numerical modelling of rock loaded to failure above ... - OnePetro

NUMERICAL MODELLING OF ROCK LOADED TO FAILURE ABOVE UNDERGROUND CAVITIES Gareth M. Swift1, Phillip W. Lloyd1, David J. Reddish1, Antony C. Waltham2, & Michael S. Rosenbaum2

ABSTRACT The stability and potential building hazard arising from cavities cut into weak Triassic sandstone are a special feature of the urban environment in the city of Nottingham, UK. Numerical modelling analyses using FLAC (Itasca, 1995), have been applied to determine the behaviour of the rock arches above these old mine workings and man-made voids. The modelling has been calibrated against results of a full-scale destructive test of a cavity roof under applied load; initial strength parameters of the sandstone were derived from laboratory testing. Physical modelling of the cavities has provided a range of data that relates cavity dimensions to failure parameters; these results have been correlated to the full-scale situation by the computer modelling. These computer modelling analyses have been combined with the field observations and geotechnical testing. These provide guidelines for assessing the integrity of the sandstone that remains in place over the voids, and which may have to be loaded by the construction of new building foundations. BACKGROUND The stability problems and potential hazards posed by existing caves in the weak Triassic sandstone underlying Nottingham city centre, have been highlighted elsewhere, (Waltham, 1992, 1993, 1996: Roodbaraky et al., 1994). Previous studies have used plaster scale models and Finite Element models (FEM), in order to attempt to simulate failure mechanisms and ultimate failure loads. These have been validated by a single full-scale test, where an expendable cavity roof in central Nottingham was loaded to failure. Further numerical models were designed to simulate the load-deformation curve recorded during the test. From this testing it should be possible to model other cavities beneath the city centre with a view to providing guidelines applicable to future constructions in the vicinity. Although the scale models accurately simulate the failure mechanisms associated with the roof beam failure above cavities of varying geometry, more confidence in the modelled failure loads was necessary. Conversely, the failure loads were more reasonably simulated using the Finite Element approach, but the mechanisms of failure could not be accurately modelled. The main problem was how to best represent the in situ mechanical behaviour of sandstone using basic FEM analyses and simple physical models. This paper describes a numerical modelling analysis of the full-scale test in order to derive a Rock Mass Factor (RMF) that could be applied to further models of other sandstone cavities. The effectiveness of this modelling strategy was considered, and the application of the deduced RMF to a series of cavities located at varying depths is documented. NUMERICAL MODELLING ANALYSES Modelling Objectives The principal objective of the numerical modelling was to accurately simulate the full-scale test on the destructibility of a cavity roof in the Triassic sandstone strata underlying the city of Nottingham. The objectives were as follows: • To accurately simulate the failure of the tested cavity roof • To derive a rock mass factor (RMF) for the sandstone • To apply this RMF approach to a variety of model geometries and in situ sandstone conditions These aims and objectives were achieved using a 2-dimensional, finite difference numerical modelling code, Fast Lagrangian Analysis of Continua (FLAC), version 3.3, (Itasca, 1995). 1 2

SChEME, University of Nottingham, University Park, Nottingham, NG7 2RD (UK) Department of Civil & Structural Engineering, Nottingham Trent University, Nottingham, NG1 4BU (UK)

Modelling Methodology The first requirement of the model is to generate the finite difference grid. Around the specific area of interest, in this case two cavities in close proximity (Figure 1), the elements defining the grid were specified as small as possible (0.1 m2), so allowing for greater detail and yielding more accurate stress re-distributions and displacements. Further away from this area, the elements were increased in size to reflect their decreasing influence on the area of interest. The boundary to the model was sufficiently remote so as not to affect the model results (50 m from the vertical boundaries and 30 m from the base of the model). The mechanical properties of the sandstone relevant to the numerical modelling investigation are the Young’s Modulus (E), Poisson’s Ratio (ν), Uniaxial Compressive Strength (UCS), Uniaxial Tensile Strength (UTS), Rock Density, and Shear Strength, expressed as Cohesion (c) and Friction Angle (φ). The available data were based upon laboratory and full-scale tests. Table 1 shows the initial mechanical properties assumed for modelling. The test site was at a sufficient depth (10 m) to be situated in relatively unweathered rock, which was strong. Strength data for other cavities is for weathered rock at shallower depths where most of these are found (Waltham, 1993).

Test site Other cavities

Table 1 : Measured mechanical properties of the Triassic sandstone ν C UCS UTS E Density MPa MPa MPa GPa g/cm3 22.68 – 33.08 1.06 – 1.54 2.2 0.25 2.3 2.95 4.12 – 10.31 0.34 – 0.50 1.2 0.25 2.3 0.94

φ o

60.8 69.4

The principal reservation with using laboratory test data for shear strength determination lies in assessing by how much the intact rock strength must be reduced to account for the engineering behaviour of the rock mass. One way of achieving this is to present the shear strength test data for the intact rock specimen as Mohr circles, and to fit a modified Hoek-Brown failure envelope to the Mohr circles (Hoek & Brown, 1980). This envelope may be further modified (reduced) by using a Rock Mass Rating (RMR) value for the rock in question, to give a rock mass failure envelope (Bieniawski, 1976). From the failure envelope it is possible to derive equivalent c and φ parameters, either for an appropriate instantaneous value of confining stress or averaged over a selected range of confining stress. The initial intact properties represented an RMR of 100. The reduced properties represent an RMR of 67. The FLAC code has a number of in-built constitutive models to represent the behaviour characteristics of the rock mass. The most appropriate model for these analyses was a Ubiquitous Joint model, which allows for the presence of an orientation of weakness (weakness plane) within a Mohr-Coulomb solid. Yield may occur in either the Mohr-Coulomb solid or along the chosen weakness planes, depending upon the state of stress, the orientation of the weakness plane as well as the material properties of both the Mohr-Coulomb solid and weakness plane. In addition to the mechanical properties already specified, mechanical properties of the planes of weakness (bedding) are also required; these are: Joint cohesion (cj), Joint friction (φj), Joint angle (β), and Joint tensile strength (σ jt). These bedding properties were taken to be 80 % of the matrix properties based upon numerous UCS tests carried out on sandstone samples by Nottingham Trent University, loaded parallel and perpendicular to bedding. Further properties required for the numerical modelling include the Bulk (K) and Shear (G) moduli which are related to the Young’s Modulus and Poisson’s Ratio. The initial intact input properties used directly within the numerical model, as well as the Young’s Modulus, Poisson’s Ratio and Rock Density defined in Table 1, are presented in Table 2. Table 2 : Numerical modelling input properties Matrix Properties Bedding Properties β φ c Ten K G cj φj σj t o o o MPa MPa MPa GPa GPa MPa Intact 2.95 60.8 1.54 1.467 0.888 2.36 48.6 1.232 180 Optimised 1.37 50 1 1.467 0.888 0.92 38 0.1 180 The numerical model is gravity loaded and the horizontal stresses were calculated using the approach suggested by Terzaghi & Richart (1952).

Numerical Modelling of Full-Scale Failure A layout of the model geometry is reproduced in Figure 1. The model consists of two superimposed cavities approximately 4 m in width, separated by a 0.5 m thick sandstone beam. The upper cavity is off-set to the right by approximately 0.5 m and is located 10 m below the model surface. The square loading pad with an edge length of 0.4 m (0.16 m2 ) is located approximately 2 m from the left hand wall of the lower cavity, as in the full-scale test. The load was applied upwards to the roof of the lower cavity, for reasons of safety and practicality in the full scale test. The load was applied so that the test was completed with 5 – 10 minutes. The full test is described in detail by Roodbaraky et al. (1994.

Load

Figure 1 : Model of in situ test showing cavity configuration and failure of roof beam under load The principal aim of this stage of the modelling programme was to simulate the failure of the 0.5 m roof beam. A complete load-deformation curve for the full-scale test was available; this showed that the rock beam failed at a load of 340 kN or 34 tonnes, with a maximum measured displacement of 25 mm. This curve is reproduced in Figure 2. Figure 2 shows the relationship between vertical displacement and vertical stress for both the real test and the optimised model. It can be seen that the simulation produced using these optimised properties reveal good agreement with the actual test; an almost identical failure load is achieved, with the same amount of deformation. Figure 3 shows the vertical stress against vertical displacement simulated in the numerical model of the beam immediately above the loading pad. This indicates that failure occurred at a load of > 2 MPa, which is equivalent to 34 tonnes, at a simulated displacement of 25 mm as measured. There is some deviation of the simulated curve from the real test data during the earlier stages of the loading, where the simulated rock mass requires greater loads to attain the same amount of measured displacement. This may be due to the fact that the stiffness properties within the FLAC model were left unchanged and the strength parameters were reduced. An important aspect of the analyses were the failure mechanisms observed within the rock beam, and whether the mechanisms accurately simulated those observed in the in situ test. Waltham (1996) describes in detail the failure mechanisms observed within the roof beam, observing that as load was imposed, the

2.5

Real Test

Vertical Stress (MPa)

FLAC Simulation 2 1.5 1 0.5 0 0

5

10

15

20

25

30

Vertical Displacement (mm)

Figure 2 : Vertical displacement against vertical stress

Vertical Stress (MPa)

2.5 2 1.5 1 0.5 0 0

10

20

30

40

50

60

70

Vertical displacement (mm)

Figure 3 : Vertical stress against vertical displacement for FLAC simulation seemingly massive sandstone rock split into three distinct beds, with the lower bed failing as a flared plug and the load was then spread onto the other beds, which had fractured as a cantilever. The mechanisms of failure simulated compared favourably with these observations. Failure developed as a flared plug above the loading pad, with shear failure in evidence adjacent to the plug and along weakness planes within the beam. Tensile failure was also observed across the bedding as the beds separated under continued loading. Ultimately, cantilevering occurred as the beds failed in tension adjacent to the loading pad. These mechanisms are illustrated in Figure 1. Table 2 shows the reduced input properties for the final ‘optimised’ model compared to the intact property values. A Rock Mass Factor can be determined for the cohesive strength of the rock matrix. This was taken to be 0.46, the ratio between the original intact cohesion and the reduced cohesion used within the optimised model. In practice the reduction in strength properties was based on Mohr envelope reductions as described earlier. FURTHER MODELLING Further analysis was conducted using the same reduction technique described for the earlier Finite Difference modelling. Intact strength and stiffness properties for the Triassic sandstone taken from other areas beneath Nottingham were used to analyse the stability of a range of cavities located at varying depths and with varying geometries. These properties as originally specified are shown in Table 1. For completeness the analysis was conducted using upper and lower bound mechanical properties based upon the upper and lower UCS and UTS values quoted for these other sites. The properties were subsequently reduced to give a more realistic representation of the in situ rock mass strength. The bedding properties were calculated on the same basis as the initial analysis (20 % reduction).

Unlike the previous analysis, the loading in this instance was provided by a 1 m2 loading pad at the surface, located centrally over a 4 m wide cavity situated at varying depths from the surface. The range of depths used was from 1 m to 4 m. RESULTS OF ANALYSIS Analysis initially took the form of studying the load-deformation curves produced from the simulations, from which it was possible to determine the load at failure, i.e. the Ultimate Bearing Pressure (UBP). This information provides quantification of the relationship between cover thickness and cavity width in terms of cavity roof stability. Studies of the failure mechanisms were also undertaken to facilitate comparison with the earlier plaster scale models. Waltham (1996) concluded from this analysis that although the mechanisms of failure were considered to be realistic, the quantitative results were not. The relationship between depth and bearing capacity of sandstone cavity roofs can be seen in Figure 4, whereas Figure 5 shows the mode of failure above a centrally loaded cavity at a depth of 3 m. Figure 4 shows the failure load (UBP) for the upper limit and the lower limit properties, and their respective trendlines. Clearly, there is a linear relationship between depth and failure load regardless of the mechanical properties used, such that at shallower depths the roof beam fails at significantly lower loads than at greater depth, where the beam is able to maintain and redistribute higher loads before ultimate failure occurs. The in situ result correlates with the lower bound values even though its rock strength was high, this may be ascribed to the small loading area applied to the very thin beam in the in situ test, such that the rock failure was more dependent on the local joints. 20

Upper

18

Lower

Failure Load (MPa)

16

Linear (Upper)

14

Linear (Lower)

12 10 8 6 4

x

2 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Roof Thickness (m)

Figure 4 : Relationship between failure load and roof thickness (the ‘x’ marks the loading capacity of the in situ test, where a 0.5 m beam was loaded until failure occurred at approximately 2 MPa) CONCLUSIONS AND FUTURE WORK The failure mechanisms observed during the full-scale in situ test were accurately simulated using the finite difference approach detailed in this paper. Further, it has also been possible to simulate the maximum load-deformation produced from the in situ test, such that the load at failure produced during the simulation was almost identical to the failure load recorded during the test, with approximately the same amount of vertical deformation in both cases. The laboratory-acquired input properties were systematically reduced in order to more accurately represent the rock mass strength, thereby facilitating a more realistic simulation of the in situ test. The final optimised properties represented a simple RMF with respect to cohesion of 0.46, that was applied to intact rock properties acquired from other samples of weak Triassic sandstone associated with other cavities beneath the city centre. Further modelling was undertaken using the reduced properties on a series of cavities at varying depths from the surface, and the results presented in Figure 4. These show that as depth increases, the bearing capacity of the roof beam likewise increases. Another important feature of this model

is that the mechanisms of failure modelled are identical to the failure process predicted using the physical models and observed in the field, thereby producing mutual validation. Future work will focus on the relationship between loading pad location and cavity stability, the effects of varying cavity geometry on cavity roof stability, and will examine the effects of loading pad size on cavity stability. Finally, the modelling approach will be applied to a cavity where remedial engineering works were placed during the initial stage of roof failure.

Figure 5 : Numerical model showing the failure mechanisms above a 4 m wide cavity at a depth of 3 m Acknowledgements The authors would like to express their appreciation to the Faculty of Construction and the Environment at the Nottingham Trent University for financial support through the Research Enhancement Fund, enabling completion of the finite element work in support of the field and laboratory testing programme. REFERENCES Bieniawski, Z.T. (1976). “Rock mass classification in rock engineering”. Proceedings of the Symposium on Exploration for Rock Engineering, Bieniawski, Z.T. (Eds), Vol. 1, pp.97-106. Hoek, E. & Brown, E.T. (1980). Underground Excavations in Rock. IMM, London. Itasca (1995). Fast Lagrangian Analysis of Continua, FLAC. Version 3.3, Volume 1-4, Itasca Consulting Group Inc., Minneapolis, Minnesota 55515, USA. Mohammad, N. (1998). Subsidence Numerical Modelling of Weak Rock Masses. Unpublished PhD Thesis, University of Nottingham. Roodbaraky, K., Chen, H., Waltham, A.C. & Johnson, D. (1994). “Finite element analysis of the failure of a sandstone cave roof in Nottingham”. Numerical Methods in Geotechnical Engineering, Smith, I.M. (Ed.), pp.409-416. Terzaghi, K. & Richart, F.E. (1952). “Stresses in rock about cavities”. Géotechnique, Vol. 3, pp.57-90. Waltham, A.C. (1992). “The sandstone caves of Nottingham”. Mercian Geologist, Vol.13,pp.3-34. Waltham, A.C. (1993). “Crownhole development in the sandstone caves of Nottingham”. Quarterly Journal of Engineering Geology, Vol. 26, pp.243-251. Waltham, A.C. (1996). “Ground subsidence over underground cavities”. Journal of the Geological Society of China, Vol. 39, No. 4, pp.605-626.