded zone) between the tensile crack and the block corner (Fig. 4b). It was directed along the plane of principal compressive stresses. Close inspection following the test, revealed a very thin crack also directed along the plane of principal compression stresses been found within the plastic zone. The model was easily divided into two parts along this surface and the tensile crack. As a result, a complex fracture surface was formed along the trajectories of the principal tensile and compressive stress planes between the corners of the block–foundation interface.

Fig. 4.

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Table 1 Parameters for the two-dimensional models tested Model type

Model construction

Model material

Block sizes b  h (cm)

Foundation

Block

A B C D E

Continuum Continuum Continuum Continuum Continuum

Gipsum Gipsum Gipsum Gipsum Gipsum

Gipsum Gipsum Gipsum Gipsum Cement

F

Compound

G

Compound

Gipsum– diatomite Gipsum

H K L

Continuum Compound Continuum

Concrete Concrete Gipsum– rubber

Model thickness, t (cm)

Loading mode

Strength parameters Uniaxial compres. sc (MPa)

Crushing resistance Rcr (MPa)

Steel

40  25 16  10 16  10 10  6 10  10 10  10 42

40 10 10 4 30 30 10

Momentless Moment Moment Moment Moment Moment Momentless

6.6 6.6 2.3 2.0 6.2 2.1 3.0

10.4 10.5 3.5 3.4 7.8 3.3 4.4

Steel

44

10

100  50 100  50 66

50 50 14

Small mom. Moment Momentless Momentless Moment

2.1 1.5 20 20 0.66

3.6 2.6 36 – 1.1

Concrete Concrete Gipsum

Note: Rcr is calculated from results of shear tests using Eqs. (4) or (4a).

Thus, the studies of stress state have shown the following. They confirmed that the initial cracks (‘1’ in Figs. 1 and 2) observed in the field tests are tensile cracks. Further, zone ‘2’ here is subjected to principal compressive stresses, which causes crushing in the foundation. 3. Investigations on continuous models Modeling investigations were carried out for more precise interpretation of the field tests results. The models were made continuous which simulated the block–foundation system with strong interface. All tests were conducted in two-dimensional loading conditions. The models were made of gypsum or concrete. Basic parameters of the tested models are given in Table 1. The models were loaded under both momentless and moment modes. The failure of a type A model loaded under a momentless mode is displayed in Fig. 5. Forces N and T were applied so that they passed through the center of the block base plane (point C). An increase of shear load was carried out at the constant normal stress on the block base sn ¼ 0.4 MPa. As can be seen in Fig. 5b the failure began, as for the field tests, by formation of tensile cracks ‘1’, developing from the re-entrant corner on the loaded side of the block in direction to the corner of the unloaded side (i.e., from the heel to the toe of the block). The ultimate state was achieved at the moment where a densely tight crack ‘2’ appeared under the unloaded side of the block, just where the crushing zone was detected in the field tests. Formation of the tight crack coincided with the peak shear stress along the block base (tp ¼ 2.0 MPa). Formation of these tight cracks was more evidently observed on the models loaded under a moment mode shown in Fig. 6 (a type B model, see Table 1). The normal force N was applied to the center of the block base plane with relative eccentricity m ¼ 0, and shear force T

Fig. 5.

was applied with relative eccentricity n ¼ 0.31, where m ¼ (0.5e/b) and n ¼ h/b (see Fig. 6a). In the tests a curvilinear or broken fracture surface occurred below the

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Fig. 6.

block base and developed between the re-entrant corners of the block sides on its interface with the foundation (points A and B in Fig. 6a). These corners are obviously points of stress singularity. The tensile crack ‘1’ propagating from the block side, where the shear load was applied, formed part of the fracture surface. The other part of the fracture surface ‘2’ propagating from the opposite block side formed a thin tight crack. The failure of the models, except for the case sn ¼ 0 (Fig. 6b), occurred without an exit of the tensile crack at the corner of unloaded block side. The length of tight crack increased with an increase in the value of the normal load N or normal stress sn. In all tests, the resultant force Rp crossed the tight crack almost in its center and transversely to the crack plane (Fig. 7). The latter caused doubts in the possibility of occurrence of such a crack under the action of shear stresses, and thus a stress analysis was carried out to confirm it. The stress state of the models loaded under a moment mode was investigated [10,13], using the finite element method, FEM. The calculations were carried out in parallel with laboratory tests. The trajectory of the crack observed in the laboratory was represented in the numerical models using Goodman joint elements [14]. Their application made it possible to trace the fracture dynamics by solving a locally nonlinear problem with an iterative approach in terms of the initial stress method. In the laboratory, a type

D model (see Table 1) was studied (Fig. 8a). The constant force N applied without eccentricity (m ¼ 0) created a normal stress on the block base of sn ¼ 1.25 MPa. An increasing shear force T was applied with relative eccentricity n ¼ 0.3. The same model parameters and loading conditions were used in the FEM calculations. The calculations showed that at a shear stress along the block base of t ¼ 0.75 MPa, formation of a tensile crack began under the influence of principal tensile stresses acting along the trajectory of the Goodman elements. The crack developed further as the shear stress t increased. The tensile crack and the stress field formed at t ¼ 1.2 MPa are displayed in Fig. 8a. Under the toe of the block (i.e., on the unloaded block side), a compression zone, where the stress state is complex, was generated. The joint, simulating a tight crack, is under the action of principal compressive stresses, which excludes the possibility of material failure due to shear. Thus, the numerical investigations confirmed that the local failure through a tight crack occurs only under principal compressive stresses (i.e., no shear). Here, a tight crack formed along a compressive plane is termed a compressive crack [15]. The same term has already been used by a number of experts [16,17]. Formation of compressive cracks in brittle materials is characterized by certain regularity. The compressive cracks, as special

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investigations shown, can be formed through other loading modes of compressed bodies: by a bend, a uniaxial or multiaxial compression. These investigations are in detail described in separate paper [18]. Formation of the compressive crack is considered here at a macrolevel, i.e., at the level of macrostresses. The mechanism of initiation and development of compressive

Fig. 7.

cracks at the microscale (i.e., the level of material microdefects) is not concerned in this paper. Compressive cracks in tests on continuous models are seen by the naked eye as very thin, hair line crack. In some cases, a compressive crack was not always visible at the moment of its formation (at t ¼ tp) and was detected visually only after unloading or dismantling of the model. In this case, the model was easily separated into two independent parts along the crack surfaces shown in Fig. 9. The walls of the compressive crack were rather clean and rough. In general, the failure of the continuous models proceeded as follows (Fig. 10). In the case of a relatively low value of sn, failure began with the formation of a tensile crack, which was initiated from the loaded block side at a certain value of shear stress, tin (point 1 on the curve in Fig. 10). The stress concentration under the unloaded block side increased, and crushing of the foundation began at the shear stress tcr (point 2). From this moment on, the tensile crack from the loaded block side and the compressive crack from the unloaded side developed toward each other. At the moment when they coalesced, the system reached the peak bearing capacity and the shear stress increased up to tpeak (point 3). In contrast, for the case of a high sn value, a compressive crack was first initiated under the unloaded block side. Complete failure came after the formation of a tensile crack under the loaded block side, together with further propagation of the compressive crack until they coalesced at tpeak. Prolonging the loading of the model in the post-peak stage resulted in (a) crushing of the compressive crack walls

Fig. 8.

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Fig. 9.

stress, s1, acting on the plane of compressive crack formation (OB in Fig. 6a), as s1p ¼ Rcr and name it as the parameter of crushing resistance or crushing strength. Let us accept that at a limiting state the peak stress s1p ¼ Rcr is uniformly distributed over the compressive crack. The force of crushing resistance, S, assuming that the compressive crack of length lcr is linear, may be written as: Z (1) S ¼ Rcr dl cr  Rcr l cr . It can be found from a solution of the three static equations (SX ¼ 0; SY ¼ 0; SM ¼ 0) that the extreme shear load on the block is expressed by the relationship: T p ¼ ½ðhtRcr Þ2 þ 2NetRcr  N 2 1=2  htRcr . Fig. 10.

(formation of crushing zone) with partial extrusion of material on the lateral surfaces of the model, (b) an additional inclination and turning of the shear block, and (c) a widening of the tensile crack. The shear resistance decreased to the value tres. The crushing zone formed, can be considered as an infill for the compressive crack, consisting of fine-grained (i.e., rock flour) crushed and shattered material. The failure at the post-peak stage had a plastic pattern. Formation of a compressive crack at the moment when the shear loading reaches the peak value has the decisive importance for estimation of the strength of the system as a whole. Let us designate the peak value of the principal

(2)

Accordingly, the peak shear stress along the block base plane or the peak shear strength is: tp ¼ ½ðnRcr Þ2 þ ð1  2mÞRcr sn  s2n 1=2  nRcr ,

(3)

or "  2 #1=2 tp ð1  2mÞsn sn 2 ¼ n þ  n Rcr Rcr Rcr

(3a)

where tp ¼ Tp/A; sn ¼ N/A; A ¼ bt is area of base plane; t the thickness of block, and m and n are the relative eccentricities of the forces N and T, i.e., m ¼ (0.5–e/b), and n ¼ h/b (Fig. 6a). Transforming Eqs. (2) and (3), it is possible to obtain the formulas for calculation of crushing resistance using the

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results of the shear tests: Rcr ¼

N 2 þ T 2p   2t eN  hT p

(4)

or Rcr ¼

s2n þ t2p 2½ð0:5  mÞsn  ntp 

(4a)

where N, Tp, t, m, n and sn are the test parameters, and tp is found in tests. Eqs. (4) and (4a) are valid for sn4tpn/ (12m). At small sn values the shear resistance, tp, is limited by the formation of tensile cracks. Values of crushing resistance parameter Rcr of various model materials and rock masses calculated using results of laboratory and field tests are tabulated in Tables 1 and 2. More details about the crushing resistance parameter Rcr are reported in Section 5. Equations (2), (3) and (3a) mathematically represent the equation of a circle. Graphically the relation (3) is displayed in Fig. 11 for different loading modes and for different strengths of the foundation material. The theoretical curves correspond well with the experimental points. As can be seen in Fig. 11, the larger the moment

created by the shear force T (i.e., the more the relative eccentricity n), the lower the peak shear strength, tp, for a given strength of the foundation material (see, for example, curves 1 and 2). This is important because concrete dams and other retaining structures are subjected to a shear load with a turning component. It can also be seen in Fig. 11 that at sn40.6sc (curve 3) the tendency for the shear strength to decrease is observed. To further widen the experimental evidence of the nature of the dependence tp ¼ f(sn), two more tests series, including the tests at a high level of normal stress sn40.5Rcr, were carried out. Models of type E and F (Table 1), representing cement and metal blocks on a gypsum foundation, were studied. Cement blocks were loaded under moment mode and metal blocks under momentless mode (Fig. 12). All models had a thickness, t, 2.5–3 times greater than the width of the blocks, b, in the shear direction. This allowed the avoidance of longitudinal splitting of the two-dimensional models at high compression loadings that did not exclude a lateral delamination (Fig. 12), while enabling the tests to be carry out under plane stress conditions. The results of all experiments in coordinates sn/Rcr and tp/Rcr are shown in Fig. 13. The test results carried out in

Table 2 Crushing resistance of dam rock foundations by results of in-situ tests Construction project

Kurpsay gravity dam (Kyrgyzstan) Krapivino gravity dam (Russia) Ingury arch dam (Georgia)

Rock type

Interbanded sandstone and argillite Diabasic porphyrite Limestone Limestone

Uniaxial compressive strength of intact rock sc (MPa)

Number of shear block tests

Average values of Deformation modulus, E 103 (Mpa)

Crushing resistance Rcr (MPa)

40–70

5

3.0

3.5

130–170

6

13.0

16.0

60–100 60–100

8 7

3.5 8.0

8.3 5.0

120–130 100–120

12 6

16.0 21.7

20.0 27.8

Sayany arch dam (Russia) Toktogul gravity dam (Kyrgyzstan)

Orthoschist Paraschist Dolomitic limestone

Naglu gravity dam (Afghanistan)

Sound gneiss Fractured gneiss Tectonic zone

90–120 – –

6 6 5

16.3 2.0 0.3

16.4 6.3 2.0

Andizhan buttress dam (Uzbekistan)

Metamorphic schist Metamorphic schist

10–100 10–100

12 8

10.0 11.0

13.0 17.0

Zeya buttress dam (Russia)

Dolerite

120–140

9

12.5

14.2

Krasnoyarsk gravity dam (Russia)

Granite, sound rock Granite, tectonic zone

110–140 –

10 10

12.0 0.4

12.3 0.9

Nurek arch dam (Tadjikistan)

Sandstone Aleurolite

80–120

3 2

15.8 2.0

16.7 4.9

Namakhvani arch dam (Georgia)

Tuff sandstone, tuff breccia

–

6

8.0

14.0

Note: (1) Almost all field tests were carried out on concrete blocks with sizes 1  1 m in plane; (2) sc is derived from laboratory testing; (3) Rcr is calculated from results of field tests using Eq. (7); (4) E was determined from results of blocks compression prior to shear loading.

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Fig. 13.

Fig. 11.

Fig. 14.

of high dams and dams on the weak rock foundations (see Section 6). 4. Investigations on compound models

Fig. 12.

the moment loading mode are recalculated for the momentless mode using Eq. (3a). The value of Rcr was determined for each test separately and then its average value was defined for each test series. As can be seen in Fig. 13, the curve tp/Rcr ¼ f(sn/Rcr) crosses the axis sn and has a maximum (tp/Rcr)max at sn/Rcr ¼ 0.5. A negative slope of the curve tp/Rcr ¼ f(sn/Rcr) corresponds to the condition, at which the normal stress, sn, is not far from the crushing resistance, Rcr, of a material. In particular, it can be attributed to the rock mass with low crushing resistance in comparison with normal pressure that is extremely important for engineering design

The two essential modes of a block–foundation system failure, i.e., the shear along a weak interface and the turning with tension and crushing of the foundation along a strong interface, have demanded investigations into the conditions at which the failure mode changes. For this, several types of compound models with different structure of the interface have been tested [19]. Models of type G (see Table 1) consisted of metal blocks placed on a gypsum foundation without an adhesive. The contact surface was smooth. The shear force, T, was applied at different heights, h, imposing small and large moments at the contact plane (Fig. 14). Failure of the compound models occurred in two ways: either by shear of the block along the contact or by its turning with formation of a deep fracture surface. A shear failure was observed at low values of the normal stress, sn. The experimental points were well approximated by a linear (Coulomb’s) relationship tp ¼ f(sn) with the

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parameters tan f ¼ 0.8 and c ¼ 0, irrespective of the loading mode. However, an increase in normal stresses resulted in a change in the failure mode. Above a certain sn value, the shearing resistance reached magnitudes such that the block and foundation began to work as a single unit. Shear deformations changed to rotational ones. Instead of a shear plane a deep crack was formed. The character of the relationship tp ¼ f(sn) changed from linear to curvilinear (thick curves in Fig. 14). The greater the moment created by shear loading, the lower the value of sn, at which the failure mode changed (curve 2 in Fig. 14), and the compound model with a weakened contact failed in a continuous manner. Let us find the critical values of sn, tan f, and c, at which shear of the block along the foundation is impossible and the system fails instead by formation of the deep cracks. With Coulomb’s formula and Eq. (3), it is possible to find the failure resistance for compound models with a weak contact interface. Failure in the turning mode occurs when, in the of absence of cohesion along the contact, c ¼ 0 (the case shown in Fig. 14): sn 4scrit n ¼

2Rcr ð0:5  m  n tan jÞ ð1 þ tan 2 fÞ

(5)

in the case where c6¼0: " #0:5  nRcr 2 ð1  2mÞRcr nRcr tan c4 tan ccrit ¼ þ 1  , sn sn sn (6) where tan c ¼ c/sn+tan f ¼ tp/sn. Some compound models with a stronger contact have also been investigated. The large-scale concrete models which have been tested in an underground chamber are shown in Fig. 15. For comparison, the tests were conducted simultaneously on continuous and compound models (type of H and K, respectively, see Table 1). The contact surface of the compound model was treated by means of metal brushes before cementing of the block. Such a treatment has created a roughness, with asperities 1–2 cm in height which can be considered insignificant for the contact of concrete dams with rocky foundation. The concrete block was placed directly on the foundation, creating with it certain cohesion. No additional adhesive solution, including cement, was applied between the block and the foundation. The tests were carried out under the momentless loading mode (m ¼ 0, n ¼ 0) at a normal stress on the plane of block base of sn ¼ 1.2 MPa. The failure of the continuous model occurred according to the already described mode (see Section 3) with formation, in the beginning, of tensile cracks and, at the final stage, of a compressive crack that only became obviously after deconstructing the model. The crushing resistance of the material calculated using Eq. (4) is Rcr ¼ 36 MPa. The failure of the compound models occurred at the first stage similarly to the failure of the continuous ones with formation of a deep tensile crack

Fig. 15.

under the loaded side at t ¼ 2.0 MPa. However, the final failure took place by shear of the block along the contact at tp ¼ 2.9 MPa. The shear coefficient tan c ¼ tp/s ¼ 2.4 was lower than the critical value calculated using Eq. (6), tan ccrit ¼ 5. The latter is proof of shear failure. 5. Strength parameters Coulomb’s parameters of shear resistance, tan f and c, represent valid strength properties for many weak surfaces in brittle materials. However, they are correct only at relatively low values of shear strength and normal stress. At certain values of the parameters as determined by Eqs. (5) and (6), shear along the weak surface does not occur and instead the brittle material works as a continuum for which the linear Coulomb’s relationship with constant parameters, tan f and c, no longer holds true. In fact, as can be seen in Fig. 11, the relationship tp ¼ f(sn) in all tests on continuous models has a nonlinear character and essentially varies with changing the loading mode. Accordingly, the shear parameters tan j and c, which can be considered as parameters of the tangent to the curve tp ¼ f(sn) or as the parameters of piecewise linear

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parts of the relationship tp ¼ f(sn), are not constant and do not satisfy Coulomb’s relationship. They essentially varied with increasing normal stress sn and loading mode (Fig. 16). Furthermore, as can be seen in Fig. 13, at sn/Rcr40.5 the curve reverses and the parameter tan j becomes negative. This contradicts the physical meaning given to the parameter tan j in the traditional shear concept as the coefficient of internal friction. The shear and normal stresses, tp and sn, of the relationship tp ¼ f(sn), which determine the shear parameters tan j and c, relate to an assumed plane for shear calculation (plane AB in Fig. 6a), along which the failure does not actually occur. Hence, the quantities tp and sn do not pertain to the stresses directly responsible for material failure; the values of these parameters calculated with these stresses do not reflect the true material resistance. The discrepancy between the supposed and actual failure stresses explains many contradictions, in particular, the curvilinear character of the relationship tp ¼ f(sn), and the variability of the shear constants tan j and c. Thus, another approach to interpret shear investigations and structural calculations based on the actual failure mechanism of brittle materials and strength criteria, reflecting this mechanism, is necessary. The dominating parameter for calculating the failure strength of the system under the turning mode is a crushing resistance, Rcr. Let us consider the parameter Rcr in more detail. It can be defined by Eqs. (4) and (4a) for a standard momentless loading mode: Rcr ¼

s2n þ t2p , sn

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each test do show the scatter inherent to experimental investigations. These values were calculated by results of the tests presented in Fig. 11 (by values sn, tp, m and n), using Eq. (4a) or (7). At the same time, the averaged Rcr is well approximated by a horizontal line, irrespective of changes in the normal stress sn and loading mode (relative eccentricities m and n). The correlation coefficient calculated for these tests closely related to zero (k=0.04). The value k=0 implies the absence of connection between parameters Rcr and sn or independence of Rcr parameter on sn parameter. So, the characteristic of crushing resistance Rcr on the average is a constant. The same result has been found in field tests. There also exists a close correlation between the crushing strength, Rcr, and the uniaxial compression strength, sc. The data presented in Fig. 18 are the results of a large number of laboratory experiments carried out with various materials (concrete, gypsum, gypsum diatomite and gypsum rubber mixes, chalk) and under various test modes. Some of the data are presented in Table 1. The linear dependence Rcr ¼ 1.47sc with correlation coefficient k ¼ 0.98 is obtained from the data shown in this table and in Fig. 18. The close connection of the parameter Rcr with the uniaxial compression strength, sc, one of the constants of

(7)

where the values sn and tp are founded from shear tests. The crushing resistance or crushing strength, Rcr, is a more meaningful physical characteristic of brittle materials for estimation of the resistance to shear loading than the shear strength parameters tan f and c. First, the parameter, Rcr, is a constant of a brittle material that is shown by a diagram in Fig. 17, although the particular Rcr values for

Fig. 16.

Fig. 17.

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Fig. 18.

brittle materials, confirms that the crushing resistance is also a material constant. The values of crushing resistance, Rcr, for various rock masses are presented in Table 2. The parameter Rcr was calculated using results of field tests carried out in the foundation of 10 dams at 16 geological sites. The test values sn and tp and Eq. (7) were used. The values of uniaxial compressive strength of intact rock, sc, found by laboratory testing of specimens from a field test site, as well as the deformation modulus of the rock mass, E, measured directly under the blocks prior to shear loading are also given in Table 2. The relation between the rock mass parameters, namely between the crushing resistance, Rcr, and the deformation modulus, E, appear to be linear: Rcr ¼ 13  104E with a correlation coefficient k ¼ 0.87 (Fig. 19). This further suggests that the parameter Rcr like deformation modulus E, is a relative constant of the rock. Thus crushing resistance, Rcr, can be used as a basis of strength and stability analysis of engineering structures on rock foundations. An example of practical application of Rcr to the stability analysis of dams is given in Section 6.1. 6. Engineering applications It is important to take into account the distinct features of shear failure of brittle materials in structural design. It is especially important to consider the turning failure

Fig. 19.

mechanism in projects involving concrete dams. For example, the undesirable seepage phenomenon, which continues to occur in the foundation of many high dams, cannot be explained through shear failure. This phenomenon leads to rupture of grouting and drainage curtains, increased seepage discharges, spouting of drainage and piezometric boreholes, a sharp increase in the uplift at the base of dams, etc. Disruption of normal operation happened in the Bratsk (height of 132 m) and Sayan (234 m) dams in Russia [20], the Inguri dam (270 m) in Georgia [20], the Ko¨lnbrein (198 m) and Schlegeis (131 m) dams in Austria [21,22], etc., and required additional expensive work after the erection had been completed and operations had begun. The origin of this undesirable phenomenon is interpreted here to be a turning failure mechanism, causing the decompression and, frequently, the rupture of a rock foundation from upstream and its compression from downstream. Engineering applications of the failure features of brittle materials have found reflection in the Russian National Construction Codes [23–25], which contains the turn method of stability calculation of retaining structures as well as the requirement to calculate the rupture and compression of rock foundations taking into account changes in filtration regime. Requirements for the construction of the interface of concrete dams with rock foundation were proposed [26].

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Two examples of engineering applications of stability analysis of concrete dams are described briefly in the following. 6.1. Stability analysis of retaining structures A method of stability analysis of concrete retaining structures termed the limit turning method has been created for calculation of the stability of such concrete structures as gravity and buttress dams, retaining and lock walls, and other retaining structures with independently working sections. It is described in detail by [23,27] and in brief consists of the following. The forces tending to turn the structure and the forces resisting it, including the force of foundation crushing resistance, S, are considered (Fig. 20). The safety or stability factor is determined through a relation between the sum of the moments of the resisting forces, Mr, and the sum of the moments of the turning forces, Mt: P Mr Fs ¼ P . (8) Mt The moments are calculated relative to the turning axis, O, located on the border of the tensile crack AO and the compressive plane OB. The position of the turning axis, O, is found from the formulae: a¼

N tRcr

(9)

d ¼ ðh2 þ 2ae  a2 Þ1=2  h

(10)

where Rcr is the crushing resistance of the foundation; N is the resultant of the vertical forces; t is the width of a

Fig. 20.

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structural section along the projected center-line or the thickness of a buttress. The crushing resistance, Rcr, is determined from field tests using Eq. (4) or (7). Rough values of Rcr can be found using analogues (e.g., see Table 2) or diagram in Fig. 14, or tables given in [23,27]. The limit turning method is similar to the classical overturning method. However, the limit turning method takes into account the possibility that the material is being crushed under the downstream face of a structure. In the classical overturning mode, the rotation of a structure around the downstream toe (point B in Fig. 20) without material failure is considered. This is possible only with an infinite value of the strength of a structure and the foundation material, which is impossible. Apart from an analytical analysis method of stability under the limit turning mode, a FEM stability analysis can also been applied based on the same principles [10]. In this case, the stepwise development of tensile cracks and formation of crushing zone would be analyzed for taking into account the crushing resistance of rock mass, Rcr. The limit turning method was approved in the designs of the Katun dam (Russia) [20], the Konstantinov dam (Ukraine) and several other structures. 6.2. Cause of Malpasset dam accident The above described mechanism of shear failure of brittle materials allows us to propose an interpretation of the cause of the serious accident involving the Malpasset dam in France [28]. The Malpasset arch dam, 60 m high, collapsed in 1959. The dam burst washed away a part of the city Frejus downstream, causing the death of more than 400 people. There were gneisses with a low deformation modulus E ¼ 1200 MPa in the dam foundation, which corresponds to a RcrE2.0 MPa (see Fig. 19). The stress state of the dam according to calculations was quite satisfactory despite the rather compliant foundation. Stability calculations performed at the design stage using the classical shear mode analysis indicated a sufficient reliability of the dam. However, at that time, the possibility of another failure mode of the dam–foundation system involving a crushing mode was not taken into account. Crushing of rock foundation under the arch dam bases has for a long time been observed by many experts [29,30]. The examination of numerous model tests has shown that arch dams failed more often owing to the fracturing of dam bodies, instead of the shear along the foundation [30–32]. Depending on such factors as the gorge and abutment topography, the dam geometry, the foundation heterogeneity, the support construction, etc., two basic modes of failure were observed: (1) division of a dam by horizontal cracks into belts (arches) with their deflection and escalating through horizontal loading into fractures in key section; (2) formation of vertical cracks with model failure owing to turning of a part of the dam, separated by

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means of one of a number of vertical cracks. In both failure modes, a crushing of the base directly under the downstream side of the dam in many cases was observed. Crushing took place simultaneously with the fracturing of the dam or preceded it [30]. With the purpose of studying the failure mechanism, special investigations of behaviors of arch foundations have been carried out [28]. The tested arches simulated elements of the Khudoni dam (Georgia). Models of the arches were made of gypsum with uniaxial compression strength sc ¼ 7.6 MPa with a foundation of gypsum– rubber mix with low strength sc ¼ 0.66 MPa. The arches were of a circular form with rectangular base (Fig. 21). Two variants of the arches were investigated: model I—an arch whose height is equal to its thickness (7  7 cm) and model II—an arch whose height is equal to its double thickness (14  7 cm). Shear tests of the single block made of the same materials (model type L, Table 1) were investigated in parallel. Model investigations have shown that the foundation under both arch bases failed by development of tensile cracks from the upstream side of the dam and formation of crushing and compressive cracks from the downstream dam side, just as in case of shear of a single block (see Section 3). As can be seen at the displacement diagrams in Fig. 21, after formation of compressive cracks at PE0.8Po, intensive nonlinear deformations with turning of the arch bases and deflection of key sections have begun that has led to fracture of the arch and, finally, to the full

failure of the model. In model II, the arch has fractured in key section at the peak load Pp ¼ 1.1Po, which caused the loss of system bearing capacity. In model I, the loss of bearing capacity (impossibility of further increase of loading) was fixed at Pp ¼ 1.03Po without fracture of the arch. However, the subsequent intensive moving of arch in key section at the same peak load Pp ¼ 1.03Po has also led finally to its fracture. The rock foundation of the Malpasset dam is characterized by a rather low crushing resistance which, as shown above, is equal RcrE2.0 MPa. At the same time, the compressive stresses under the toe of the downstream face involve an operational load of approximately 6–7 MPa. This, obviously, was the reason for the beginning of the rock foundation crushing. From the upstream face of the dam, as many experts note, a tensile crack developed [33]. It produced an excess uplift, U, additional turning of the dam, and a further crushing of the foundation (Fig. 22). The crushing commencement of the rock mass and the intensive deformations of the dam foundation resulted in nonlinear deformations of the dam body, i.e., conditions not taken into account in the dam design. This caused an overstressing of the spatially worked construction, its fracturing, and in the end, complete failure of the dam. 7. Conclusions The failure of brittle materials by shear takes place along weak surfaces characterized by both low values of shear

Fig. 21.

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References

Fig. 22.

strength (tan f and c) and low levels of normal stress sn. At sn 4scrit (Eq. (5)) and tan c4tan ccrit (Eq. (6)) these n surfaces do not play a significant role and the brittle material behaves as a continuum, for which the failure mode diverges from that of a shear mode. Continuous brittle materials under a compressive shear load fail in the turning (bend) mode with the formation of tensile cracks and compressive cracks (crushing zone), which develop, respectively, along the principal planes of tensile and compressive stresses and form a common, combined fracture surface of a curvilinear or irregular (step path) shape. The formation of compressive cracks (crushing zones) is the culminating moment of the failure, coinciding with the peak shear load. The bearing capacity of a sheared body is determined by the material crushing resistance Rcr. The turning mechanism of brittle material failure is the basis of a limit turning method used for calculation of the stability of concrete dams and other retaining structures on rock foundation, as well as for the coupling design of structure interfaces with foundations. Acknowledgments The author wishes to expresses his gratitude to E. Gaziev, L. Lapin, E. Erlikhman, E. Tiden A. Kudryavtsev, V. Yanitskiy, K. Ter-Mikaelyan, E. Kalustyan, G. Kaganov and I. Evdokimova for the great help in carrying out of field and laboratory investigations. He thanks the reviewers for useful remarks and suggestions that helped to improve the paper.

[1] Fishman YuA. Shear resistance along rock mass discontinuities: results of large-scale field tests. Int J Rock Mech Min Sci 2004;41: 1029–34. [2] Fishman YuA. Computing the stability and strength of rock foundations under concrete gravity dams. Hydrotech Construct 1976; 10(5):450–61. [3] Laitai E. The influence of interlocking rock discontinuities on compressive strength (model experiments). Rock Mech Eng Geol 1967;6:331–43. [4] Rosa SA, Selensky BD. Investigations of mechanical properties of hydraulic structure rock foundations. Moscow: Energia; 1967. [5] Fishman YuA. Investigation into the mechanism of the failure of concrete dams rock foundations and their stability analysis. In: Proc 4th cong ISRM, Montreux, 1979. p. 147–52. [6] Atkinson BK. Fracture mechanics of rock. San Diego: Academic Press; 1987. [7] Ingraffea AR. Theory of crack initiation and propagation in rock. In: Atkinson BK, editor. Fracture mechanics of rock. London: Academic Press; 1987. p. 71–107. [8] Whittaker BN, Singh RN, Sun G. Rock fracture mechanics. Amsterdam: Elsevier; 1992. [9] Shah SP, Swartz SE, Quyang C. Fracture mechanics of concrete: application of fracture mechanics to concrete, rock and other quasibrittle materials. New York: Wiley; 1995. [10] Orekhov BG, Zertsalov MG. Fracture mechanics in engineering structures and rock masses. Rotterdam: Balkema; 2001. [11] Fishman YuA, Panfilov VS, Sarabeiev VE. Photoelastic model studies of failure processes in rock foundations during shearing of blocks. Isvestiya VNIIG, 111, Leningrad, 1976. p. 74–81. [12] Parton VZ, Morozov EM. Mechanics of elasto-plastic fracture. New York: Hemisphere Pub; 1989. [13] Zertsalov MG, Ivanov VA, Tolstikov VV. The abstract of the program of complex ‘‘Crack’’. In: Soil mechanics, foundations and bases, vol. 5, 1988. [14] Goodman RE, Taylor RL, Brekke TA. Model for mechanics of jointed rock. J Soil Mech Found Div ASCE 1968;94(SM3): 637–59. [15] Fishman YuA. Formation of compression cracks in brittle materials. Proc Eurock Conf, Aachen 2000;1:617–22. [16] Kawamoto T. Macroscopic shear failure of jointed and layered brittle media. In: Proc 2nd cong ISRM, Belgrade, vol. 2, 1970. p. 215–21. [17] Goldstein RV, Osipenko NM. Some aspects of fracture mechanics of ice cover. In: Proc 7th int conf Port Ocean Eng under Arctic Conditions (POAC-83), Helsinki, vol. 37, 1983. p. 132–43. [18] Fishman YuA. Features of compressive failure of brittle materials. Int J Rock Mech Min Sci 2008;45:993–8. [19] Fishman YuA, Lapin LW. Shearing resistance along the concrete structure resting on the rock foundation. Felsbau 1983;1:70–2. [20] Fishman YuA. Limiting states of rock foundations of gravity and arch dams. Felsbau 1991;9:96–102. [21] Demmer W, Ludescher H. Measures taken to reduce uplift and seepage at Ko¨lnbrein dam. In: Proc int congn (ICOLD large dams), vol. 15, 1985. p. 1371–4. [22] Stauble H. The behaviour of the Schlegeis arch dam and the measures taken to improve it. In: Proc int conf safety of dams, Coimbra, 1984. p. 115–21. [23] Foundations of hydraulic structures. Designing requirements. National Construction Code (CHN& 2.02.02-85). Moscow, 1986. [24] Plain and reinforced concrete dams. Designing requirements. National Construction Code (CHN& 2.06.06-85). Moscow, 1986. [25] Retaining walls and shipping locks. Designing requirements. National Construction Code (CHN& 2-06-07-87). Moscow, 1987. [26] Fishman YuA. Current requirements imposed of the contact of gravity and buttress dams with rock foundations. Power Tech Eng (formerly Hydrotech Construct) 1980;14(11):1220–5.

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[27] Fishman YuA. New method of stability analysis of concrete dams on rock foundations. In: Proc ISRM Symp Rock Mech & Power Plants, Madrid, 1988. p. 459–62. [28] Fishman YuA. Once again about causes of Malpasset dam failure. In: Proc 9th cong ISRM, Paris, 1999. p. 461–4. [29] Fumagalli E. Statical and geomechanical models. Wien, 1973. [30] Kulgavij YaK. Analysis of failure patterns of large arch dam models. Hydrotech Construct 1968;4:35–9.

[31] Oberti G. Le`tude rationnelle et economique des grands barrages en beton par lu´tilisation des esseis sur modeles. In: Proc 5th cong ICOLD, Paris, 1955. p. 71. [32] Semenova KG, Karavayev AV, Antonov SS. Investigation of arch dams on models from brittle materials. In: Proc hydraul struct 5, Gosenergoizdat, Moscow, 1963. p. 27–35. [33] Jaeger C. Rock mechanics and engineering. Cambridge: Cambridge University Press; 1972.