Rock Mechanics, Nelson & Laubach (eds) © 1994 Balkema, Rotterdam, ISBN 90 54103808

An investigation of rock joint models on prediction of joint behavior under pseudostatic cyclic shear loads

Sui-Min (Simon) Hsiung, Amitava Ghosh & Asadul H.Chowdhury Center for Nuclear Waste Regulatory Analyses, San Antonio, Tex., USA

Commonly used empirical representations of jointed rock behavior under cyclic loading conditions reside in the Mohr-Coulomb, Barton-Bandis, and Continuously-Yielding models. These models were developed based on data obtained under unidirectional loading conditions, and their ability to predict joint performance under cyclic loading conditions has not been tested. Analysis has indicated that these models are not capable of predicting joint behavior observed in the laboratory during pseudostatic shear reversal.

1 INTRODUCTION

In 1987, the United States Congress designated Yucca Mountain, in southern Nevada, as the only site to be characterized to determine its suitability for a high-level nuclear waste repository. The proposed repository horizon is about 300 m beneath the surface, in a densely welded prominently vertically and subvertically jointed tuff. An important phenomenon that could affect the preclosure and postclosure performances of a repository is repeated ground motion due to seismic activities (Kana et aI., 1991). The fundamental failure mechanism for an excavation in a jointed rock mass subjected to repetitive seismic loading is the accumulation of shear displacement along joints (Hsiung et aI., 1992).

Conditions for joint slip or for sliding of individual blocks from the boundaries of excavations are governed by the shear strengths of the discontinuities concerned. The conditions for behavior to be considered explicitly in assessing the response of the jointed rock mass surrounding an opening include: (1) pseudostatic loadings, (2) dynamic loadings, and (3) repetitive episodes of dynamic loadings arising from a series of earthquakes or underground nuclear explosions.

The Mohr-Coulomb, Barton-Bandis, and Continuously-Yielding models are three commonly used empirical representations of rock joint behavior. These models were developed based primarily on data taken under unidirectional pseudostatic loading conditions. The ability of these models to predict joint performance under cyclic pseudostatic loading conditions as well as the second and third conditions listed above has not been tested. This paper assesses these models and their ability to predict joint behavior under cyclic pseudostatic shear loads by comparing with experimental results.

2 LABORATORY OBSERVATION OF JOINT CYCLIC SHEAR BEHAVIOR

Two important, distinct features have been identified through the examination of laboratory-scale direct shear experiments on rock joints of Apache Leap (in Arizona) tuff under cyclic loading conditions; one is that the shear strength upon reverse shearing is smaller than that during forward

111

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,-:tS

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Figure 1. A typical experimental result of joint (a) shear stress response and (b) dilation under pseudostatic cyclic loading condition as a function of normal stress.

shearing, and the other is that the joint dilation resulting from forward shearing recovers during reverse shearing (Hsiung et aI., 1993). These two features of joint behavior are independent of the environmental conditions under which the jOint is tested. Figure l(a) shows a typical shear stress response of a rock joint under various normal stress levels. The test sequence followed an ascending order with respect to normal stress. The curve with the I-MPa normal stress (the first cycle of shearing) illustrates the shear behavior of an originally undamaged (fresh) joint and shows a distinct peak shear strength at the early stage of the shear cycle. The shear strength of the joint gradually reduces to a residual value as shear displacement increases. No distinct peak shear strength was observed for other cycles of shearing. Another feature that can be observed in this figure is that the shear strength during the reverse shearing is smaller than that during the forward shearing. The relation between normal displacement (dilation) and shear displacement for the same specimen is given in Figure 1(b). This figure indicates that joint dilation at various levels of normal stress increases with shear displacement during forward shearing. For reverse shearing, there is some degree of hysteresis in the jOint dilation, with the reverse-shearing dilation curve being below that of forward shearing and decreasing towards zero. In general, for repeated shear cycles, the amount of joint dilation decreased with increasing normal stress. The same behavior has also been reported by other researchers (ling et al., 1992; Wibowo et aI., 1992; Huang et aI., 1993) for rock joint replicas under cyclic pseudostatic loads.

3 EVALUATION OF MOHR-COULOMB JOINT MODEL

The simplest model for joint strength and deformation is the Mohr-Coulomb model, a linear deformation model that includes a shear failure criterion for a rock joint

'To = C + lIn tan t/l

where 'To is shear strength along the joint, lIn is normal stress across the joint, C is cohesion, and t/l is friction angle. Once 'To is reached, the joint deformation is perfectly plastic. This equation suggests that the joint strength is the same in forward and reverse shearing directions. The joint shear response is governed by a constant shear stiffness K.

(2) tI.'T = K. tl.u.e

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where /:;.T is incremental shear stress and /:;.u/ is an elastic component of the incremental shear displacement. Based on Eqs. (I) and (2), /:;.T becomes zero after the condition I TI = To is reached, where T is the shear stress on the joint.

The Mohr-Coulomb joint model in its basic form does not consider joint wear and dilation behavior. However, the dilation behavior may be added. For example. the dilation may be restricted (that is, the dilation angle ift is zero) until shear stress has reached the shear strength of the joint, that is, joint dilation starts after the joint begins to deform plastically. Since there is no wear of the joint, joint dilation angle should remain constant with shear displacement. The dilation angle becomes zero after a critical shear displacement is reached. A form of the model has been given by ITASCA Consulting Group, Inc., 1992

(3) if ITI < To, then ift 0

and

(4) if ITI To and I <!: ucs• then ift = 0

where Us is the joint shear displacement and ucs is the critical shear displacement. Eq. (4) suggests that joint dilation should continue to increase even during reverse shearing.

Figure 2(a) shows the predicted joint shear behavior using the Mohr-Coulomb model compared to the actual experimental results. For clarity. results using normal stresses of 1. 3, and 5 MPa are shown. Results using intermediate normal stresses 2 and 4 MPa are omitted. A friction angle of 40.7 0 and zero cohesion were assumed for each computation. A shear stiffness, K., of 16.65 GPafm and a dilation angle, ift, 3.6 0 were used when the normal stress, un' was 1 MPa. A K. of 9.18 GPafm and a ift of 2.6 0 were used when un was 3 MPa. For 5 MPa normal stress, 10.14 GPafm and 2.3 0 were the values of K. and ift, respectively. These values were determined from the experimental results of each cycle. It is ch;:ar from the figure that the Mohr-Coulomb joint model simulates the residual shear phenomenon in the forward direction quite well but cannot simulate the peak behavior. As there is no damage accumulation. the predicted strength is independent of shear displacement. Although the Mohr-Coulomb model simulates the 'first-order' shear behavior in the forward direction very well, it overestimates the strength in the reverse direction. This has been observed in each of the sixteen specimens tested.

Figure 2(b) shows the dilation response of the joint when a constant dilation an!1le of 3.6 0 was

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Figure 2. Comparison of simulated joint (a) shear response and (b) dilation using Mobr-Coulomb model with experimental results.

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used with a normal stress of 1 MPa. The figure shows that the Mohr-Coulomb model predicts the dilation behavior in the forward direction reasonably well. In the reverse direction, the dilation curve predicted by Eq. (4), shown as Modell, does not show any dilation recovery (joint compaction) with increasing shear displacement. But the joint compaction can be simulated numerically if the dilation angle is assumed negative while shearing in the reverse direction. This is shown as Model 2 in Figure 2(b). In this case, the dilation curve for reverse direction overlaps the dilation curve for forward direction, as shown in Figure 2(b).

4 EVALUATION OF BARTON-BANDIS JOINT MODEL

The Barton-Bandis model was proposed with the intent to take into consideration the effect of joint surface roughness on joint deformation and strength. The nonlinear Joint strength criterion can be expressed as (Barton et al., 1985)

(5) To = an tan [JRC log10 [J~S 1 + !/I,] where JRC is joint roughness coefficient, JCS is joint wall compressive strength, and !/I, is residual joint friction angle. The attrition of the surface roughness or reduction of the JRC is represented in a piece-wise linear manner (Barton et al., 1985). When the joint is not sheared, it has a maximum JRC value, JRCpe<lk' As the joint is sheared, the JRC decreases as a result of attrition of asperities on the joint surface. The peak shear stress (shear strength) is reached at a joint shear displacement up that can be calculated using the following equation (Barton and Bandis, 1982)

(6) n JRC L [ ] 0.33

Up = 500 Ln

where Ln is the length of a test specimen. Based on Eq. (5), the joint shear stress under a constant normal stress depends solely on the

JRC value. When a joint is sheared in one direction, its roughness is reduced from JRCpeak to a value, say JRCc• If, at this point, the direction of the shear is reversed, the initial shear stress required for the Joint to be sheared in the reverse direction is controlled by the JRCc value, following Eq. (5), which serves as the maximum JRC in the reverse direction. In other words, the Barton-Bandis model assumes that the shear strength at initiation of the shear in the reverse direction is equal to the shear strength when the forward shearing was stopped. Also, Eq. (5) suggests that joint wear will stop after the JRC becomes zero. This condition will be reached when the ratio of the actual shear displacement u. to the shear displacement lip is greater than 100 (Barton et al., 1985). Once the JRC becomes zero, the joint shear essentially resumes the Mohr­Coulomb type of behavior.

The Barton-Bandis joint model also recognizes the dilatant nature of joints and suggests that the angle of dilation should be a function of the JRC. The relation between the JRC and dilation angle 1ft is (Barton et al., 1985)

(7) 1ft = 0.5 JRC log10 [J~S 1 This equation indicates that, as joint surface roughness wears, its angle of dilation decreases. In other words, the rate of dilation becomes smaller as joint shearing progresses. The dilation angle will eventually become zero, that is, there will be no further dilation if uiup becomes greater than 100. Judging from the nature of Eq. (7), 1ft is always positive. Joint dilation will continue to increase, although at a gradually slower rate, even after the direction of shear has been reversed.

Figure 3(a) illustrates the shear response of the joint for a JCS of 106.8 MPa, a JRC of 11.9, and a residual friction angle, !/Ir. of 28.4° with normal stresses of 1, 3, and 5 MPa. Responses for

114

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Figure 3. Comparison of simulated joint (a) shear response and (b) dilation using Barton-Bandis model with experimental results.

intermediate an of 2 and 4 MPa are not shown in the figure for clarity. JRC was back-calculated from Eq. (5) after measuring ICS and <Pr independently. The damage accumulation model reduces the IRC value as shear displacement increases. The figure shows that the predicted shear stress in the first cycle is quite close to the actual during forward shearing but overestimates the strength in the reverse direction. The predicted shear strength at the beginning of reverse shearing is exactly the same as the strength at the end of forward shear. This phenomenon is true for all subsequent cycles. This figure also shows that the damage to the joint surface in the Barton-Bandis model accumulates much faster than that observed in actual experiments. As a result, the rough joint surfaces quickly become smooth (Le., JRC value becomes zero) and the shear response is controlled by the residual friction angle afterwards. Consequently, the strengths achieved in subsequent cycles with higher normal stresses are significantly lower than that observed in actual experiments.

Figure 3(b) shows the dilation of the jOint for only the first cycle when the applied normal stress was 1 MPa. The figure shows that the Barton-Bandis model overestimates the dilation in the forward direction. Dilation in the reverse direction was calculated using Eq. (7) and is shown as Modell. This curve does not show any joint compaction. Contrary to the experimental observation, Model 1 actually shows further increase in dilation while the top block is moved in the reverse direction. Curve designated as Model 2 was calculated assuming negative of the dilation angle in the reverse direction. This curve shows compaction of the joint in reverse shearing but the dilation recovery is far from that observed. The rate of damage accumulation in the Barton-Bandis model is so high that the IRC at the end of forward shear is significantly less than the initial value. As result, the dilation recovery calculated using Eq. (7) while shearing in reverse direction is not sufficient to return the dilation close to zero, as observed in experiments.

5 EVALUATION OF CONTINUOUSLY-YIELDING JOINT MODEL

The Continuously-Yielding model for rock joint deformation was developed by ITASCA Consulting Group, Inc. (ITASCA Consulting Group, Inc. 1992; Cundall and Lemos, 1988). The

115

model can simulate progressive damage of the joint surface during shear and displays irreversible nonlinear behavior from the onset of shear loading. The shear stress increment is

(8) (1 - Tiro) t:.T = F KS t:.US = KS t:.us (1 - r)

where t:.u. is the incremental shear displacement and the shear stiffness Ks may be a function of normal stress. F is a factor that governs the shear stiffness. It depends on the difference between the actual stress curve and the bounding strength curve To' The factor r in Eq. (8) is initially zero. It restores the elastic stiffness immediately after load reversal. That means r is set to Tiro (F becomes 1). In practice, r is restricted to a maximum value of 0.75 to avoid numerical noise when the shear stress is approximately equal to the bounding strength, To' The bounding strength, To' is

(9) To '" (Tn tan ¢m sign (t:.us)

where ¢m is the friction angle that would apply if the joint is to dilate at the maximum dilation angle and is initially equal to the joint initial friction angle, ¢mo' As damage accumulates, ¢m is continuously reduced according to

(10) ¢m = (¢mo - ¢) e(-u: I R) - ¢

where u! is the plastic shear displacement and ¢ is the basic friction angle of the rock surface. R is a material parameter with a dimension of length that expresses the joint roughness. A large value of R produces slower reduction of ¢m and a higher peak stress. The peak is reached when the bounding strength equals the shear stress. After the peak, the joint is in the softening region and the value of F becomes negative.

Based on Eq. (9), joint bounding shear strength under a constant normal stress depends solely on the friction angle, tPm• When a joint is sheared in one direction and its friction angle, ¢m' is reduced from its initial value tPmo to a value, say tPj ; and at this point, the direction of the shear is reversed, the corresponding bounding shear strength in the reversed direction, according to Eq. (9), is controlled by the tPp which serves as the maximum ¢m in the reverse direction. In other words, the Continuously-Yielding model assumes that the maximum bounding shear strength during reverse shearing is the same as the bounding shear strength at the end of the forward shearing process.

The formulation of joint dilation angle in the Continuously-Yielding model is expressed as

(11) ~ = tan-I [I;~ 1 - tP

Dilation takes place whenever the shear stress is at the bounding shear strength level, and is obtained from the friction angle, ¢m (Cundall and Lemos, 1988). Examining Eq. (11) reveals that the joint dilation, as treated in the Continuously-Yielding model, should continue to increase regardless of whether a joint is sheared in the forward or reverse direction. However, the rate of dilation decreases gradually and is governed by the value of ¢m' After the bounding shear strength is reached, Eq. (11) can be rewritten as

(12) ~ = (tPmo - ¢) e(-u: I R)

Note again the increasing nature of joint dilation as shown in Eq. (12).

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Figure 4. Comparison of simulated joint (a) shear response and Yielding model with experimental results.

(b)

dilation using Continuously-

Assuming ajoint with a constant shear stiffness, K., of 19.68 GPalm, initial friction angle, "p""" of 48.9°, basic friction angle, "p, of 36.0·, and roughness parameter, R, of 0.0105 m, the predicted joint shear strength using the Continuously-Yielding model subjected to normal stresses, (111' of 1,3, and 5 MPa is illustrated in Figure 4(a). These parameters were determined using the data from the first cycle of forward shearing only. The damage accumulation model adjusts the "pm value as shear displacement increases.

Figure 4{a) shows that the Continuously-Yielding joint model predicts the shear strength quite well during the first cycle, although it overpredicts the strength during reverse shearing. The predicted shear strength in the reverse direction is almost the same as that in the forward direction. This is a general trend observed in the study. The figure shows another important feature of the joint model. The shear strength predicted by the Continuously-Yielding joint model quickly reaches the residual strength value when compared with the experimental curves. In other words, joint roughness degradation occurs more rapidly in the Continuously-Yielding joint model than in the actual experiment. As a result, numerically the joint becomes two plane surfaces whose behavior is controlled by the basic friction angle following Mohr-Coulomb friction law. Consequently, in subsequent cycles of the experiment, the predicted shear strength is always smaller than the experimental results.

Figure 4(b) shows the dilation response of the joint using the same parameters. For clarity, only the first cycle «(1n equal to 1 MPa) is shown. The predicted dilation curve in the figure shows overprediction in the beginning and underprediction thereafter. The dilation angle quickly decreases to zero with shear displacement. Due to rapid degradation of the joint roughness, the predicted dilation at the end of forward shearing in the first cycle is smaller than observed experimentally. In the reverse direction, the dilation curve does not return to zero. It stays almost at the same level as the peak. value during forward shearing (Modell). This confirms that the roughness of the joint is completely worn off during forward shearing. Due to the same reason, the dilation recovery is minimal when the dilation angle is taken as negative in the reverse direction (Model 2).

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6 DISCUSSION AND CONCLUSION

Careful examination of the respective equations representing the Mohr-Coulomb, Barton-Bandis, and Continuously-Yielding rock joint models has revealed that all three models were essentially developed for unidirectional shearing. They assume the same joint behavior in both forward and reverse shearing directions. With this assumption, the same shear strength criterion is applicable in both shearing conditions. This is not consistent with the measured shear strength in laboratory experiments. The shear strength during reverse shearing was always smaller than that in the forward direction. Rates of damage accumulation for both Barton-Bandis and Continuously­Yielding models are much faster than that observed in experiments. As a result, the joint surfaces are calculated to become smooth at relatively small shear displacement. Consequently, the strength in subsequent cycles is significantly underestimated.

The Mohr-Coulomb model calculates the dilation based on a constant dilation angle. Dilation starts when peak shear strength is reached and ceases when a critical shear displacement is reach~. The Barton-Bandis and Continuously-Yielding models directly link joint dilation with shear displacement through the roughness properties. Since a joint continues to wear during shearing, the dilation angle used to calculate joint dilation continues to decrease. The constant decrease of dilation angle implies that the joint will eventually stop dilating when it is completely worn. All three models inherently assume an increase of joint dilation with increasing shear displacement irrespective of the direction of shearing. This is inconsistent with the joint dilation behavior observed in the laboratory.

ACKNOWLEDGEMENTS

The authors thank Dr. Wesley C. Patrick and Dr. Budhi Sagar for their valuable advice and suggestion during preparation of this paper. This report was prepared to document work performed by the CNWRA for the U.S. Nuclear Regulatory Commission (NRC) under Contract NRC-02-93-005. The activities reported here were performed on behalf of the NRC Office of Nuclear Regulatory Research, Division of Regulatory Applications. This report is an independent product of the CNWRA and does not necessarily reflect the views or regulatory position of the NRC.

REFERENCES

Barton, N.R., and S.C. Bandis. 1982. Effects of block size on the shear behavior of jointed rock. 23rd U.S. Symposium on Rock Mechanics. Berkeley, CA.

Barton, N.R., S.C. Bandis, and K. Bakhtar. 1985. Strength, deformation and conductivity coupling of rock joints. International Journal of Rock Mechanics and Mining Sciences ell: Geomechanics Abstracts 22(3): 121-140.

Cundall, P.A., and LV. Lemos. 1988. Numerical simulation of fault instabilities with the continuously-yielding joint model. Proceedings of the 2nd International Symposium on Rockbursts and Seismicity in Mines. Minneapolis, MN: University of Minnesota.

Hsiung, S.M., D.O. Kana, M.P. Ahola, A.H. Chowdhury, and A. Ghosh. 1993. Laboratory Characterization of Rock Ioints. Report CNWRA-93-013. San Antonio, Texas: Center for Nuclear Waste Regulatory Analyses. .

Hsiung, S.M., W. Blake, A.H. Chowdhury, and T.L Williams. 1992. Effects of Mining-Induced Seismic Events on a Deep Underground Mine. PAGEOPH 139(3/4): 741-762.

Huang, X., B.C. Hairnson, M.E. Plesha, and X. Qiu. 1993. An Investigation of the Mechanics of Rock Ioints--Part I. Laboratory Investigation. International Journal of Rock Mechanics and Mining Sciences ell: Geomechanics Abstracts 30(3): 257-269.

ITASCA Consulting Group, Inc. 1992. UDEC Universal Distinct Element Code Version 1.8 Volume I: User's Manual. Minneapolis, MN: ITASCA Consulting Group, Inc.

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Jing, L., E. Nordlund, and O. Stephansson. 1992. An Experimental Study on the Anisotropy and Stress-Dependency of the Strength and Deformability of Rock Joints. International Journal of Rock Mechanics and Mining Sciences &: Geomechanics Abstracts 29(6): 535-542.

Kana, D.D., B.H.G. Brady, B.W. Vanzant, and P.K. Nair. 1991. Critical Assessment of Seismic and Geomechanics Literature Related to a High-Level Nuclear Waste Underground Repository. NUREG/CR-5440. Washington, DC: U.S. Nuclear Regulatory Commission. 161.

Wibowo, J.T., B. Amadei, S. Sture, and A.B. Robertson. 1992. Shear Response of a Rock Joint Under Different Boundary Conditions: An Experimental Study. Conference of Fractured and Jointed Rock Masses. Preprints. June 3-5, 1992. Lake Tahoe, California.

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