Heuristic study on the influence of LSS on the failure types of different cross-sectional-

In this section, an attempt of applying the study results of cross-sectional shape effect (Chapter 3), combined with the insights gained from the study of LSS (Sections 5.1 to 5.2.3) to improve pillar design, is illustrated by a simplified numerical experiment. It is known that in underground mines, if the Local Mine Stiffness (LMS) of a pillar is softer than the post-peak stiffness of the pillar, pillar-burst will be encountered (Salamon, 1970). The influence of LMS on the failure types of pillars can be simplified and modeled by applying uniaxial compression loading to a rock specimen. In such a case, LMS is conceptually manifested by LSS.

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This study aims at evaluating the instability of different cross-sectional-shaped specimens with the same slenderness ratio under a relatively soft LSS loading condition. Hence, three different cross-sectional-shaped specimens with a slenderness ratio of 2.0, the same as those used in Chapter 3 (Figure 3-6), are adopted in the simulation models (Figure 5-23). In addition, steel properties (E = 200 GPa,  = 0.3) are assigned to the loading platens and  = 0.1 is assumed for the rock specimen-steel platen contacts (Figure 5-23); thus, the contact frictional behavior in this numerical experiment is the same as that in the study of cross-sectional shape in Chapter 3.

The rock failure types in Chapter 3 are stable because the LSS of the steel platen loading condition in those cases is much stiffer than the critical LSS loading condition of the rock specimens (). In this study, LSS is prescribed to a relatively soft LSS; thus, unstable rock failures might occur. In this fashion, the instability of different shaped rock specimens under the critical LSS loading condition can be examined by comparing their post-peak stress–strain curves with those under the stiff LSS loading condition (Figure 3-8).

The post-peak deformation behavior of the cylindrical specimen shown in Figure 3-8 is referred to choose the prescribed value for LSS. The post-peak stiffness of the cylindrical specimen determined from the stress–strain curve (Ep) is roughly 62 GPa ( = 1.217 GN/m). The calculation of Ep based on the stress–strain curve might be subjective because the post-peak stress–strain curve of rocks can never be a perfectly straight line (refer to the discussion on Ep in Section 5.2.2). Therefore, to observe at least one unstable rock failure case under the critical LSS loading condition for the cylindrical specimen (Equation 2-6), a relatively smaller value of 60 GPa is assumed for the actual Ep of the cylindrical specimen. In such a case, the prescribed LSS is equal to 1.178 GN/m (Table 5-6). Because the loading platens in different simulation models

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have different cross-sectional shapes, the height of different loading platens is varied to ensure that their LSS is equal to 1.178 GN/m (Table 5-6).

Figure 5-23 Simulation model of a rock specimen under a relatively soft LSS loading condition (cylindrical specimen as an illustration).

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Table 5-6 Calculation of the post-peak stiffness of the rock specimen () and the LSS of loading platens Ep or E (GPa) Diameter or side length (m) Cross-sectional area (m2) Total height (m)  or LSS (GN/m) Rock specimen 62 0.050 0.0020 0.10 1.217 Cylindrical platen 200 0.052 0.0021 0.18×2 1.178 Square platen 200 0.052×0.052 0.0027 0.23×2 1.178 Rectangular platen 200 0.036×0.072 0.0026 0.22×2 1.178

Figure 5-24 presents the complete stress–strain curves of different cross-sectional-shaped specimens under the stiff and prescribed LSS loading conditions in uniaxial compression.

0.0 0.2 0.4 0.6 0.8 0 30 60 90 120 150

Cylinder specimen (critical LSS)

Square specimen (critical LSS) Rectangular specimen (critical LSS)

Cylinder specimen (stiff LSS)

Square specimen(stiff LSS) Rectangular specimen(stiff LSS)

Str es s ( MPa ) Strain (%) Unloading behavior of prescribed LSS Excess energy

Figure 5-24 Complete stress–strain curves of different cross-sectional-shaped specimens under stiff (solid curves) and prescribed (dash curves) LSS loading conditions.

Similar to the modeling results shown in Figure 3-8, the post-peak stress–strain curves of the specimens under the prescribed LSS loading condition are influenced by their cross-sectional shapes. Moreover, the post-peak stress–strain curves under the prescribed LSS loading condition

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are all flatter than those under the stiff LSS loading condition with the same cross-sectional shape. In particular, compared with other shaped specimens, the rectangular prismatic specimen shows a much flatter post-peak stress–strain curve under the prescribed LSS loading condition than that under the stiff LSS loading condition. The large difference of post-peak stress–strain curves indicate excess energy stored in the loading platens under the prescribed LSS loading condition might be released to the rectangular prismatic specimen, thus the rock failure type in this case can be unstable (Section 5.2.2).

Table 5-7 compares the LSRI of different cross-sectional-shaped specimens under the prescribed LSS loading condition together with their peak strengths. The peak strengths of different shaped specimens are very close. The calculated LSRI values confirm that the failure types of the square and cylindrical shaped specimens under the prescribed LSS loading condition are semi-stable (Manouchehrian and Cai, 2015). Under the same prescribed LSS, compared with the square and cylinder specimens, the failure type is more violent for a rectangular prismatic specimen with the same slenderness ratio. Hence, it is inferred that for the same cross-section area and slenderness ratio, rectangular-shaped rock specimens are vulnerable to rock instability.

Table 5-7 LSRI values and peak strengths of different cross-sectional-shaped specimens under the prescribed LSS loading condition

Cylindrical specimen Square prismatic specimen Rectangular prismatic specimen LSRI 4.1 3.0 17.6

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5.3

Summary

Great efforts had been made by other researchers to develop stiff laboratory test machines to capture post-peak stress–strain curves of rock for rock engineering design; however, the influence of the LSS of a stiff test machine on the post-peak deformation behavior of rock is poorly understood. This is in part due to the difficulty in varying LSS to conduct laboratory tests using rock specimens with nearly identical mechanical properties. Numerical experiments provide a solution to address this problem.

This numerical experiment focuses on studying the post-peak stress–strain curves of stable rock failure under the test condition of different LSS values. Based on the modeling results, Figure 5-25 illustrates conceptually that for stable rock failure to occur under a finite LSS (e.g., rock laboratory testing using a stiff test machine), both Et and Ein contribute to Er, and Er is always smaller than ErB. Furthermore, the modeling results suggest that Ein is affected by the test machine with a finite LSS (> ). Consequently, the descending slope of the post-peak stress– strain curve of a rock under a finite LSS loading condition is always steeper than that under the ideal loading condition (LSS = ∞). Furthermore, Ein and Er are varied with LSS, and it is therefore concluded that the post-peak stress–strain curves of stable rock failure is influenced by LSS.

This conclusion drawn by the numerical modeling is partially supported by the incomplete laboratory test results on nearly identical material specimens, which showed that different stress– strain curves of concrete (Hudson et al., 1972b based on Whitney, 1943) and ice (Sinha and Frederking, 1979) were obtained by different test machines. Most importantly, the modeling

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results confirmed the laboratory test results of Bieniawski et al. (1969), revealing that the descending slopes for the post-peak stress–strain curves of rocks were dependent on LSS.

In this study, the influence of LSS on the stable post-peak failure of rock was investigated. Future study will consider the influence of LSS on different rock failure types, especially the unstable post-peak failure, which is strongly related to the strain energy stored in the test machine in laboratory testing. A heuristic study is presented in Appendix A.

Deformation 0 L o a d

Base case of rock (LSS = ∞)

Er

Et Ein

Er

Rock behavior obtained (finite LSS > )

< LSS < ∞

Peak load

ErB

Unloading behavior of test machine (finite LSS > )

Implication:

Er≠ErB

Figure 5-25 Comparison of the load–deformation curve of the base case (LSS = ) with that under a finite LSS > .

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Chapter 6