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Upper-bound and lower-bound solutions for the axisymmetric compression of a concrete plug
International Journal of Rock Mechanics & Mining Sciences 83 (2016) 1–13
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International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Upper-bound and lower-bound solutions for the axisymmetric compression of a concrete plug Olivier Deck a,n, Cyrille Balland b, Jacques Morel b, Remi de la Vaissière c a
Ecole des Mines de Nancy, Université de Lorraine, Nancy, France INERIS, Ecole des Mines de Nancy, France c ANDRA, France b
art ic l e i nf o
a b s t r a c t
Article history: Received 29 January 2015 Received in revised form 22 September 2015 Accepted 22 December 2015
Deep radioactive waste deposits require specific solutions to close underground galleries and avoid any radioactive migration in the environment. One investigated solution consists of a swelling clay plug that induces a swelling pressure of approximately 7 MPa on the walls of the galleries. The clay plug is confined by a concrete plug to maintain and drive the swelling pressure towards the walls of the galleries. The concrete plug must therefore be designed to endure such a pressure. This paper addresses the mechanical strength of a cylindrical concrete plug with a ring-shaped tooth, a free face and a uniform compressive stress over the other face. Two experimental tests over 1/43 reduced samples are first performed and show two collapse mechanisms: dome collapse and tooth-shear collapse. Upper- and lower-bound solutions are investigated based on the Mohr–Coulomb criterion from nonlinear criteria of concrete. The choices of parameters for the friction angle (φ) and cohesion (C) are discussed in relation to the uniaxial compressive and tensile strength of the concrete and the magnitude of the stresses in the model. The results of the simplified analytical, numerical and experimental approaches are then compared, with good agreement found between the approaches. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Limit analysis Concrete Radioactive waste deposit Finite element model Mohr–Coulomb
1. Introduction 1.1. Objectives The concept of gallery sealing, studied in the framework of deep radioactive waste deposits, consists of a sealing core based on swelling clayed rock (bentonite). The natural resaturation of the core generates a swelling pressure against the drift wall that induces a compaction of the host rock EDZ (excavation damaged zone)1,2 and achieves a low permeability of the system (up to 10 11 m/s). Stiff and resistant concrete plugs on both sides provide mechanical confinement of the core during its resaturation. The plugs must ensure the best possible and radial swelling, i.e., the best possible confinement between the core and the host rock. The objective of the NSC experiment (NSC stands for Noyau de Scellement, Fig. 1) performed in the Andra (French National Radioactive Waste Management Agency) underground research laboratory in Bure, France, is to evaluate the behaviour and hydraulic performance of a sealing core and its vicinity.3 n Correspondence to: GeoRessources Laboratory, Ecole des Mines de Nancy, Université de Lorraine, CNRS, CREGU, Nancy, F-54042, France. E-mail address: [email protected] (O. Deck).
http://dx.doi.org/10.1016/j.ijrmms.2015.12.003 1365-1609/& 2015 Elsevier Ltd. All rights reserved.
The core, 5 m long and nearly 5 m in diameter, is confined by a concrete block. The stiff concrete plug aims to provide mechanical confinement, thus ensuring a condition of zero displacement at the interfaces with the host rock (claystone) and the swelling clay plug. The concrete plug will thus be subject to high mechanical stresses (estimated between 2.5 and 6.4 MPa, depending on the effective density of the swelling clay plug). The plug has been designed to resist the thrust of the sealing bentonite plug with its own weight as well as the induced friction between the solid, the gallery and an anchor. This anchor consists of an outgrowth of solid concrete in an excavated part of the gallery called the “tooth”. The diameter of the gallery is 4.6 m at its narrowest part and 5.6 m at its excavated part. The Young's modulus of the concrete plug should be greater than 43 GPa. The construction consists of several successive passes with a final pass under pressure to ensure contact between the highest part of the plug and the roof of the gallery. This type of experiment has been performed only once in the underground laboratory of Manitoba in Canada4 while monitoring the damage to the concrete structure and the rock vicinity. However, this experiment was performed in granite, which has a much better resistance to breakage than argillites. A concrete block could be considered to be a core with stiffness comparable with the rock, whereas in our case, it is a rigid core in a soft rock. The mechanical
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Fig. 1. NSC experimental set-up.1
behaviour expected of the assembly is thus different. As a result, before full-scale implementation, initial investigations were performed over 1/43 reduced samples of the concrete plug, with a comparison of the experimental tests and the analytical (collapse load theorems) or numerical investigations performed to assess the ultimate strength of the plug, i.e., the maximal bentonite pressure that may support the plug. This paper presents a synthesis of this comparison. 1.2. Theoretical background Limit equilibrium analysis or collapse load theorems to assess the stability or ultimate strength have been used in geotechnical engineering for decades to study the stability of earth slopes, the ground pressure behind embankments, or foundation design. Collapse load theorems are justified for elastic-perfectly plastic bodies and allow for the procurement of an upper and lower bound of the real theoretical but unknown ultimate load, which leads to body collapse.5 The lower-bound theorem specifies that collapse will not occur for a given load if a statically admissible state of stresses can be found; i.e., if the state of stress justifies both equations of equilibrium, the boundary conditions of the stresses, and the yield criteria. The upper-bound theorem specifies that collapse must occur for a given load if a kinematically admissible velocity field is found; i.e., if the velocity field is compatible with the plastic flow rules and the displacement boundary conditions and if the rate of work of the external forces on the body exceeds the rate of internal energy dissipation.6 The application of the upper-bound theorem generally requires splitting the studied body into several blocks, separated by slip lines where failure is localised. Blocks are assumed to involve rigid movements to avoid any elastic strain energy in the blocks and concentrate the energy dissipation over the slip surfaces. The use of the lower-bound theorem is possible for any body and yield criteria, but requires the determination of an analytical expression of a statically allowed state of stress that is sometimes
impossible to achieve. The use of the upper-bound theorem is also possible in many contexts, but requires a linear yield criterion, such as Tresca, von Mises, or Mohr–Coulomb, to be easily applied; linear slip lines are then compatible with the plastic flow rules when considering block translations. The application of the upperbound theorem for nonlinear yield criteria may be investigated using a finite element formulation.7 Some specific configurations that display a unique block collapse may also be analytically solved, as in the case of tunnel stability using the Hoek and Brown yield criteria.8 More general resolutions are possible with the use of the finite element method.9 Lower- and upper-bound theorems are widely used to assess the ultimate strength of a concrete structure.10–14 The case of axisymmetric problems is seldom investigated in limit analysis. Kobayashi and Thomsen15 investigated the case of axisymmetric compression and extrusion on conical slip surfaces. However, the considered velocity fields do not correspond to rigid movements, and a term of elastic strain energy is introduced. The next sections describe an application of limit analysis to assess the ultimate load that may be applied to the concrete plug previously described. Four results are compared: two of these results are the lower- and upper-bound ultimate load calculated using limit analysis, the third result is a numerical value computed using a finite element model, and the fourth result is an experimental value. An original aspect of this study is the use of a linear criterion for the upper-bound limit analysis, adjusted over a nonlinear yielding criterion used for the other investigations. We first introduce the nonlinear yield criterion used in this analysis (a parabolic criterion well adapted for concrete) as well as the proposed procedure for linearising this nonlinear criterion, as is generally addressed in the field of rock mechanics16; the nonlinear failure surface of Hoek and Brown17,18 is continuously approximated (linear approximation) by the tangent Mohr–Coulomb envelope.19 Second, we present a laboratory investigation based on the reduced physical model and the measured strains. Third, the
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Fig. 2. Values of C and φ required to fit the Mohr–Coulomb yield criterion over the parabolic yield criterion, for different values of Rc and Rt ¼ Rc/10: (a) under the assumption s1 ¼ s2 o s3 and (b) under the assumption s1 o s2 ¼s3.
application of the simplified analytical approach and numerical modelling to predict the stress and strain fields is presented. Finally, a comparison between the analytical and numerical and observed results from the specific laboratory experiment are discussed.
2. Nonlinear yield criteria and the linear approximation In this study, the investigated body is a cylindrical concrete plug loaded with a uniform pressure, and an axisymmetric assumption can be made. Concrete is basically modelled with parabolic yield criteria defined with two parameters that correspond to the uniaxial compressive strength Rc and the uniaxial tensile strength Rt:
Fp ( σ ) = J2 ( σ ) +
( RC − RT ) . I ( σ ) − RC ⋅RT 1
3 3 ( σ1 − σ 2 )2 + ( σ1 − σ 3 )2 + ( σ 3 − σ 2 )2 J2 ( σ ) = 6 I1 ( σ ) = σ1 + σ 2 + σ 3
(1)
where s1, s2 and s3 are the three principal stresses, with negative values for the compressive stresses and positives values for Rc and Rt. Because of the difficulty to apply the upper-bound theorem for such yield criteria, the parabolic criteria may be approximated by a Mohr–Coulomb criterion for any specific average state of stress:
Fmc ( σ̲ ) = Fmc ( σ1, σ 3 ) = ( σ 3 − σ1) + ( σ1 + σ 3 ) sin φ − 2C cos φ
(2)
where s1 is the major principal stress, and s3 is the minor principal stress, with both being negative for compressive stresses (s1 rs2 rs3); φ is the friction angle; and C is the cohesion. In this study, an associated flow rule is considered for the Mohr–Coulomb criterion. Given specific values of the principal stresses sI, sII and sIII, it is possible to define C and φ so that the two criteria are locally equivalent:
∇ ( Fmc ( σ̲ ) ) ∇ ( Fmc ( σ̲ ) )
=
∇ ( Fp ( σ̲ ) )
⎡ 1 ⎢ ∇ ( Fmc ( σ I , σ III ) ) 2 ⎢⎣ ∇ ( Fmc ( σ I , σ III ) )
+
⎤ ∇ ( Fmc ( σ II , σ III ) ) ⎥ = ∇ ( Fmc ( σ II , σ III ) ) ⎥⎦
∇ ( Fp ( σ̲ ) ) ∇ ( Fp ( σ̲ ) )
(4)
When further assumptions are provided, it is possible to give a theoretical expression for both C and φ, in relation to Rc and Rt. In Section 3, it will be shown and discussed that the axisymmetric geometry and loading of the concrete plug induce similar values for two of the three major principal stresses. In that case, if we assume s1 ¼s2 o s3, or s1 o s2 ¼y3, then Eq. (3) gives the following:
⎧ (3+sin φ) ⎪ σ 3 = σ1 + (Rc − Rt ) 4 sin φ ⎪ ⎪ ⎪ ( σ1 − σ 3 )2 ( R c − R t ) 2σ + σ − R c R t = 0 ⎨ + ( 1 3) ⎪ 3 3 3 ⎪ ⎪ ( σ − σ ) + 2C cos φ ( R c − Rt ) − ( σ + σ ) sin φ = 0 ⎪ 3 1 3 ⎩ 1 3 for σ1 = σ 2 < σ 3 ⎧ (3−sin φ) ⎪ σ 3 = σ1 + ( R c − R t ) 4·sin φ ⎪ ⎪ ⎪ ( σ1 − σ 3 )2 ( R c − R t ) σ + 2σ − R c R t = 0 ⎨ + ( 1 3) ⎪ 3 3 3 ⎪ ⎪ ( σ − σ ) + 2C cos φ ( R c − Rt ) − ( σ + σ ) sin φ = 0 ⎪ 3 1 3 ⎩ 1 3 for σ1 < σ 2 = σ 3
(5)
The resolution of these three equations gives a value for C and
φ for a given value of s1 (Eq. (6)) or a given value of s3 (Eq. (7)),
∇ ( Fp ( σ̲ ) )
Fp ( σ̲ ) = 0 Fmc ( σ̲ ) = 0
Because the Mohr–Coulomb criterion involve edges, a singular situation must be considered when the given specific values of the principal stresses correspond to one edge, i.e., when two principal stresses are equal. In that case, one average value is calculated to define the normal direction to the Mohr–Coulomb criterion. As an example, if considering the case where sI ¼sII≠sIII with s1 ¼sI ¼sII the major principal stress and s3 ¼yIII the minor one, then first part in Eq. (3) must be replaced by
(3)
where ∇ is the gradient operator, and || || is the Euclidean norm. The first part of Eq. (3) indicates that the two criteria are tangent.
which is a more traditional choice. As an illustration, Fig. 2 shows the values of C and φ for different values of y3 and y. Fig. 3 shows a 3D plot of the two criteria for values taken from Eq. (5). It can be observed that Eq. (3) is well satisfied by a specific state of stresses but differences may be significant for other states.
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Fig. 3. Illustration of the Mohr–Coulomb (A) and parabolic yield criteria (B) fitted with Eq. (5) for a specific state of stresses s1 o s2 ¼ s3 (C) and s1 ¼s2 o s3 (fitting for s1 ¼ 30 MPa, Rc ¼ 30 MPa, Rt ¼3 MPa).
O. Deck et al. / International Journal of Rock Mechanics & Mining Sciences 83 (2016) 1–13 ⎧ ⎡ 4K – ( R c − R t ) ⎤ ⎪ φ = Arccsc ⎢ ⎥ ⎪ ⎢⎣ 3 ( R c − R t ) ⎥⎦ ⎪ ⎪ K + ( K + 2σ 1)· sin φ ⎨ for σ 1 = σ 2 < σ 3 ⎪C = 2 · cos φ ⎪ ⎪ ⎪ with K = ( R t − R c ) + ( R c + R t )2 − 12 ( R c − R t )· σ 1 ⎩ ⎧ ⎡ 4K + ( R c − R t ) ⎤ ⎪ φ = Arcsin ⎢ ⎥ ⎪ ⎢⎣ 3 ( R c − R t ) ⎥⎦ ⎪ ⎪ K + ( K + 2σ 1)· sin φ ⎨C = for σ 1 < σ 2 = σ 3 2 · cos φ ⎪ ⎪ ⎪ ( Rt − R c ) + R c2 + Rt2 − R c ·Rt − 3 ( R c − Rt ). σ 1 ⎪ with K = ⎩ 2
3.1. Experimental design
(6)
⎧ ⎡ 4K – ( R c − R t ) ⎤ ⎪ φ = Arccsc ⎢ ⎥ ⎪ ⎢⎣ 3 ( R c − R t ) ⎥⎦ ⎪ ⎪ ⎨ K − ( K − 2σ 3 )· sin φ for σ 1 = σ 2 < σ 3 ⎪C = 2 · cos φ ⎪ ⎪ 2 ⎪ ⎩ with K = ( R c − R t ) − R c +(R t − R c ) (R t + 3σ 3 ) ⎧ ⎡ 4K + ( R c − R t ) ⎤ ⎪ φ = Arcsin ⎢ ⎥ ⎪ ⎢⎣ 3 ( R c − R t ) ⎥⎦ ⎪ ⎪ K − ( K − 2σ 3 )· sin φ ⎨C = for σ 1 < σ 2 = σ 3 2 · cos φ ⎪ ⎪ ⎪ ( R c − Rt ) + R c2 + 2R c ( Rt − 6σ 3 ) + Rt (Rt + 12σ 3 ) ⎪ with K = ⎩ 2
5
(7)
3. Laboratory tests Two laboratory tests on a reduced scale model were conducted in the GeoResources Laboratory to study the behaviour of the concrete plug. The principle of these tests was to monitor the behaviour and final rupture of a specimen scale 1:43 with the same mechanical characteristics as the 1:1 future plug. The geometrical characteristics of the specimen are shown in Fig. 4. These tests do not take into account the effects of scales20 and are based on the hypothesis of a perfect rigid rock around the core because a concrete plug is incorporated into a steel cell. This situation is obviously not the case in the site in the underground laboratory experimental gallery, where the vicinity is the host rock characterized by a damaged zone (EDZ).
A steel cell was specifically developed for these tests. The cell exactly takes the shape of the in-situ host rock around the concrete plug. The bentonite core pressure on the life-size model is reproduced by a pressurisation chamber connected to a hydraulic servo syringe (0–60 MPa). The pressurising chamber provides a hydrostatic pressure, avoiding any type of puncture that could be induced by a conventional press. The 1:43 scale concrete plug is cast directly in the cell together with four strain gauges positioned in the plane passing through the top of the key (tooth). For the first test, the strain gauges have all been radially oriented to the specimen at mid-radius (position R1 in Fig. 4). For the second test, two gauges were positioned at the centre of the specimen: the radial (position R0) and the other axial (position A0). The other two gauges were placed respectively at mid-radius; the first radial (position R1) and the other axial (position A1). Please see the experimental protocol for further details. Preliminary analytical calculations suggest that the failure of the specimen should occur between 15 MPa and 40 MPa, depending on the input data. The test is performed with a stresscontrolled velocity of 0.5 MPa/min. The concrete used in situ consists of four different aggregates (0/2 mm, 0/4 mm, 4/8 mm and 8/20 mm) in addition to cement and adjuvants. Given the size of the sample (100 mm), the aggregate 8/20 could cause discontinuous phenomena and has thus not been used to make both samples. However, three standard uniaxial tests were performed after eight days of drying to check for any variations in concrete strength as a function of the aggregate chosen. The uniaxial compressive strengths measured for the different mixtures vary between 10.3 and 13.7 MPa. The type of component aggregate in the concrete thus does not seem to have a significant influence over its compressive strength. Following these preliminary experiments, the first sample (Fig. 5) was incorporated with aggregates of 0–4 mm, while the second sample was incorporated with granulates of 0–8 mm. For each specimen, the drying time before testing this time was 28 days. The compressive strength was assumed to be equal to Rc ¼30 MPa, while the traction strength is assumed equal to 3 MPa, based on a traditional 1/10 ratio between both strengths. With the
Fig. 4. Dimensions of the experimental set-up with the positions of the gauges (on the left) and the pressure cell (on the right).
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extrapolation from the Rc at 8 days, the European standard EN1992-1-1 give us an Rc at 28 days that is close to 20 MPa, which is less than the target value (Eq. (8)). Because concrete is known to be very heterogeneous in strength, we decided to keep the number of 30 MPa, which is the target value.
R c (28days) =
R c (8days) ⎧ ⎡ 28 exp ⎨ s ⎢ 1 − 8 ⎩ ⎣
1 ⎤⎫ 2
( ) ⎥⎦⎬⎭
(8)
with s ¼0.38, a coefficient relative to the concrete class. The tests were voluntarily stopped just after the collapse, as soon as the pressure setpoint could not be held by the hydraulic syringe, to preserve the integrity of the specimens and to visualise the fracture planes. For this purpose, the two collapsed samples were sawn off for autopsy. 3.2. Mechanical observations The first result of these tests is the occurrence of two different failure modes: a dome-shaped failure mode for sample 1 (Fig. 5), and the other failure mode with the sheared key on the entire periphery of the specimen. This difference will be investigated with analytical and numerical analysis. The starting point of the fracture is the same for both tests: in the downstream corner of the key. It therefore appears that the behaviour differences depend on the mode of crack propagation in the sample. The other direct result of these tests is the difference in the ultimate load. The first sample, with the smallest particle size, broke when the pressure reached 17 MPa, while the second sample broke at 38 MPa (Figs. 6 and 7). For test 1, the amplitude of the measured strains is divergent for gauges placed at symmetrical locations (position R1). This result shows the difficulty and uncertainty to clearly describe the whole deformation of the sample. Test 1 gauges should
theoretically show the same deformation. However, the response of the gauges is the consequence of the sample heterogeneity and the uncertainties regarding the exact strain position due to the technical difficulties of incorporating the gauges in the matrix. In fact, the aggregate size is similar to the gauge dimension, and the gauge may record some local behaviour that may be different from the one occurring in a purely continuous media. The charge and discharge cycles conducted at the beginning of confinement show a clear elastic mechanical behaviour. The cycles were not pursued with the most significant pressures to not disturb the measuring resistance to breakage by fatigue phenomena that are not expected on site. The main objective is to highlight the rupture phenomena and their level. The loading and unloading cycles show no deformations for test 1 and no hysteresis cycles for test 2. An estimate of the modulus of deformation from these cycles does not seem serious. Qualitatively, however, the curves show that the monitored strains in sample 1 are greater than in sample 2. One gauge placed in the R1 position in sample 1 shows a strain up to 7000 microstrain, while the similar gauge in sample 2 placed at the same location and direction shows only 2000 microstrain. Comparing these two gauges in the same range of stress (0–17 MPa) shows an even larger contrast. Sample 1 is thus more deformed than sample 2 during the loading and reached its breaking point earlier. The difference in size of the granulates offers the first explanation of the difference between both tests. However, two failure modes appear to be possible for this sample geometry. This mechanical aspect is discussed below in the numerical modelling Section 5.
4. Upper- and lower-bound solutions This section is dedicated to the analytical assessment of the ultimate load based on limit equilibrium analysis.
Fig. 5. Sample 1 (on the top) and sample 2 (on the bottom) after the test condition (full view and section); the red dotted line highlights the exposed fractures.
O. Deck et al. / International Journal of Rock Mechanics & Mining Sciences 83 (2016) 1–13
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Fig. 6. Deformation curves as a function of the pressure for test 1; the four gauges have the same position R1 with a π/2 lag around the sample.
Fig. 7. Deformation curves as a function of pressure for test 2.
4.1. Lower-bound solution The lower-bound solution is investigated with a statically allowable state of stresses. A parabolic yield criteria is considered with Rc ¼30 MPa and Rt ¼3 MPa. A cylindrical coordinate system (r, θ, z) is considered with the origin 0, as plotted in Fig. 8. Pressure p is assumed to be positive. Because of the axisymmetric geometry, loading, and boundary conditions, all stresses must be independent of θ; srθ and sθz are null, sθθ is always a principal stress, and the equilibrium equations reduce to:
∂σrr ∂σrz σrr − σθθ + + =0 r ∂r ∂z ∂σzz ∂σrz σrz + + =0 ∂z ∂r r
(compressive, with p a positive value). This state of stresses respects both Eqs. (8) and (9) and the boundary conditions of the region. More specifically, no sliding may occur on the right border because sθθ is principal and no shear stresses are calculated over the border. In region 4, all stresses are assumed to be null. This assumption leads to a free-stress boundary condition under region 2. In region 2, the boundary conditions impose constant values of szz and srz on the upper and lower circular boundaries:
σzz,2 (z=H ) = − p σzz,2 (z=0) = 0
(9)
σrz,2 (z=H ) = 0 σrz,2 (z=0) = 0
(10)
The medium is divided into four regions (Fig. 8), in which four statically admissible stress fields must be founded. These stress fields must be consistent with the equations of equilibrium and the boundary conditions of the stresses. Consequently, most of the stress tensor components must be continuous between two adjacent regions. The following stress field is considered: In region 1, szz, sθθ and srr are all assumed to be constant, principal, and equal. The upper boundary imposes szz,1 ¼ p
(11)
These boundary conditions may be satisfied by considering a 3d degree polynomial expression for szz,2 that satisfies Eq. (11):
σzz,2 (z ) = −
6·p ⎛ z 3 H·z 2 ⎞ ⎟ .⎜ − 3 2 ⎠ H ⎝ 3
(12)
By substituting Eq. (12) in Eq. (10), it can be found that
σrz,2 (z, r ) = −
3·p·r·z·(H − z ) H3
+
k r
(13)
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Fig. 9. 3D plot of Fp(s) over the entire region 2 (axes r and z) for the statically admissible state of stresses that give the maximal value of the load p.
to be a lower bound of the strength capacity for a compressive strength Rc ¼30 MPa (Table 1). Others values can be assessed if the compressive strength is assumed to be greater. Fig. 8. Discretization of the casing into four zones for the lower-bound solution.
where k is a constant of integration equal to 0 to satisfy Eq. (11). Finally, sθθ and srr are assumed to be equal and can be determined by substituting Eq. (13) into Eq. (9):
σrr,2 (z, r ) = σθθ,2 (z, r ) = −
3·p·r 2 2H 3
. (2z − H ) + k1 (z )
(14)
where k1(z) is a function of z. A polynomial function is used for k1 (z), and the state of stress is now compared to the yield criteria (Eq. (1)) to obtain the maximal possible value for pressure p. When a 0 degree polynomial is used, the greater value of p is found to be equal to 8.6 MPa for k1 ¼ 26.14. When a first degree is used, a value of p ¼16.6 MPa is found for k1 ¼1.6–1695z. When other degrees are tested, the value of p remains the same. For the first degree polynomial, the value of Fp(s) (Eq. (1)) is plotted over the whole region 2 (Fig. 9). This value of p will correspond to a lower bound if a statically admissible state of stresses may also be found in region 3. In this region, sθθ,3 and srr,3 are still assumed to be equal, and szz,3 is assumed to be constant. Eq. (7) implies srz,3(r,z) to be an inverse function of r, and continuity between region 2 and 3 is used to obtain the final expression:
σrz,3 (z, r ) =
3·p·R2·(z 2 − Hz ) H3r
⎛ r ⎞⎞ 3pR2 (2z − H ) ⎛ . ⎜ 1 − 2.Ln ⎜ ⎟ ⎟+k1 (z ) ⎝ R ⎠⎠ ⎝ 2H 3
4.2.1. Theoretical developments The upper-bound solution investigation requires a linear yield criterion. The Mohr–Coulomb yield criterion is used and calibrated over the parabolic criterion to allow for comparison with the previous lower-bound solution (Fig. 3). A kinematically admissible velocity field is required to obtain an upper-bound solution. For this purpose, the plug is divided into a number of rigid regions separated by slip surfaces, where both the Mohr–Coulomb criterion and the plastic flow rule are respected. The first condition signifies that Eq. (2) must be satisfied over the entire slip surface. The second condition signifies that the relative velocity of two adjacent regions separated by a given slip surface is not parallel to the surface but inclined with an angular equal to the dilatancy angle. For an associated flow rule with the Mohr–Coulomb criterion, the dilatancy angle is set equal to the friction angle. Thus, if Vij corresponds to the relative velocity of zone i compared to zone j, then the plastic flow rules requires an angle φ between Vij and the slip surface. Because of the axisymmetric geometry, Vij is assumed to be directed into the plane of any vertical section that includes the symmetry axis. Vij can then be split into normal and tangential velocity components vij and uij:
uij /vij = tan φ
(15)
Eqs. (6) and (12) imply that srr is a logarithmic function of the variable r. Continuity of srr between regions 2 and 3 finally is used to obtain
σrr,3 (z, r )=σθθ,3 (z, r )=
4.2. Upper-bound solution for the Mohr–Coulomb yield criterion
(16)
where k1(z) is the linear function used in Eq. (14). Eqs. (14) and (15) give no conditions about the value of szz,3; thus, any value may be considered. The value that gave the maximal strength is found to be equal to 5 MPa (compressive stresses). The yield criterion (Eq. (1)) is then respected over the whole zone (Fig. 10). This stress is compatible with the boundary conditions if the interface is assumed to be stronger than the zone. More specifically, tangential stresses are not assumed to be limited along these borders because the teeth are wedged into the steel cell and that any sliding is then assumed to be impossible. In conclusion, the ultimate capacity of pmin ¼ 16.6 MPa is found
(17)
where vij and uij are the tangential and normal components, respectively, to a slip surface of the velocity vector, and the rate of energy dissipation per unit slip area dij is calculated using:
dij = C·vij
(18)
where C is the cohesion of the slip surface, and vij is the norm of the relative tangential velocity between the two zones i and j. The in-plane section of the plug is subdivided into 3 zones (Fig. 12). Zone I is the in-plane triangular key (tooth). Zone I is assumed to be fixed, and VI ¼0. Zone II is the in-plane triangular shape zone that includes a part of the upper boundary where the load p is applied. A vertical slip surface is then introduced between zones I and II and an inclined surface, with angle α formed between zones II and III. When the ultimate load is reached, zone II is assumed to move with a velocity VII equal to the relative velocity VI/II between zones I and II. To satisfy the flow rule, VI/II is inclined at an angle φ with respect to the slip surface. Because the problem is axisymmetric, the velocity of zone II cannot be associated to a
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of each zone. From the figure, each relative tangential velocity can be deduced:
‖VII ‖ =
sin (α − φ) ‖VIII ‖ sin (α )
vII / I = cos φ‖VII ‖ =
vIII / II =
Fig. 10. 3D plot of Fp(s) over the entire region 3 for the statically admissible state of stresses that give the maximal value of the load p.
Table 1 Results of the lower bound of the ultimate load p for different mechanical strengths (Rc and Rt). Rc [MPa]
Rt [MPa]
p [MPa]
30 40 50
3 4 5
16.6 22.2 27.7
rigid movement and will cause tangential strains. Zone III is the inplane trapezoidal shape zone that includes the free lower boundary and a part of the upper one. Only a vertical velocity is kinematically admissible because horizontal displacements are incompatible with the boundary conditions. VIII is then vertical, but the relative velocity between zones II and III, VII/III, must be inclined at an angle φ with respect to the slip surface. In Fig. 11, a hodograph shows the velocity and relative velocity
(19)
sin (α − φ) cos φ‖VIII ‖ sin (α )
sin (2φ) ‖VIII ‖ 2 sin (α )
(20)
(21)
the angle α may be any angle between the maximal and minimal values. The maximal value is arctan(R/L) because, for a greater value, the cylindrical zone II is not hollow and VII is not kinematically admissible. The minimum value is φ because the hodograph cannot be closed for a smaller value. The ultimate load p may then be determined by equalizing the working rate of the external forces and the rate of internal energy dissipation .4 The load p is the single external force that works for the investigated velocity field. The working rate is then equal to:
(
)
= pπD2||VIII || + p·π · R2 − D2 ||VII || cos φ ⎞ ⎛ sin (α − φ) cos φ⎟ ||VIII || = pπ ⎜ D 2 + R 2 − D 2 ⎠ ⎝ sin (α )
(
)
(22)
where D is the radius of the upper part of region III. The internal energy dissipation is calculated by integrating the rate of energy dissipation per unit of area (Eq. (18)) over all of the slip surfaces:
= II / I + III / II = 2πCRHvII / I + C = 2πCRH +C
(
(
π R 2 − D2 sin (α )
)v
III / II
sin (α − φ) cos φ sin (α )
)
π R2 − D2 sin 2φ 2 sin2 (α )
‖VIII ‖
Fig. 11. Definitions of the 3 zones used to obtain an admissible velocity field and the associated hodograph.
(23)
10
O. Deck et al. / International Journal of Rock Mechanics & Mining Sciences 83 (2016) 1–13
An upper bound of the ultimate load p may then be assessed by combining Eqs. (22) and (23):
=
(24)
4.2.2. Application The upper-bound method is applied, and the ultimate load p may be calculated for any value of α∈[φ, arctan(R/L)]. However, the values of C and φ must be chosen in relation to the stress state because the parabolic criteria is linearised. Different assumptions are made for the minor principal stress. s3 is set to be equal to 10 MPa, 20 MPa, Rc, or p. The two major stresses s2 and s1 are assumed to be equal. The numerical results, which will be described in Section 5, will show that the more realistic assumption is s3 equal to 10 MPa and s2 and s1 are not perfectly equal, but are similar. When s3 is assumed to be equal to p, an iterative procedure is applied to find an upper bound of the ultimate load. For given values of Rc, Rt and α, and the first assumption s3 ¼ Rc, the two values of C and φ are calculated as in Fig. 2b. The value of p is then calculated using Eq. (24). If p is different than the initial value of s3, then the procedure is repeated with the new value s3 ¼ p until the ultimate load is found to be equal to initial value of s3. Different ultimate loads can be obtained for each value of α. A second analysis shows that the smaller value of p is found for α ¼ φ. The final results are shown in Table 2, which indicates that the maximal value of p is not strongly dependant on the assumption regarding s3.
5. Numerical results A finite element model is developed to compare the analytical and numerical results. The objectives are as follows: to compare the ultimate load p with those assessed by the upper- and lowerbound methods; to compare the slip lines used by the upperbound method with the yielding areas given by the numerical model; to compare the stress state given by the numerical model with those considered by the lower-bound solution or with assumption about s1, s2 and s3 considered by the upper-bound solution; and to compare the strain curves with those given by the monitored samples. The computations are performed using Cesar software (ITECH, France). Fig. 12 shows the model with its mesh and boundary conditions. Horizontal displacements are not fixed over the vertical right boundary because the model displays a horizontal displacement directed to the axisymmetric axis. A null displacement would then induce tensile radial stresses that cannot exist in the real experiment because the interface between the concrete and steel has no tension resistance. These displacements are consistent with the direction of VII defined in the upper-bound analysis (Fig. 11). Six nodes triangular elements are used. Different mesh sizes were tested, and an average length of 2.5 mm was found to give robust results. The ultimate load is calculated for two yielding criteria of the concrete. First a parabolic criteria is used with Rc ¼30 MPa and Rt ¼3 MPa; this case corresponds to the more realistic behaviour of the concrete, and this is the same criteria used for the lower-bound solution. Second, an associated Mohr–Coulomb criterion is used with C¼ 20.25 MPa and φ ¼19.5° (values taken from Table 2 when s1 ¼s2 os3 and s3 ¼ 10 MPa); this case is the same criteria used for the upper-bound solution with values of C and φ, so that the Mohr–Coulomb criterion fits the parabolic one. External load p is imposed with 2 MPa increments. A set of iterations is performed between each increment when yielding
occurs. The ultimate load is reached when the calculations no longer converge. The results of the numerical model are illustrated in Fig. 13. The ultimate load depends on the yielding criteria used for the concrete. An ultimate load of 43 MPa is found for the Mohr–Coulomb criterion when the parabolic criterion gives 48 MPa. These values are more comparable with those given by real test 2 (38 MPa) than by test 1 (17 MPa). For a similar case, the lower- and upper-bound solutions gave an ultimate load of 16.6 and 45 MPa, respectively. A first conclusion is that the lower-bound solution appears to strongly underestimate the theoretical ultimate loads, while the upper-bound solution appears to give a closer solution. However, the theoretical solution that seems to be well approximated by both the numerical model and the upper-bound solution are greater than the two observed ultimate stresses. Fig. 13 shows values of strains in the same position and directions as the gauges used in the experiment (A0, A1, R0 and R1 position). The two results are quite identical, but different from the results given in the experiment (Figs. 6 and 7). Only gauge R1 (Fig. 7) is similar between the two results. For the other gauges, the values and/or signs are different. These discrepancies can probably be partially explained by the difficulty encountered for placing the gauges into the fresh concrete when the samples were manufactured. There is no proof that the gauge did not move. Fig. 13 shows the isovalues of the plastic strain norm that determine the sliding planes along which failure occurs. The plastic strain norm is calculated with the “Frobenius norm” of the plastic strain tensor « ε » and is equal to the square root of the trace of the ε.ε matrix. Similarities are found between the numerical result and the sliding planes defined in Section 4.2 for the upper-bound solution. A conical slip surface can be identified, similar to the one considered in the upper-bound analysis. Moreover, the large yielding area that appears in the tooth is consistent with the second slip surface also considered. However, when failure occurs, none of the two slip lines are extended over the whole model, as in Fig. 11, and a contact between the two appears. These similarities may explain why the results of the numerical model and the upper-bound method are close. Table 2 Results of the upper bound of the ultimate load p for different mechanical strengths (Rc and Rt) and different values of s3 under the assumption s1 ¼ s2 os3. Initial mechanical strengths Equivalent Mohr–Coulomb parameters
Ultimate load
C [MPa]
φ [deg]
p [MPa]
Hyp s3 ¼ 10 30 3 40 4 50 5
20.25 26.3 32.3
19.5 20.4 21
45.24 60.5 75.9
Hyp s3 ¼ 20 30 3 40 4 50 5
22.2 28.4 34.4
16.9 18 18.8
45.7 60.45 75.4
Hyp s3 ¼ Rc 30 3 40 4 50 5
23.9 31.9 39.9
15.2 15.2 15.2
46.8 62.4 78
Hyp s3 ¼ p 30 40 50
25.3 35 44
14.1 13.4 13.1
38.4 61 81
Rc [MPa]
Rt [MPa]
3 4 5
O. Deck et al. / International Journal of Rock Mechanics & Mining Sciences 83 (2016) 1–13
Fig. 12. Mesh and boundary conditions of the numerical model under the axisymmetric assumption.
6. Discussion The results show that the upper- and lower-bound theorems as well as the numerical modelling may be used to assess the ultimate load capacity of the studied concrete plug, which has some discrepancies with the experimental values. However, all these different approaches display some advantages and disadvantageous
11
that can be highlighted. The experimental results may first appear to be the most credible because no theoretical assumptions are made. However, the results are logically scattered due to the concrete heterogeneity. The mechanical behaviour between two similar samples is known to be very scattered due to the unique distribution of aggregates for each sample, the existence of randomly distributed micro-cracks in the cement matrix and the difference in the cement hydration. As a consequence, the experimental tests are not sufficient unless a large number of measurements can be performed, and the experimental results must be interpreted to give the most interesting information. Experimental interpretation involves a comparison of the results with some predictions. Numerical analysis is a powerful method to obtain the theoretical prediction of the ultimate load of any system. Numerical analysis can account for many complex mechanical behaviours and boundary conditions of mechanical systems, for which no theoretical and analytical solution may be found. Due to the development of even more complex yielding criteria, numerical results suffer two disadvantages. First, numerical results must be validated, i.e., compared with other results (experimental if available), to increase the confidence in their use. Second, numerical results must be interpreted, as is the case with experimental results, to give the most interesting information. Analytical solutions, such as those described in this paper, based on the upper- and lower-bound theorems are a good tool to help provide these interpretations. As shown in Section 4, the limit analysis suffers some important weakness because it requires linear yielding criteria to be easily used and because the results are only given over an interval, which may be large. In that case, the
Fig. 13. Isovalues of the plastic strain norm for the ultimate load and the Mohr–Coulomb yield criterion (left, for C¼ 8.4 MPa and φ¼ 31.8°) or the parabolic one (right).
12
O. Deck et al. / International Journal of Rock Mechanics & Mining Sciences 83 (2016) 1–13
Fig. 14. Comparison of the stresses on Section 1 for: (a) the lower-bound solution and (b) for the numerical model for a load p ¼ 46 MPa.
the sample, is used to predict where micro-cracks are the most likely to occur, even if the concrete plug is not expected to break out as the samples. As a final application, the results will also be used to design the concrete reinforcement of the real concrete plug (Fig. 15).
7. Conclusion
Fig. 15. Concrete block in the drift sealing experiment NSC (for Noyau de Scellement, Fig. 1) performed in the Andra (French National Radioactive Waste Management Agency) underground research laboratory in Bure, France.3
analytical solution must not be seen as a competitive approach to precisely assess the ultimate load, but as a powerful method of interpretation. Based on this study and utilising the lower- and upper-bound theorems, the ultimate load of the concrete casing may here be interpreted as two simple mechanisms. The first mechanism corresponds to a vault mechanism in which the pressure p is transferred to the key. This behaviour is the most intuitive and corresponds to the lower-bound stress state. The second mechanism is less intuitive, as it corresponds to the influence of the dilatancy angle of concrete when a failure starts to occur in the key. The shearing of the key involves a centipede displacement of the external part of the casing that leads in turn to an axial expulsion of the internal part. The final comparison between the analytical and numerical results clearly shows that the real behaviour is probably a mix between these two simple mechanisms. These two mechanisms can now be used to organise the discussion between all engineers and researchers that study the behaviour of the casing. The sealing drift experiment (NSC) is currently underway with the real-size concrete plug (Fig. 14). The concrete plug will be monitored with geophones to detect and locate micro-cracks with their acoustic emissions. The results of this study, which focus on
This study compared the ultimate load of a concrete plug among different types of assessments: experimental, analytical and numerical. The two investigated samples were reduced-scale concrete plugs of potential concrete plugs that would be used in a drift sealing concept in the framework of an underground storage for nuclear waste. A steel cell was developed for these tests. Concrete samples were built inside and then tested with a uniform pressure until collapse of the concrete. The analytical and numerical results are based on similar yielding criteria: Mohr–Coulomb or parabolic. For the analytical results, the upper- and lower-bound solutions are described. The results show some good agreements and discrepancies among all the results. The analytical upper-bound method gives results close to the numerical results. Both approaches appear to overestimate the real ultimate load, whereas the lower-bound solution appears to underestimate the real ultimate load. Discrepancies between the experimental and theoretical results are probably due to local stress concentrations in the concrete sample because granulates inside the sample have significant diameters compared to the sample diameter. Moreover, the introduction of gauges inside the sample is also responsible for some of the heterogeneities. As a final conclusion, the authors think that the upper-bound solution gives a good approximation of the ultimate load for a homogeneous material. For small concrete samples, excessive heterogeneities lead to a strong reduction in the ultimate load. These reduced-scale results cannot be extrapolated to real scale 1 concrete blocks, such as those used in the NSC experiment, especially because scale 1 blocks may be considered to be homogeneous, and the expected stress on the block wall is well below the resistance line. However, the reduced-scale results may help to understand the behaviour of the concrete block, together with an acoustic emission survey that has been installed for this experiment, to monitor potential damage that could occur in the concrete plug during the swelling of the core.
Acknowledgements The authors thank the Laboratory of Rock Mechanics of GeoResources for the logistic and technical support in the experimental tests and Andra for providing funding.
O. Deck et al. / International Journal of Rock Mechanics & Mining Sciences 83 (2016) 1–13
References 1. Armand G, Noiret A, Zghondi J, Seyedi DM. Short- and long-term behaviors of drifts in the Callovo-Oxfordian claystone at the Meuse/Haute-Marne underground research laboratory. J Rock Mech Geotech Eng. 2013;5:221–230. 2. Blümling P, Bernier F, Lebon P, Martin DC. The excavation damaged zone in clay formations time-dependent behaviour and influence on performance assessment. Phys Chem Earth. 2007;32:588–599. 3. De la Vaissière R, Conil N, Morel J, Leveau F, Gatabin C, Garcia-Sineriz JL, Habos H, Rey M, Piedevache M, Helminger B, Balland C. Design and construction of a large-scale sand-bentonite seal in in the Meuse/Haute Marne underground research laboratory: NSC experiment. In: Proceedings of the International Conference on the Performance of Engineered Barriers, Hannover, 6–7 February; 2014. 4. Young RP, Collins DS. Seismic studies of rock fracture at the underground research laboratory, Canada. Int J Rock Mech Min Sci. 2001;38:787–799. 5. Salençon J. Calcul à la rupture et analyse limite. In Presses de l'ENPC; 1983. 6. Davis O, Selvadurai APS. Plasticity and Geomechanics. Cambridge: Cambridge University Press; 2002. 7. Lyamin AV, Yu HS, Sloan SW, Hossain M. Lower bound limit analysis for jointed rocks using the Hoek-Brown criterion. Aust Geomech J. 1998;33:46–62. 8. Fraldi M, Guarracino F. Limit analysis of collapse mechanisms in cavities and tunnels according to the Hoek–Brown failure criterion. Int J Rock Mech Min Sci. 2009;46:665–673. 9. Turgeman S, Pastor J. Limit analysis: a linear formulation of the kinetic approach for axisymmetric mechanic problem. Int J Numer Anal Methods Geomech. 1982;6:109–128. 10. Pisano AA, Fuschi P, De Domenico D. Peak loads and failure modes of steel-
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reinforced concrete beams: Predictions by limit analysis. Eng Struct. 2013;54:477–488. Joergensen HB, Hoang LC. Tests and limit analysis of loop connections between precast concrete elements loaded in tension. Eng Struct. 2013;52:558–569. Chang ZT, Bradford MA, Gilbert RI. Limit analysis of local failure in shallow spherical concrete caps subjected to uniform radial pressure. Thin-Walled Struct. 2010;48(6):373–378. Milani G, Milani E, Trall A. Upper bound limit analysis model for FRP-reinforced masonry curved structures. Part I: unreinforced masonry failure surfaces. Comp Struct. 2009;87(23–24):1516–1533. Krabbenhoft K, Damkilde L. Lower bound limit analysis of slabs with nonlinear yield criteria. Comp Struct. 2002;80(27–30):2043–2057. Kobayashi S, Thomsen EG. Upper and lower bound solutions to axisymmetric compression and extrusion problems. Int J Mech Sci. 1965;7:127–143. Sofianos AI, Nomikos PP. Equivalent Mohr–Coulomb and generalized Hoek– Brown strength parameters for supported axisymmetric tunnels in plastic or brittle rock. Int J Rock Mech Min Sci. 2006;43(5):683–704. Hoek E, Brown ET. Underground Excavations in Rock. London: Institute of Mining and Metallurgy; 1980. Hoek E, Brown ET. Practical estimates of rock mass strength. Int J Rock Mech Min Sci. 1998;34(8):1165–1186. Lee YK, Bobet A. Instantaneous friction angle and cohesion of 2-D and 3-D Hoek–Brown rock failure criteria in terms of stress invariants. Rock Mech Rock Eng. 2014;47:371–385. Zervos A, Papanastasiou P, Vardoulakis I. Modelling of localisation and scale effect in thick-walled cylinders with gradient elastoplasticity. Int J Solids Struct. 2001;38(30–31):5081–5095.
Contents lists available at ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Upper-bound and lower-bound solutions for the axisymmetric compression of a concrete plug Olivier Deck a,n, Cyrille Balland b, Jacques Morel b, Remi de la Vaissière c a
Ecole des Mines de Nancy, Université de Lorraine, Nancy, France INERIS, Ecole des Mines de Nancy, France c ANDRA, France b
art ic l e i nf o
a b s t r a c t
Article history: Received 29 January 2015 Received in revised form 22 September 2015 Accepted 22 December 2015
Deep radioactive waste deposits require specific solutions to close underground galleries and avoid any radioactive migration in the environment. One investigated solution consists of a swelling clay plug that induces a swelling pressure of approximately 7 MPa on the walls of the galleries. The clay plug is confined by a concrete plug to maintain and drive the swelling pressure towards the walls of the galleries. The concrete plug must therefore be designed to endure such a pressure. This paper addresses the mechanical strength of a cylindrical concrete plug with a ring-shaped tooth, a free face and a uniform compressive stress over the other face. Two experimental tests over 1/43 reduced samples are first performed and show two collapse mechanisms: dome collapse and tooth-shear collapse. Upper- and lower-bound solutions are investigated based on the Mohr–Coulomb criterion from nonlinear criteria of concrete. The choices of parameters for the friction angle (φ) and cohesion (C) are discussed in relation to the uniaxial compressive and tensile strength of the concrete and the magnitude of the stresses in the model. The results of the simplified analytical, numerical and experimental approaches are then compared, with good agreement found between the approaches. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Limit analysis Concrete Radioactive waste deposit Finite element model Mohr–Coulomb
1. Introduction 1.1. Objectives The concept of gallery sealing, studied in the framework of deep radioactive waste deposits, consists of a sealing core based on swelling clayed rock (bentonite). The natural resaturation of the core generates a swelling pressure against the drift wall that induces a compaction of the host rock EDZ (excavation damaged zone)1,2 and achieves a low permeability of the system (up to 10 11 m/s). Stiff and resistant concrete plugs on both sides provide mechanical confinement of the core during its resaturation. The plugs must ensure the best possible and radial swelling, i.e., the best possible confinement between the core and the host rock. The objective of the NSC experiment (NSC stands for Noyau de Scellement, Fig. 1) performed in the Andra (French National Radioactive Waste Management Agency) underground research laboratory in Bure, France, is to evaluate the behaviour and hydraulic performance of a sealing core and its vicinity.3 n Correspondence to: GeoRessources Laboratory, Ecole des Mines de Nancy, Université de Lorraine, CNRS, CREGU, Nancy, F-54042, France. E-mail address: [email protected] (O. Deck).
http://dx.doi.org/10.1016/j.ijrmms.2015.12.003 1365-1609/& 2015 Elsevier Ltd. All rights reserved.
The core, 5 m long and nearly 5 m in diameter, is confined by a concrete block. The stiff concrete plug aims to provide mechanical confinement, thus ensuring a condition of zero displacement at the interfaces with the host rock (claystone) and the swelling clay plug. The concrete plug will thus be subject to high mechanical stresses (estimated between 2.5 and 6.4 MPa, depending on the effective density of the swelling clay plug). The plug has been designed to resist the thrust of the sealing bentonite plug with its own weight as well as the induced friction between the solid, the gallery and an anchor. This anchor consists of an outgrowth of solid concrete in an excavated part of the gallery called the “tooth”. The diameter of the gallery is 4.6 m at its narrowest part and 5.6 m at its excavated part. The Young's modulus of the concrete plug should be greater than 43 GPa. The construction consists of several successive passes with a final pass under pressure to ensure contact between the highest part of the plug and the roof of the gallery. This type of experiment has been performed only once in the underground laboratory of Manitoba in Canada4 while monitoring the damage to the concrete structure and the rock vicinity. However, this experiment was performed in granite, which has a much better resistance to breakage than argillites. A concrete block could be considered to be a core with stiffness comparable with the rock, whereas in our case, it is a rigid core in a soft rock. The mechanical
2
O. Deck et al. / International Journal of Rock Mechanics & Mining Sciences 83 (2016) 1–13
Fig. 1. NSC experimental set-up.1
behaviour expected of the assembly is thus different. As a result, before full-scale implementation, initial investigations were performed over 1/43 reduced samples of the concrete plug, with a comparison of the experimental tests and the analytical (collapse load theorems) or numerical investigations performed to assess the ultimate strength of the plug, i.e., the maximal bentonite pressure that may support the plug. This paper presents a synthesis of this comparison. 1.2. Theoretical background Limit equilibrium analysis or collapse load theorems to assess the stability or ultimate strength have been used in geotechnical engineering for decades to study the stability of earth slopes, the ground pressure behind embankments, or foundation design. Collapse load theorems are justified for elastic-perfectly plastic bodies and allow for the procurement of an upper and lower bound of the real theoretical but unknown ultimate load, which leads to body collapse.5 The lower-bound theorem specifies that collapse will not occur for a given load if a statically admissible state of stresses can be found; i.e., if the state of stress justifies both equations of equilibrium, the boundary conditions of the stresses, and the yield criteria. The upper-bound theorem specifies that collapse must occur for a given load if a kinematically admissible velocity field is found; i.e., if the velocity field is compatible with the plastic flow rules and the displacement boundary conditions and if the rate of work of the external forces on the body exceeds the rate of internal energy dissipation.6 The application of the upper-bound theorem generally requires splitting the studied body into several blocks, separated by slip lines where failure is localised. Blocks are assumed to involve rigid movements to avoid any elastic strain energy in the blocks and concentrate the energy dissipation over the slip surfaces. The use of the lower-bound theorem is possible for any body and yield criteria, but requires the determination of an analytical expression of a statically allowed state of stress that is sometimes
impossible to achieve. The use of the upper-bound theorem is also possible in many contexts, but requires a linear yield criterion, such as Tresca, von Mises, or Mohr–Coulomb, to be easily applied; linear slip lines are then compatible with the plastic flow rules when considering block translations. The application of the upperbound theorem for nonlinear yield criteria may be investigated using a finite element formulation.7 Some specific configurations that display a unique block collapse may also be analytically solved, as in the case of tunnel stability using the Hoek and Brown yield criteria.8 More general resolutions are possible with the use of the finite element method.9 Lower- and upper-bound theorems are widely used to assess the ultimate strength of a concrete structure.10–14 The case of axisymmetric problems is seldom investigated in limit analysis. Kobayashi and Thomsen15 investigated the case of axisymmetric compression and extrusion on conical slip surfaces. However, the considered velocity fields do not correspond to rigid movements, and a term of elastic strain energy is introduced. The next sections describe an application of limit analysis to assess the ultimate load that may be applied to the concrete plug previously described. Four results are compared: two of these results are the lower- and upper-bound ultimate load calculated using limit analysis, the third result is a numerical value computed using a finite element model, and the fourth result is an experimental value. An original aspect of this study is the use of a linear criterion for the upper-bound limit analysis, adjusted over a nonlinear yielding criterion used for the other investigations. We first introduce the nonlinear yield criterion used in this analysis (a parabolic criterion well adapted for concrete) as well as the proposed procedure for linearising this nonlinear criterion, as is generally addressed in the field of rock mechanics16; the nonlinear failure surface of Hoek and Brown17,18 is continuously approximated (linear approximation) by the tangent Mohr–Coulomb envelope.19 Second, we present a laboratory investigation based on the reduced physical model and the measured strains. Third, the
O. Deck et al. / International Journal of Rock Mechanics & Mining Sciences 83 (2016) 1–13
3
Fig. 2. Values of C and φ required to fit the Mohr–Coulomb yield criterion over the parabolic yield criterion, for different values of Rc and Rt ¼ Rc/10: (a) under the assumption s1 ¼ s2 o s3 and (b) under the assumption s1 o s2 ¼s3.
application of the simplified analytical approach and numerical modelling to predict the stress and strain fields is presented. Finally, a comparison between the analytical and numerical and observed results from the specific laboratory experiment are discussed.
2. Nonlinear yield criteria and the linear approximation In this study, the investigated body is a cylindrical concrete plug loaded with a uniform pressure, and an axisymmetric assumption can be made. Concrete is basically modelled with parabolic yield criteria defined with two parameters that correspond to the uniaxial compressive strength Rc and the uniaxial tensile strength Rt:
Fp ( σ ) = J2 ( σ ) +
( RC − RT ) . I ( σ ) − RC ⋅RT 1
3 3 ( σ1 − σ 2 )2 + ( σ1 − σ 3 )2 + ( σ 3 − σ 2 )2 J2 ( σ ) = 6 I1 ( σ ) = σ1 + σ 2 + σ 3
(1)
where s1, s2 and s3 are the three principal stresses, with negative values for the compressive stresses and positives values for Rc and Rt. Because of the difficulty to apply the upper-bound theorem for such yield criteria, the parabolic criteria may be approximated by a Mohr–Coulomb criterion for any specific average state of stress:
Fmc ( σ̲ ) = Fmc ( σ1, σ 3 ) = ( σ 3 − σ1) + ( σ1 + σ 3 ) sin φ − 2C cos φ
(2)
where s1 is the major principal stress, and s3 is the minor principal stress, with both being negative for compressive stresses (s1 rs2 rs3); φ is the friction angle; and C is the cohesion. In this study, an associated flow rule is considered for the Mohr–Coulomb criterion. Given specific values of the principal stresses sI, sII and sIII, it is possible to define C and φ so that the two criteria are locally equivalent:
∇ ( Fmc ( σ̲ ) ) ∇ ( Fmc ( σ̲ ) )
=
∇ ( Fp ( σ̲ ) )
⎡ 1 ⎢ ∇ ( Fmc ( σ I , σ III ) ) 2 ⎢⎣ ∇ ( Fmc ( σ I , σ III ) )
+
⎤ ∇ ( Fmc ( σ II , σ III ) ) ⎥ = ∇ ( Fmc ( σ II , σ III ) ) ⎥⎦
∇ ( Fp ( σ̲ ) ) ∇ ( Fp ( σ̲ ) )
(4)
When further assumptions are provided, it is possible to give a theoretical expression for both C and φ, in relation to Rc and Rt. In Section 3, it will be shown and discussed that the axisymmetric geometry and loading of the concrete plug induce similar values for two of the three major principal stresses. In that case, if we assume s1 ¼s2 o s3, or s1 o s2 ¼y3, then Eq. (3) gives the following:
⎧ (3+sin φ) ⎪ σ 3 = σ1 + (Rc − Rt ) 4 sin φ ⎪ ⎪ ⎪ ( σ1 − σ 3 )2 ( R c − R t ) 2σ + σ − R c R t = 0 ⎨ + ( 1 3) ⎪ 3 3 3 ⎪ ⎪ ( σ − σ ) + 2C cos φ ( R c − Rt ) − ( σ + σ ) sin φ = 0 ⎪ 3 1 3 ⎩ 1 3 for σ1 = σ 2 < σ 3 ⎧ (3−sin φ) ⎪ σ 3 = σ1 + ( R c − R t ) 4·sin φ ⎪ ⎪ ⎪ ( σ1 − σ 3 )2 ( R c − R t ) σ + 2σ − R c R t = 0 ⎨ + ( 1 3) ⎪ 3 3 3 ⎪ ⎪ ( σ − σ ) + 2C cos φ ( R c − Rt ) − ( σ + σ ) sin φ = 0 ⎪ 3 1 3 ⎩ 1 3 for σ1 < σ 2 = σ 3
(5)
The resolution of these three equations gives a value for C and
φ for a given value of s1 (Eq. (6)) or a given value of s3 (Eq. (7)),
∇ ( Fp ( σ̲ ) )
Fp ( σ̲ ) = 0 Fmc ( σ̲ ) = 0
Because the Mohr–Coulomb criterion involve edges, a singular situation must be considered when the given specific values of the principal stresses correspond to one edge, i.e., when two principal stresses are equal. In that case, one average value is calculated to define the normal direction to the Mohr–Coulomb criterion. As an example, if considering the case where sI ¼sII≠sIII with s1 ¼sI ¼sII the major principal stress and s3 ¼yIII the minor one, then first part in Eq. (3) must be replaced by
(3)
where ∇ is the gradient operator, and || || is the Euclidean norm. The first part of Eq. (3) indicates that the two criteria are tangent.
which is a more traditional choice. As an illustration, Fig. 2 shows the values of C and φ for different values of y3 and y. Fig. 3 shows a 3D plot of the two criteria for values taken from Eq. (5). It can be observed that Eq. (3) is well satisfied by a specific state of stresses but differences may be significant for other states.
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O. Deck et al. / International Journal of Rock Mechanics & Mining Sciences 83 (2016) 1–13
Fig. 3. Illustration of the Mohr–Coulomb (A) and parabolic yield criteria (B) fitted with Eq. (5) for a specific state of stresses s1 o s2 ¼ s3 (C) and s1 ¼s2 o s3 (fitting for s1 ¼ 30 MPa, Rc ¼ 30 MPa, Rt ¼3 MPa).
O. Deck et al. / International Journal of Rock Mechanics & Mining Sciences 83 (2016) 1–13 ⎧ ⎡ 4K – ( R c − R t ) ⎤ ⎪ φ = Arccsc ⎢ ⎥ ⎪ ⎢⎣ 3 ( R c − R t ) ⎥⎦ ⎪ ⎪ K + ( K + 2σ 1)· sin φ ⎨ for σ 1 = σ 2 < σ 3 ⎪C = 2 · cos φ ⎪ ⎪ ⎪ with K = ( R t − R c ) + ( R c + R t )2 − 12 ( R c − R t )· σ 1 ⎩ ⎧ ⎡ 4K + ( R c − R t ) ⎤ ⎪ φ = Arcsin ⎢ ⎥ ⎪ ⎢⎣ 3 ( R c − R t ) ⎥⎦ ⎪ ⎪ K + ( K + 2σ 1)· sin φ ⎨C = for σ 1 < σ 2 = σ 3 2 · cos φ ⎪ ⎪ ⎪ ( Rt − R c ) + R c2 + Rt2 − R c ·Rt − 3 ( R c − Rt ). σ 1 ⎪ with K = ⎩ 2
3.1. Experimental design
(6)
⎧ ⎡ 4K – ( R c − R t ) ⎤ ⎪ φ = Arccsc ⎢ ⎥ ⎪ ⎢⎣ 3 ( R c − R t ) ⎥⎦ ⎪ ⎪ ⎨ K − ( K − 2σ 3 )· sin φ for σ 1 = σ 2 < σ 3 ⎪C = 2 · cos φ ⎪ ⎪ 2 ⎪ ⎩ with K = ( R c − R t ) − R c +(R t − R c ) (R t + 3σ 3 ) ⎧ ⎡ 4K + ( R c − R t ) ⎤ ⎪ φ = Arcsin ⎢ ⎥ ⎪ ⎢⎣ 3 ( R c − R t ) ⎥⎦ ⎪ ⎪ K − ( K − 2σ 3 )· sin φ ⎨C = for σ 1 < σ 2 = σ 3 2 · cos φ ⎪ ⎪ ⎪ ( R c − Rt ) + R c2 + 2R c ( Rt − 6σ 3 ) + Rt (Rt + 12σ 3 ) ⎪ with K = ⎩ 2
5
(7)
3. Laboratory tests Two laboratory tests on a reduced scale model were conducted in the GeoResources Laboratory to study the behaviour of the concrete plug. The principle of these tests was to monitor the behaviour and final rupture of a specimen scale 1:43 with the same mechanical characteristics as the 1:1 future plug. The geometrical characteristics of the specimen are shown in Fig. 4. These tests do not take into account the effects of scales20 and are based on the hypothesis of a perfect rigid rock around the core because a concrete plug is incorporated into a steel cell. This situation is obviously not the case in the site in the underground laboratory experimental gallery, where the vicinity is the host rock characterized by a damaged zone (EDZ).
A steel cell was specifically developed for these tests. The cell exactly takes the shape of the in-situ host rock around the concrete plug. The bentonite core pressure on the life-size model is reproduced by a pressurisation chamber connected to a hydraulic servo syringe (0–60 MPa). The pressurising chamber provides a hydrostatic pressure, avoiding any type of puncture that could be induced by a conventional press. The 1:43 scale concrete plug is cast directly in the cell together with four strain gauges positioned in the plane passing through the top of the key (tooth). For the first test, the strain gauges have all been radially oriented to the specimen at mid-radius (position R1 in Fig. 4). For the second test, two gauges were positioned at the centre of the specimen: the radial (position R0) and the other axial (position A0). The other two gauges were placed respectively at mid-radius; the first radial (position R1) and the other axial (position A1). Please see the experimental protocol for further details. Preliminary analytical calculations suggest that the failure of the specimen should occur between 15 MPa and 40 MPa, depending on the input data. The test is performed with a stresscontrolled velocity of 0.5 MPa/min. The concrete used in situ consists of four different aggregates (0/2 mm, 0/4 mm, 4/8 mm and 8/20 mm) in addition to cement and adjuvants. Given the size of the sample (100 mm), the aggregate 8/20 could cause discontinuous phenomena and has thus not been used to make both samples. However, three standard uniaxial tests were performed after eight days of drying to check for any variations in concrete strength as a function of the aggregate chosen. The uniaxial compressive strengths measured for the different mixtures vary between 10.3 and 13.7 MPa. The type of component aggregate in the concrete thus does not seem to have a significant influence over its compressive strength. Following these preliminary experiments, the first sample (Fig. 5) was incorporated with aggregates of 0–4 mm, while the second sample was incorporated with granulates of 0–8 mm. For each specimen, the drying time before testing this time was 28 days. The compressive strength was assumed to be equal to Rc ¼30 MPa, while the traction strength is assumed equal to 3 MPa, based on a traditional 1/10 ratio between both strengths. With the
Fig. 4. Dimensions of the experimental set-up with the positions of the gauges (on the left) and the pressure cell (on the right).
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extrapolation from the Rc at 8 days, the European standard EN1992-1-1 give us an Rc at 28 days that is close to 20 MPa, which is less than the target value (Eq. (8)). Because concrete is known to be very heterogeneous in strength, we decided to keep the number of 30 MPa, which is the target value.
R c (28days) =
R c (8days) ⎧ ⎡ 28 exp ⎨ s ⎢ 1 − 8 ⎩ ⎣
1 ⎤⎫ 2
( ) ⎥⎦⎬⎭
(8)
with s ¼0.38, a coefficient relative to the concrete class. The tests were voluntarily stopped just after the collapse, as soon as the pressure setpoint could not be held by the hydraulic syringe, to preserve the integrity of the specimens and to visualise the fracture planes. For this purpose, the two collapsed samples were sawn off for autopsy. 3.2. Mechanical observations The first result of these tests is the occurrence of two different failure modes: a dome-shaped failure mode for sample 1 (Fig. 5), and the other failure mode with the sheared key on the entire periphery of the specimen. This difference will be investigated with analytical and numerical analysis. The starting point of the fracture is the same for both tests: in the downstream corner of the key. It therefore appears that the behaviour differences depend on the mode of crack propagation in the sample. The other direct result of these tests is the difference in the ultimate load. The first sample, with the smallest particle size, broke when the pressure reached 17 MPa, while the second sample broke at 38 MPa (Figs. 6 and 7). For test 1, the amplitude of the measured strains is divergent for gauges placed at symmetrical locations (position R1). This result shows the difficulty and uncertainty to clearly describe the whole deformation of the sample. Test 1 gauges should
theoretically show the same deformation. However, the response of the gauges is the consequence of the sample heterogeneity and the uncertainties regarding the exact strain position due to the technical difficulties of incorporating the gauges in the matrix. In fact, the aggregate size is similar to the gauge dimension, and the gauge may record some local behaviour that may be different from the one occurring in a purely continuous media. The charge and discharge cycles conducted at the beginning of confinement show a clear elastic mechanical behaviour. The cycles were not pursued with the most significant pressures to not disturb the measuring resistance to breakage by fatigue phenomena that are not expected on site. The main objective is to highlight the rupture phenomena and their level. The loading and unloading cycles show no deformations for test 1 and no hysteresis cycles for test 2. An estimate of the modulus of deformation from these cycles does not seem serious. Qualitatively, however, the curves show that the monitored strains in sample 1 are greater than in sample 2. One gauge placed in the R1 position in sample 1 shows a strain up to 7000 microstrain, while the similar gauge in sample 2 placed at the same location and direction shows only 2000 microstrain. Comparing these two gauges in the same range of stress (0–17 MPa) shows an even larger contrast. Sample 1 is thus more deformed than sample 2 during the loading and reached its breaking point earlier. The difference in size of the granulates offers the first explanation of the difference between both tests. However, two failure modes appear to be possible for this sample geometry. This mechanical aspect is discussed below in the numerical modelling Section 5.
4. Upper- and lower-bound solutions This section is dedicated to the analytical assessment of the ultimate load based on limit equilibrium analysis.
Fig. 5. Sample 1 (on the top) and sample 2 (on the bottom) after the test condition (full view and section); the red dotted line highlights the exposed fractures.
O. Deck et al. / International Journal of Rock Mechanics & Mining Sciences 83 (2016) 1–13
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Fig. 6. Deformation curves as a function of the pressure for test 1; the four gauges have the same position R1 with a π/2 lag around the sample.
Fig. 7. Deformation curves as a function of pressure for test 2.
4.1. Lower-bound solution The lower-bound solution is investigated with a statically allowable state of stresses. A parabolic yield criteria is considered with Rc ¼30 MPa and Rt ¼3 MPa. A cylindrical coordinate system (r, θ, z) is considered with the origin 0, as plotted in Fig. 8. Pressure p is assumed to be positive. Because of the axisymmetric geometry, loading, and boundary conditions, all stresses must be independent of θ; srθ and sθz are null, sθθ is always a principal stress, and the equilibrium equations reduce to:
∂σrr ∂σrz σrr − σθθ + + =0 r ∂r ∂z ∂σzz ∂σrz σrz + + =0 ∂z ∂r r
(compressive, with p a positive value). This state of stresses respects both Eqs. (8) and (9) and the boundary conditions of the region. More specifically, no sliding may occur on the right border because sθθ is principal and no shear stresses are calculated over the border. In region 4, all stresses are assumed to be null. This assumption leads to a free-stress boundary condition under region 2. In region 2, the boundary conditions impose constant values of szz and srz on the upper and lower circular boundaries:
σzz,2 (z=H ) = − p σzz,2 (z=0) = 0
(9)
σrz,2 (z=H ) = 0 σrz,2 (z=0) = 0
(10)
The medium is divided into four regions (Fig. 8), in which four statically admissible stress fields must be founded. These stress fields must be consistent with the equations of equilibrium and the boundary conditions of the stresses. Consequently, most of the stress tensor components must be continuous between two adjacent regions. The following stress field is considered: In region 1, szz, sθθ and srr are all assumed to be constant, principal, and equal. The upper boundary imposes szz,1 ¼ p
(11)
These boundary conditions may be satisfied by considering a 3d degree polynomial expression for szz,2 that satisfies Eq. (11):
σzz,2 (z ) = −
6·p ⎛ z 3 H·z 2 ⎞ ⎟ .⎜ − 3 2 ⎠ H ⎝ 3
(12)
By substituting Eq. (12) in Eq. (10), it can be found that
σrz,2 (z, r ) = −
3·p·r·z·(H − z ) H3
+
k r
(13)
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Fig. 9. 3D plot of Fp(s) over the entire region 2 (axes r and z) for the statically admissible state of stresses that give the maximal value of the load p.
to be a lower bound of the strength capacity for a compressive strength Rc ¼30 MPa (Table 1). Others values can be assessed if the compressive strength is assumed to be greater. Fig. 8. Discretization of the casing into four zones for the lower-bound solution.
where k is a constant of integration equal to 0 to satisfy Eq. (11). Finally, sθθ and srr are assumed to be equal and can be determined by substituting Eq. (13) into Eq. (9):
σrr,2 (z, r ) = σθθ,2 (z, r ) = −
3·p·r 2 2H 3
. (2z − H ) + k1 (z )
(14)
where k1(z) is a function of z. A polynomial function is used for k1 (z), and the state of stress is now compared to the yield criteria (Eq. (1)) to obtain the maximal possible value for pressure p. When a 0 degree polynomial is used, the greater value of p is found to be equal to 8.6 MPa for k1 ¼ 26.14. When a first degree is used, a value of p ¼16.6 MPa is found for k1 ¼1.6–1695z. When other degrees are tested, the value of p remains the same. For the first degree polynomial, the value of Fp(s) (Eq. (1)) is plotted over the whole region 2 (Fig. 9). This value of p will correspond to a lower bound if a statically admissible state of stresses may also be found in region 3. In this region, sθθ,3 and srr,3 are still assumed to be equal, and szz,3 is assumed to be constant. Eq. (7) implies srz,3(r,z) to be an inverse function of r, and continuity between region 2 and 3 is used to obtain the final expression:
σrz,3 (z, r ) =
3·p·R2·(z 2 − Hz ) H3r
⎛ r ⎞⎞ 3pR2 (2z − H ) ⎛ . ⎜ 1 − 2.Ln ⎜ ⎟ ⎟+k1 (z ) ⎝ R ⎠⎠ ⎝ 2H 3
4.2.1. Theoretical developments The upper-bound solution investigation requires a linear yield criterion. The Mohr–Coulomb yield criterion is used and calibrated over the parabolic criterion to allow for comparison with the previous lower-bound solution (Fig. 3). A kinematically admissible velocity field is required to obtain an upper-bound solution. For this purpose, the plug is divided into a number of rigid regions separated by slip surfaces, where both the Mohr–Coulomb criterion and the plastic flow rule are respected. The first condition signifies that Eq. (2) must be satisfied over the entire slip surface. The second condition signifies that the relative velocity of two adjacent regions separated by a given slip surface is not parallel to the surface but inclined with an angular equal to the dilatancy angle. For an associated flow rule with the Mohr–Coulomb criterion, the dilatancy angle is set equal to the friction angle. Thus, if Vij corresponds to the relative velocity of zone i compared to zone j, then the plastic flow rules requires an angle φ between Vij and the slip surface. Because of the axisymmetric geometry, Vij is assumed to be directed into the plane of any vertical section that includes the symmetry axis. Vij can then be split into normal and tangential velocity components vij and uij:
uij /vij = tan φ
(15)
Eqs. (6) and (12) imply that srr is a logarithmic function of the variable r. Continuity of srr between regions 2 and 3 finally is used to obtain
σrr,3 (z, r )=σθθ,3 (z, r )=
4.2. Upper-bound solution for the Mohr–Coulomb yield criterion
(16)
where k1(z) is the linear function used in Eq. (14). Eqs. (14) and (15) give no conditions about the value of szz,3; thus, any value may be considered. The value that gave the maximal strength is found to be equal to 5 MPa (compressive stresses). The yield criterion (Eq. (1)) is then respected over the whole zone (Fig. 10). This stress is compatible with the boundary conditions if the interface is assumed to be stronger than the zone. More specifically, tangential stresses are not assumed to be limited along these borders because the teeth are wedged into the steel cell and that any sliding is then assumed to be impossible. In conclusion, the ultimate capacity of pmin ¼ 16.6 MPa is found
(17)
where vij and uij are the tangential and normal components, respectively, to a slip surface of the velocity vector, and the rate of energy dissipation per unit slip area dij is calculated using:
dij = C·vij
(18)
where C is the cohesion of the slip surface, and vij is the norm of the relative tangential velocity between the two zones i and j. The in-plane section of the plug is subdivided into 3 zones (Fig. 12). Zone I is the in-plane triangular key (tooth). Zone I is assumed to be fixed, and VI ¼0. Zone II is the in-plane triangular shape zone that includes a part of the upper boundary where the load p is applied. A vertical slip surface is then introduced between zones I and II and an inclined surface, with angle α formed between zones II and III. When the ultimate load is reached, zone II is assumed to move with a velocity VII equal to the relative velocity VI/II between zones I and II. To satisfy the flow rule, VI/II is inclined at an angle φ with respect to the slip surface. Because the problem is axisymmetric, the velocity of zone II cannot be associated to a
O. Deck et al. / International Journal of Rock Mechanics & Mining Sciences 83 (2016) 1–13
9
of each zone. From the figure, each relative tangential velocity can be deduced:
‖VII ‖ =
sin (α − φ) ‖VIII ‖ sin (α )
vII / I = cos φ‖VII ‖ =
vIII / II =
Fig. 10. 3D plot of Fp(s) over the entire region 3 for the statically admissible state of stresses that give the maximal value of the load p.
Table 1 Results of the lower bound of the ultimate load p for different mechanical strengths (Rc and Rt). Rc [MPa]
Rt [MPa]
p [MPa]
30 40 50
3 4 5
16.6 22.2 27.7
rigid movement and will cause tangential strains. Zone III is the inplane trapezoidal shape zone that includes the free lower boundary and a part of the upper one. Only a vertical velocity is kinematically admissible because horizontal displacements are incompatible with the boundary conditions. VIII is then vertical, but the relative velocity between zones II and III, VII/III, must be inclined at an angle φ with respect to the slip surface. In Fig. 11, a hodograph shows the velocity and relative velocity
(19)
sin (α − φ) cos φ‖VIII ‖ sin (α )
sin (2φ) ‖VIII ‖ 2 sin (α )
(20)
(21)
the angle α may be any angle between the maximal and minimal values. The maximal value is arctan(R/L) because, for a greater value, the cylindrical zone II is not hollow and VII is not kinematically admissible. The minimum value is φ because the hodograph cannot be closed for a smaller value. The ultimate load p may then be determined by equalizing the working rate of the external forces and the rate of internal energy dissipation .4 The load p is the single external force that works for the investigated velocity field. The working rate is then equal to:
(
)
= pπD2||VIII || + p·π · R2 − D2 ||VII || cos φ ⎞ ⎛ sin (α − φ) cos φ⎟ ||VIII || = pπ ⎜ D 2 + R 2 − D 2 ⎠ ⎝ sin (α )
(
)
(22)
where D is the radius of the upper part of region III. The internal energy dissipation is calculated by integrating the rate of energy dissipation per unit of area (Eq. (18)) over all of the slip surfaces:
= II / I + III / II = 2πCRHvII / I + C = 2πCRH +C
(
(
π R 2 − D2 sin (α )
)v
III / II
sin (α − φ) cos φ sin (α )
)
π R2 − D2 sin 2φ 2 sin2 (α )
‖VIII ‖
Fig. 11. Definitions of the 3 zones used to obtain an admissible velocity field and the associated hodograph.
(23)
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An upper bound of the ultimate load p may then be assessed by combining Eqs. (22) and (23):
=
(24)
4.2.2. Application The upper-bound method is applied, and the ultimate load p may be calculated for any value of α∈[φ, arctan(R/L)]. However, the values of C and φ must be chosen in relation to the stress state because the parabolic criteria is linearised. Different assumptions are made for the minor principal stress. s3 is set to be equal to 10 MPa, 20 MPa, Rc, or p. The two major stresses s2 and s1 are assumed to be equal. The numerical results, which will be described in Section 5, will show that the more realistic assumption is s3 equal to 10 MPa and s2 and s1 are not perfectly equal, but are similar. When s3 is assumed to be equal to p, an iterative procedure is applied to find an upper bound of the ultimate load. For given values of Rc, Rt and α, and the first assumption s3 ¼ Rc, the two values of C and φ are calculated as in Fig. 2b. The value of p is then calculated using Eq. (24). If p is different than the initial value of s3, then the procedure is repeated with the new value s3 ¼ p until the ultimate load is found to be equal to initial value of s3. Different ultimate loads can be obtained for each value of α. A second analysis shows that the smaller value of p is found for α ¼ φ. The final results are shown in Table 2, which indicates that the maximal value of p is not strongly dependant on the assumption regarding s3.
5. Numerical results A finite element model is developed to compare the analytical and numerical results. The objectives are as follows: to compare the ultimate load p with those assessed by the upper- and lowerbound methods; to compare the slip lines used by the upperbound method with the yielding areas given by the numerical model; to compare the stress state given by the numerical model with those considered by the lower-bound solution or with assumption about s1, s2 and s3 considered by the upper-bound solution; and to compare the strain curves with those given by the monitored samples. The computations are performed using Cesar software (ITECH, France). Fig. 12 shows the model with its mesh and boundary conditions. Horizontal displacements are not fixed over the vertical right boundary because the model displays a horizontal displacement directed to the axisymmetric axis. A null displacement would then induce tensile radial stresses that cannot exist in the real experiment because the interface between the concrete and steel has no tension resistance. These displacements are consistent with the direction of VII defined in the upper-bound analysis (Fig. 11). Six nodes triangular elements are used. Different mesh sizes were tested, and an average length of 2.5 mm was found to give robust results. The ultimate load is calculated for two yielding criteria of the concrete. First a parabolic criteria is used with Rc ¼30 MPa and Rt ¼3 MPa; this case corresponds to the more realistic behaviour of the concrete, and this is the same criteria used for the lower-bound solution. Second, an associated Mohr–Coulomb criterion is used with C¼ 20.25 MPa and φ ¼19.5° (values taken from Table 2 when s1 ¼s2 os3 and s3 ¼ 10 MPa); this case is the same criteria used for the upper-bound solution with values of C and φ, so that the Mohr–Coulomb criterion fits the parabolic one. External load p is imposed with 2 MPa increments. A set of iterations is performed between each increment when yielding
occurs. The ultimate load is reached when the calculations no longer converge. The results of the numerical model are illustrated in Fig. 13. The ultimate load depends on the yielding criteria used for the concrete. An ultimate load of 43 MPa is found for the Mohr–Coulomb criterion when the parabolic criterion gives 48 MPa. These values are more comparable with those given by real test 2 (38 MPa) than by test 1 (17 MPa). For a similar case, the lower- and upper-bound solutions gave an ultimate load of 16.6 and 45 MPa, respectively. A first conclusion is that the lower-bound solution appears to strongly underestimate the theoretical ultimate loads, while the upper-bound solution appears to give a closer solution. However, the theoretical solution that seems to be well approximated by both the numerical model and the upper-bound solution are greater than the two observed ultimate stresses. Fig. 13 shows values of strains in the same position and directions as the gauges used in the experiment (A0, A1, R0 and R1 position). The two results are quite identical, but different from the results given in the experiment (Figs. 6 and 7). Only gauge R1 (Fig. 7) is similar between the two results. For the other gauges, the values and/or signs are different. These discrepancies can probably be partially explained by the difficulty encountered for placing the gauges into the fresh concrete when the samples were manufactured. There is no proof that the gauge did not move. Fig. 13 shows the isovalues of the plastic strain norm that determine the sliding planes along which failure occurs. The plastic strain norm is calculated with the “Frobenius norm” of the plastic strain tensor « ε » and is equal to the square root of the trace of the ε.ε matrix. Similarities are found between the numerical result and the sliding planes defined in Section 4.2 for the upper-bound solution. A conical slip surface can be identified, similar to the one considered in the upper-bound analysis. Moreover, the large yielding area that appears in the tooth is consistent with the second slip surface also considered. However, when failure occurs, none of the two slip lines are extended over the whole model, as in Fig. 11, and a contact between the two appears. These similarities may explain why the results of the numerical model and the upper-bound method are close. Table 2 Results of the upper bound of the ultimate load p for different mechanical strengths (Rc and Rt) and different values of s3 under the assumption s1 ¼ s2 os3. Initial mechanical strengths Equivalent Mohr–Coulomb parameters
Ultimate load
C [MPa]
φ [deg]
p [MPa]
Hyp s3 ¼ 10 30 3 40 4 50 5
20.25 26.3 32.3
19.5 20.4 21
45.24 60.5 75.9
Hyp s3 ¼ 20 30 3 40 4 50 5
22.2 28.4 34.4
16.9 18 18.8
45.7 60.45 75.4
Hyp s3 ¼ Rc 30 3 40 4 50 5
23.9 31.9 39.9
15.2 15.2 15.2
46.8 62.4 78
Hyp s3 ¼ p 30 40 50
25.3 35 44
14.1 13.4 13.1
38.4 61 81
Rc [MPa]
Rt [MPa]
3 4 5
O. Deck et al. / International Journal of Rock Mechanics & Mining Sciences 83 (2016) 1–13
Fig. 12. Mesh and boundary conditions of the numerical model under the axisymmetric assumption.
6. Discussion The results show that the upper- and lower-bound theorems as well as the numerical modelling may be used to assess the ultimate load capacity of the studied concrete plug, which has some discrepancies with the experimental values. However, all these different approaches display some advantages and disadvantageous
11
that can be highlighted. The experimental results may first appear to be the most credible because no theoretical assumptions are made. However, the results are logically scattered due to the concrete heterogeneity. The mechanical behaviour between two similar samples is known to be very scattered due to the unique distribution of aggregates for each sample, the existence of randomly distributed micro-cracks in the cement matrix and the difference in the cement hydration. As a consequence, the experimental tests are not sufficient unless a large number of measurements can be performed, and the experimental results must be interpreted to give the most interesting information. Experimental interpretation involves a comparison of the results with some predictions. Numerical analysis is a powerful method to obtain the theoretical prediction of the ultimate load of any system. Numerical analysis can account for many complex mechanical behaviours and boundary conditions of mechanical systems, for which no theoretical and analytical solution may be found. Due to the development of even more complex yielding criteria, numerical results suffer two disadvantages. First, numerical results must be validated, i.e., compared with other results (experimental if available), to increase the confidence in their use. Second, numerical results must be interpreted, as is the case with experimental results, to give the most interesting information. Analytical solutions, such as those described in this paper, based on the upper- and lower-bound theorems are a good tool to help provide these interpretations. As shown in Section 4, the limit analysis suffers some important weakness because it requires linear yielding criteria to be easily used and because the results are only given over an interval, which may be large. In that case, the
Fig. 13. Isovalues of the plastic strain norm for the ultimate load and the Mohr–Coulomb yield criterion (left, for C¼ 8.4 MPa and φ¼ 31.8°) or the parabolic one (right).
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Fig. 14. Comparison of the stresses on Section 1 for: (a) the lower-bound solution and (b) for the numerical model for a load p ¼ 46 MPa.
the sample, is used to predict where micro-cracks are the most likely to occur, even if the concrete plug is not expected to break out as the samples. As a final application, the results will also be used to design the concrete reinforcement of the real concrete plug (Fig. 15).
7. Conclusion
Fig. 15. Concrete block in the drift sealing experiment NSC (for Noyau de Scellement, Fig. 1) performed in the Andra (French National Radioactive Waste Management Agency) underground research laboratory in Bure, France.3
analytical solution must not be seen as a competitive approach to precisely assess the ultimate load, but as a powerful method of interpretation. Based on this study and utilising the lower- and upper-bound theorems, the ultimate load of the concrete casing may here be interpreted as two simple mechanisms. The first mechanism corresponds to a vault mechanism in which the pressure p is transferred to the key. This behaviour is the most intuitive and corresponds to the lower-bound stress state. The second mechanism is less intuitive, as it corresponds to the influence of the dilatancy angle of concrete when a failure starts to occur in the key. The shearing of the key involves a centipede displacement of the external part of the casing that leads in turn to an axial expulsion of the internal part. The final comparison between the analytical and numerical results clearly shows that the real behaviour is probably a mix between these two simple mechanisms. These two mechanisms can now be used to organise the discussion between all engineers and researchers that study the behaviour of the casing. The sealing drift experiment (NSC) is currently underway with the real-size concrete plug (Fig. 14). The concrete plug will be monitored with geophones to detect and locate micro-cracks with their acoustic emissions. The results of this study, which focus on
This study compared the ultimate load of a concrete plug among different types of assessments: experimental, analytical and numerical. The two investigated samples were reduced-scale concrete plugs of potential concrete plugs that would be used in a drift sealing concept in the framework of an underground storage for nuclear waste. A steel cell was developed for these tests. Concrete samples were built inside and then tested with a uniform pressure until collapse of the concrete. The analytical and numerical results are based on similar yielding criteria: Mohr–Coulomb or parabolic. For the analytical results, the upper- and lower-bound solutions are described. The results show some good agreements and discrepancies among all the results. The analytical upper-bound method gives results close to the numerical results. Both approaches appear to overestimate the real ultimate load, whereas the lower-bound solution appears to underestimate the real ultimate load. Discrepancies between the experimental and theoretical results are probably due to local stress concentrations in the concrete sample because granulates inside the sample have significant diameters compared to the sample diameter. Moreover, the introduction of gauges inside the sample is also responsible for some of the heterogeneities. As a final conclusion, the authors think that the upper-bound solution gives a good approximation of the ultimate load for a homogeneous material. For small concrete samples, excessive heterogeneities lead to a strong reduction in the ultimate load. These reduced-scale results cannot be extrapolated to real scale 1 concrete blocks, such as those used in the NSC experiment, especially because scale 1 blocks may be considered to be homogeneous, and the expected stress on the block wall is well below the resistance line. However, the reduced-scale results may help to understand the behaviour of the concrete block, together with an acoustic emission survey that has been installed for this experiment, to monitor potential damage that could occur in the concrete plug during the swelling of the core.
Acknowledgements The authors thank the Laboratory of Rock Mechanics of GeoResources for the logistic and technical support in the experimental tests and Andra for providing funding.
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11. 12.
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