# Stability Assessment of Rock Mass System under Multiple Adjacent Structures

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

_{ci}is the uniaxial compressive strength of the intact rock. σ

_{1}and σ

_{3}are the major and minor principal stresses, respectively, which are considered as positive when tensile in nature. m

_{b}, s, and a are the dimensionless material parameters, defined as

_{i}is the value of m

_{b}for intact rock. The value of m

_{i}is related to the minerology and texture of the intact rock, varying from 4 for very fine weak rock (e.g., claystone) to 33 for coarse igneous light-colored rocks (e.g., granite). The value of m

_{i}can be obtained from experiments. GSI is the geological strength index which features the quality of the rock mass [21]. The value of GSI ranges from about 10 for extremely poor rock masses to 100 for intact rock. D is the disturbance factor which depends on the extent of weathering and blast damage of the rock mass. The value of D varies from 0 for the undisturbed in situ rock masses to 1 form extremely disturbed rock masses. The modified HB criterion is applicable to intact rock or heavy jointed rock mass that can be considered is isotropic and homogeneous and therefore is proper to estimate the bearing capacity of surface footings on rock.

_{σ}is the bearing capacity factor, which is related to the HB material parameters, namely, m

_{i}, GSI, and D. However, the value of σ

_{ci}is very little effect on the footing capacity [16,18].

## 3. Problem Statement

_{ci}, m

_{i}, GSI, and D. The σ

_{ci}value is kept equal to 20 MPa [22] and the m

_{i}values are taken as 7, 10, 15, 17, and 25 for five types of rocks [23]. The values of GSI and D are within the range of 10 to 90 and 0 to 1, respectively. The rock mass has unit weight γ, which is quantified by σ

_{ci}/γB. The value of σ

_{ci}/γB varies from 125 to ∞ that indicates the weightless rock mass [24]. The ground is loaded with a uniform surcharge pressure q and its effect is represented by q/σ

_{ci}. The value of q/σ

_{ci}ranges between 0 to 1, which cover most problems of practical interest [25]. The vertical downward load is simultaneously applied to the center of each footing and all footings are collapsed exactly at the identical bearing load Q

_{u}.

_{σ(mul)}is the bearing capacity factor for an interfering footing in a group of multiple footings and N

_{σ(iso)}is the bearing capacity factor for single isolated footing with the same width of the interfering footing. From this, the ultimate bearing capacity of an intervening footing is determined using the following expression:

## 4. Finite Element Limit Analysis

## 5. Results and Comparison

_{i}, a comparison of bearing capacity N

_{σ0}obtained in this study with the analyses of Serrano et al. [28] using the slip line method, Merifield et al. [24] using the finite element limit analysis, Chakraborty and Kumar [29] using lower bound finite element limit analysis, and Keshavarz and Kumar [25] using the stress characteristics method. A comparative representation of all these values is given in Table 1. The present results remain very close to the existing solutions. Additionally, the values of N

_{σ0}for isolated foundation with rough and smooth roughness for GSI = 50, m

_{i}= 20 are determined to be 1.722 and 1.709, respectively, which are 1.2% and 3.0% in error compared with those from Keshavarz and Kumar [25].

_{u}value of two interfering footings (S/B = 1) on rock mass, together with other numerical analysis for same footing and soil conditions. The Figure 5 shows the variation of q

_{u}values with GSI for the case of m

_{i}= 7 and 10. As shown in the Figure 5, the results of the numerical analysis are very close to those obtained with distinct element method of Javid et al. [18]. However, the value of q

_{u}in upper bound solutions by Shamloo et al. [19] was under-predicted, and the difference increases with increasing GSI. Given that the result of present study is included in the result of Javid et al. [15] and Shamloo et al. [19], it can be seen that agree well with previous study.

_{ci}/γB = infinite, q/σ

_{ci}= 0 and m

_{i}= 7, 15, and 25. Figure 6 shows that the ξ value decreases as S/B increases. It is important to note that the lower GSI, the greater ξ value, however the greater rate at which ξ values decreases as adjacent space increases. Furthermore, as m

_{i}increases, the ξ values according to each GSI increases in the same adjacent space. All ξ values decrease as S/B increases, reached value of 1. In the case of ξ = 1, the bearing capacity of multiple footing has no adjacent distance influence, meaning that the behavior of isolated footing is performed. In the present study, the space ratio corresponding to ξ = 1 is defined as S

_{max}/B. S

_{max}/B varies depending on GSI and m

_{i}, respectively, and details of this are illustrated in Figure 7.

_{max}/B for different combinations GSI and m

_{i}is shown in Figure 7 for the case of D = 0, σ

_{ci}/γB = infinite, q/σ

_{ci}= 0. Figure 7 shows that S

_{max}/B decreases with increasing GSI, but for a given GSI, S

_{max}/B increases with increasing m

_{i}. It should be noted that the maximum value of S

_{max}/B is 12.76 for GSI = 10, m

_{i}= 25 and the minimum value of S

_{max}/B is 2.41 for GSI = 90, m

_{i}= 7. In addition, the S

_{max}/B can be described as exponential function of GSI. Expressed in terms of the S

_{max}/B:

_{i}. As stated in Equation (9), it is obvious that the value of S

_{max}/B for multiple footings on rock mass drops towards decreases with increasing of GSI. It is worth noting that these coefficients are accurate for interpolation only, and should not be used for extrapolation beyond 10 < GSI < 90. The relations of a with m

_{i}have the form of (R

^{2}= 0.991 and R

^{2}= 0.997, respectively)

_{i}= 10, σ

_{ci}/γB = infinite, q/σ

_{ci}= 0 and GSI = 10, 50, and 90. Figure 8 shows that the value of ξ decreases as D increases. It can also be seen that as GSI increases, the value of ξ decreases and this trend increases as D decreases.

_{ci}/γB is shown in Figure 9 for the case of m

_{i}= 10, D = 0, q/σ

_{ci}= 0 and GSI = 10, 50, and 90. Figure 9 shows that the greater σ

_{ci}/γB for a given value of σ

_{ci}/γB, the greater the value of ξ. However, it should be noted that the smaller GSI, the significant the difference in the value of ξ according to σ

_{ci}/γB, but the greater GSI, the smaller the difference in the value of ξ. This indicated that for the case of GSI = 90, the unit weight of rock mass has little effect on the value of ξ of multiple footings. In addition, as GSI decreases, the efficiency factor of σ

_{ci}/γB except 125 decreases. For the case of σ

_{ci}/γB = 125, the value of ξ increases slightly when GSI increases from 10 to 50, but when it is greater than 50, the value of ξ decreases.

_{ci}is shown in Figure 10 for the case of m

_{i}= 10, D = 0, σ

_{ci}/γB = infinite and GSI = 10, 50, and 90. Figure 10 shows that decrease in the value of ξ when surcharges are applied. Given that the value of ξ resulting from the applied of surcharge is smaller than the value of ξ in the without of surcharge, this implies that surcharge reduces the value of ξ. As the GSI increases, the ξ values continues to decrease for q/σ

_{ci}= 0 and 0.001, while for q/σ

_{ci}= 0.1 and 1.0, the ξ values increases. For the case of q/σ

_{ci}= 0.01, the ξ values varies slightly with the change in GSI. It should also be noted that increasing GSI from 10 to 50 increases ξ values, however decreasing ξ value from 50 to 90.

_{ci}/γB = infinite, q/σ

_{ci}= 0 and m

_{i}= 7, 15, and 25. Figure 11 shows that rough bases always maintain ξ value greater than smooth bases. It can be seen that the varies in ξ values due to roughness becomes smaller as GSI increases and vice versa increases as m

_{i}increases. However, it should be noted that m

_{i}has a small effect on the ξ value and that GSI is significant.

_{i}= 10, D = 0, σ

_{ci}/γB = infinite and q/σ

_{ci}= 0. Figure 12 shows that unlike isolated footings, the failure mechanism of multiple footings does not lead to surface and is formed below the surface. As the S/B of multiple footing increases, the failure mechanism of multiple footings changes close to failure mechanism of independent foundations. In the case of S/B = 2, adjacent effects are expressed between the spacing of multiple footing and failure mechanisms are formed at relatively low depths. These effects can be seen to become decreases as S/B increases. For rough bases, failure mechanism of multiple footing is formed in the external rather than center of the footing. On the other hand, for smooth bases, it should be noted that failure mechanism of multiple footings is formed simultaneously at center of footing and at the external.

## 6. Conclusions

_{i}and σ

_{ci}/γB. An increase in surface surcharge leads to a decrease in ξ and its trend is predominant for smaller GSI. The value of ξ for rough footings is higher than smooth footings and the effect of the interference on the bearing capacity for rough base is greater than that for smooth base. The maximum spacing for which the nearby footings influence each other is determined and the closed-form expression is proposed. The zone of failure mechanism decreases dramatically for the interfering footings, implying a significant interaction between the adjacent footing and hence an increase in the bearing capacity of individual footings.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The concept of representative element volume (REV) for rock mass system (modified from [3]).

**Figure 2.**Design process of engineering problem for rock mass with an emphasis on numerical methods.

**Figure 5.**Comparison of q

_{u}of two interfering footingsith S/B = 1 for q = 0, σ

_{ci}= 10 MPa and D = 0: (

**a**) m

_{i}= 7; and (

**b**) m

_{i}= 10.

**Figure 6.**Variation of ξ with S/B and GSI for D = 0, σ

_{ci}/γB = ∞ and q/σ

_{ci}= 0: (

**a**) m

_{i}= 7; (

**b**) m

_{i}= 15; and (

**c**) m

_{i}= 25.

**Figure 7.**Variation of S

_{max}/B with GSI and mi for multiple strip footings with D = 0, σ

_{ci}/γB = ∞ and q/σ

_{ci}= 0.

**Figure 8.**Variation of ξ with S/B and D for m

_{i}= 10, σ

_{ci}/γB = ∞ and q/σ

_{ci}= 0: (

**a**) GSI = 10; (

**b**) GSI = 50; and (

**c**) GSI = 90.

**Figure 9.**Variation of ξ with S/B and σ

_{ci}/γB for m

_{i}= 10, D = 0 and q/σ

_{ci}= 0: (

**a**) GSI = 10; (

**b**) GSI = 50; and (

**c**) GSI = 90.

**Figure 10.**Variation of ξ with S/B and q/σ

_{ci}for m

_{i}= 10, D = 0 and σ

_{ci}/γB = ∞: (

**a**) GSI = 10; (

**b**) GSI = 50; and (

**c**) GSI = 90.

**Figure 11.**Variation of ξ with S/B and GSI = 10 and 90 for rough and smooth bases footings with D = 0, σ

_{ci}/γB = ∞, q/σ

_{ci}= 0: (

**a**) m

_{i}= 7; (

**b**) m

_{i}= 15; and (

**c**) m

_{i}= 25.

**Figure 12.**Failure mechanisms of multiple strip footings for GSI = 30, m

_{i}= 10, D = 0, σ

_{ci}/γB = ∞ and q/σ

_{ci}= 0 with different combinations of S/B and base roughness: smooth base with (

**a**) S/B = 2; (

**b**) S/B = 4; and (

**c**) S/B = 6; rough base with (

**d**) S/B = 2; (

**e**) S/B = 4; and (

**f**) S/B = 6.

**Table 1.**Comparisons of N

_{σ0}values obtained from this study and literature for single isolated strip footings on rock mass.

GSI | m_{i} | SE | ME | CK | KK | This Study |
---|---|---|---|---|---|---|

10 | 10 | 0.072 | 0.077 | 0.075 | 0.077 | 0.075 |

20 | 0.159 | 0.156 | 0.151 | 0.153 | 0.151 | |

30 | 0.259 | 0.238 | 0.230 | 0.237 | 0.226 | |

30 | 10 | 0.393 | 0.397 | 0.388 | 0.393 | 0.390 |

20 | 0.716 | 0.713 | 0.701 | 0.702 | 0.685 | |

30 | 1.038 | 0.988 | 1.015 | 1.007 | 0.960 | |

50 | 10 | 1.031 | 1.037 | 1.028 | 1.025 | 1.024 |

20 | 1.760 | 1.765 | 1.739 | 1.742 | 1.722 | |

30 | 2.458 | 2.367 | 2.406 | 2.426 | 2.366 | |

70 | 10 | 2.434 | 2.444 | 2.415 | 2.415 | 2.437 |

20 | 3.998 | 4.012 | 3.978 | 3.961 | 3.904 | |

30 | 5.470 | 5.491 | 5.437 | 5.417 | 5.389 | |

90 | 10 | 5.741 | 5.758 | 5.724 | 5.689 | 5.683 |

20 | 9.100 | 9.125 | 9.086 | 9.015 | 9.015 | |

30 | 12.237 | 12.270 | 12.198 | 12.114 | 12.085 |

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**MDPI and ACS Style**

Kim, B.S.; Lee, J.K.
Stability Assessment of Rock Mass System under Multiple Adjacent Structures. *Systems* **2022**, *10*, 5.
https://doi.org/10.3390/systems10010005

**AMA Style**

Kim BS, Lee JK.
Stability Assessment of Rock Mass System under Multiple Adjacent Structures. *Systems*. 2022; 10(1):5.
https://doi.org/10.3390/systems10010005

**Chicago/Turabian Style**

Kim, Bo Sung, and Joon Kyu Lee.
2022. "Stability Assessment of Rock Mass System under Multiple Adjacent Structures" *Systems* 10, no. 1: 5.
https://doi.org/10.3390/systems10010005