Relief methods. Jacking methods. Strain recovery methods. Borehole breakout methods. Case studies and comparison between different methods. Monitoring of stress change. The state of stress in the earth's crust: from local measurements to the world stress map. Using stresses in rock engineering, geology and geophysics. We operate a 30 day money back guarantee. If you are unhappy with the product for whatever reason, please contact us to arrange a return and refund. As shipping costs are not retrievable, we are unable to refund shipping costs. We use an automated eBay feedback response system.

If you are happy with the product, please leave positive feedback and we will automatically leave positive feedback for you. If you are unhappy with the transaction for any reason, please contact us first to resolve. The horizontal stress need not be zero at surface, whereas the vertical stress must be zero.

Haxby and Turcotte incorporate thermally induced expansion and contraction of the rock mass depending on the geothermal gradient and its depth of burial. This is an important paper, although the author does not agree with their assertion that the removal of overburden pressure has a compressive effect on the rock mass. The full derivation of the effects of erosion and isostatic uplift appears in Handley , since it differs slightly from that of Haxby and Turcotte The equation for horizontal stress changes due to subsidence or uplift and thermomechanical effects derived in Handley is given by:.

These effects are all linear, which will later be shown to be an important feature of stresses near the surface above 10 km deep. Table IV contains the mechanical and thermal effects due to erosion of -1 km of continental crust using Equation [12] and the equation derived by Haxby and Turcotte for comparison. Equation [12] predicts that the horizontal stress rate with erosion is slightly more than the vertical stress rate in the case of sandstone, while for limestone, granite, gabbro, quartzite, and marble the horizontal stress rate is less than the vertical stress rate.

The Haxby and Turcotte analysis results in far greater horizontal stress rates than does Equation [12]. The latter is probably more plausible because the Haxby and Turcotte equation predicts strongly compressive stress states at surface, unless the horizontal stress at depth is always considerably less than the vertical stress see Figure If there are lithostatic stresses at depth, the Haxby and Turcotte result precludes the development of vertical jointing in rock see Price, and Figure 13 - a phenomenon that is seen everywhere.

Although this is probably not true earthquakes can originate at depths much greater than this , for the purposes of this paper it is assumed true in stable continental conditions such as those in southern Africa, and is therefore the basis of the plot in Figure Equation [12] predicts that the horizontal stress rates are fairly close to the vertical stress rates, allowing the horizontal stress to be either mildly tensile or mildly compressive at surface, depending on the state of stress before erosion and uplift takes place, and also on the rock properties.

This allows for the formation of vertical joints as well as the observation that there are often compressive horizontal stresses at surface, if the rock mass is in a more or less lithostatic stress state before erosion. Both phenomena are nearly always present at surface, suggesting that the rock mass is most often in a lithostatic stress state or close to it when deeply buried. Rocks from this depth exposed at surface would then be marginally in horizontal tension or compression if Equation [12] is correct, which is the case observed all over the world.

It is well known that the continental rock mass near surface is seldom in a lithostatic state of stress at least above m, as confirmed by the measurements in the consistent database , which suggests that the erosional model is important and significant, but it is not the only mechanism at work in determining crustal stresses near surface. The model appears to be too simplistic, possibly for the following reasons:. The correlation coefficients for the stress components versus depth for different collections of measurements appear in Table V. The correlations are very good, even though the data comes from different locations across southern Africa, from rock masses with differing geological histories, but all in a similar stage of uplift through erosion and mantle upwelling McCarthy and Rubidge, The 'A'- and 'B'-graded data selected by Stacey and Wesseloo show significantly improved correlation coefficients.

Because of the good correlations one should conclude that there is a linear relationship between the measured stress components and depth, and that the thermomechanical model presented above may have some merit near surface if some adjustments are made to the boundary conditions. It is unlikely to be correct for the whole section through continental crust, and this should be the subject of further geophysical research in the long-term.

A near-surface pre-mining stress model showing a linear relationship with depth can be constructed from Table VI and the ideas of Price , Voight , Gay , and Haxby and Turcotte The author first chooses a depth of m at which to define the best-fit stress state because conditions will not be significantly different from the depth of the deepest measurement in the database at m below surface. The stress state at m is determined for each horizontal stress using the intercepts and slopes of the leastsquares best-fit lines for the individual measurements from the consistent database in Table VI.

The same is done for the vertical stress component, this time using the slope and intercept for the least-squares best-fit line determined for the overburden stress determined from each measurement site. Again, the individual measurements from the consistent database were used. The reason for this choice is that the slopes determined for all the other datasets in Table VI are probably to too low for known densities of common crustal rocks see vertical stress rates for different rock types in Table IV , with the exception of the 'A'- and 'B'-rated data from Stacey and Wesseloo , where the slope is possibly too high.

This point should be investigated in future research. A negative term h - multiplied by the stress gradient with depth is added, so that a linear plot of the stress component versus depth is obtained. The slopes and intercepts are rounded so that the equations provide results to the nearest megapascal. The resulting equations are:. The consistent database stress measurements together with the straight lines from Equation [13] appear plotted together in Figure The naming of the principal stresses in Equation [13] separates the vertical component from the two horizontal components because Figure 14 shows that both a h1 and a h2 are greater than a o b near the surface, and both are less than a ob at depth.

This makes the normal naming convention of principal stresses impossible to apply. Like the other models already described, the denudation model fails to provide a good visual fit to the data. The directions of the horizontal stresses cannot be specified in Figure 14 , but local conditions on a mine will often indicate their directions unambiguously. There is still much work to be done in this area before any conclusions about the validity of one result or the other can be made.

For the present, the scarcity and variability of the data and its being sourced from different geological environments may obscure the true patterns. These factors will be considered when building a generic pre-mining stress model. This model is derived from the deep-level gold mines of South Africa, where early stress measurements provided guidance for the assumption that the maximum principal stress was vertical, with the two horizontal principal stresses usually assumed equal and to be about half the vertical stress, i.

This constant of proportionality became known as the k-ratio, which is defined as:. Subsequent stress measurements suggested that the horizontal stresses ranged between 0.

## Estimations of Regional Stress Based on Measured Local Stress

The equations for the simple tabular model are given by:. The platinum mines in South Africa also use this model, but in some cases assume the horizontal stresses to be equal to or greater than the vertical stress component. From the few measurement data available from the gold mines, it appears that the stress tensor has a major principal component perpendicular to the bedding, the intermediate principal component is horizontal, parallel to the strike of the strata, and the minor principal component is parallel to the dip and dip direction of the strata.

This pattern is faintly visible in the Carletonville Goldfield, and accounts for the observations that strike-stabilizing pillars are unstable - they punch into the footwall in the back areas - while dip-stabilizing pillars are stable, hence the success of sequential grid mining with dip pillars Handley et al. The vertical component of stress parallel to the gravity vector must be equal to the overburden weight, so this puts a limit on the size of two tilted components of the principal stress tensor, namely the component perpendicular to the strata, and the component parallel to strata dip.

There are variations to this, especially near faults and dykes, which have been confirmed by observed changes in mining-induced seismicity near faults and dykes. It is therefore possible that non-zero shear stresses develop on vertical surfaces such as dyke boundaries and faults. There is virtually no physical information on this except for papers on stress measurements near dykes Leeman, ; Deacon and Swan, ; and Gay, Figure 16 contains the plot of measured k -ratios versus depth, inferred k -ratios obtained from the denudation model, and maximum and minimum possible k -ratios from the Hoek-Brown Failure Criterion see Figure 17 for minimum Hoek-Brown parameters of crustal strength derived from the measured stress data.

It is apparent that the measured k- ratio is definitely not constant with depth, and that there is a large spread in values. The superimposed denudation k -ratio curves on the data in Figure 17 produce a qualitatively better fit to the data than any of the other models, even though the linear stress curves produced by the denudation model in Figure 14 do not fit the data any better than any of the other stress models shown.

Voight's denudation model k-ratio fits the k-ratio obtained from the least-squares best-fit curve of the maximum measured horizontal stress data and the leastsquares best-fit curve of the measured vertical stress data see Equation [13] for the parameters of these curves. The Hoek-Brown-derived limits of the k -ratio provide maximum and minimum limits to the k -ratio data for all depths after the concepts introduced earlier in Figure 10 , and the derivation of the limit parameters discussed below.

Proposed pre-mining stress model. A good pre-mining stress model should recognize two facts: 1 that all stress states from the lithostatic state to a state of tensile or compressive crustal yield exist in every rock mass, and 2 the denudation model discussed above and encapsulated in Equation [13] and Figures 13 and 14 must provide a reasonable approximation of near-surface stress states.

These assertions are supported by geological structure everywhere, which suggests that crustal rocks have been subject to successive stress states ranging from the tensile yield limit to the compressive yield limit several times in the geological past. The tensile yield limit is manifested by joints, igneous intrusions, and normal faults, which could have developed in many different directions as a result of several separate episodes over geological time.

Likewise, the compressive limit is imprinted on the rock mass in the form of reverse faults, folding, and mountain building. In addition to these extreme states, the rock would have been subject to every stress state in between. All rock masses in southern Africa exhibit geological structure consistent with both crustal stress extremes, in that both the tensile and compressive features are nearly always present.

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The observed variability of stresses in the crust is so high that the probability of measuring a stress state close to the crustal strength, either in tension or compression, must be reasonably good. In addition, there can be considerable variation in the vertical stress due to rock mass structure.

If the consistent database contains measurements of crustal stress states close to compressive and tensile failure, then fitting yield curves to the outermost measurements may provide a good indication of actual limits to crustal stress. These parameters were found by fitting Hoek-Brown yield curves to the outlying stress measurements.

By definition, the m- and s -values must be the same for both sets of curves because they describe one continental rock mass composed of many different rock formations. Assuming the s -values are the same, and placing the curves such that they pass though the centre of gravity of the extreme values, we can find m- values by the solution of the equations. The results appear in Table VII. They also provide good estimates of the k -ratio limits, shown in Figure These results are thus accepted for the crustal rock mass and used to plot the stress limits in Figure The consistent database stress measurements appear in the plot to provide a visual indication of the stress limit curve fits to the data.

The fact that the m -values for an assumed fixed s -value are similar but not the same can be considered to be the result of natural variability in the rock mass. In addition, the database used seems to provide a relatively good picture of the crustal stress extremes, and supports the supposition that the data-set contains stress measurements close to the crustal stress limits. The fits of limit curves to the stress data have not contributed much more towards a pre-mining stress model. However, partitioning the space between the lithostatic stress line and the stress limit curves and determining the probability that a stress state will be found in any of the partitions will contribute much more toward a generic pre-mining stress model for southern Africa.

Figure 17 shows the divisions, with the stress data measurements from the individual measurements from the consistent database superimposed. The counts were done on the individual measurements from the consistent database, since including the 50 averages would have resulted in double counting. This procedure was repeated for the depth ranges m, m, and m, and plotted as the probability curves in Figures Figure 18 shows that the pre-mining stress above m tends to be lithostatic, although there is variation in all three of the components on both sides of the lithostatic line.

The maximum horizontal stress tends to be bigger than the vertical overburden stress between 0 and m below surface. The minimum horizontal principal stress is generally equal to the vertical stress between surface and m. From m to m there is still a peak around the lithostatic stress line for the vertical stress component, while the two horizontal stress components are now more spread out, with the maximum horizontal stress being larger than the overburden stress, and the minimum horizontal stress being smaller than the overburden stress. Between m and m there are only 19 consistent stress measurements, which are insufficient to provide any definite trends.

It appears that the vertical and maximum horizontal components are equal, and that they peak weakly on the lithostatic line.

The minimum horizontal stress tends to be lower than the other two components. Much more data will be necessary to clarify these trends. The most common stress state in the lithosphere is likely to be lithostatic, with increasing deviations from this state at shallower depths, decreasing to the purely lithostatic stress state once molten rocks are reached in the mantle. The depth range of the stress measurements in Figures 18 to 20 is too small to show this trend, but they do show a tendency toward the lithostatic state. Certainly, igneous rocks are liquid when they are emplaced, hence their stress states will always start in the lithostatic state.

Cooling and other effects will alter this stress state later. The same can be said for sedimentary rocks: the pore-water and sediment mixture will almost certainly behave as if it were a liquid prior to solidification, resulting in a more-or-less lithostatic stress state at the time of formation - in direct contradiction of the rigid confinement model discussed above and found not to be in agreement with the stress measurements.

The initial stress state at solidification lithifi-cation is likely to be lithostatic until it is altered by other geological processes after formation. Metamorphic rocks may also start in the lithostatic stress state if they have undergone partial melting. Sometimes the fabric of metamorphic rocks for example the alignment of platy minerals such as mica show that the rock was subjected to anisotropic pressure, and that the rock was probably able to sustain the deviatoric stress at the time.

Such rocks probably were not in the lithostatic stress state once they had cooled. The way to use the probabilities in Table VIII is to first select the depth range in which the mine or part of the mine falls for mines deeper than m, use the m range. This means that, since stress variations appear to exist in rock at all scales, variations such as these will be seen both on small and large scales.

This will be manifested in the haulage by zones centimetres to metres, and even tens of metres long, where ground conditions are bad and the tunnel needs extra support. This will happen only where the stresses are extreme enough to cause rock instability near the tunnel periphery. These poor areas could be interspersed by good ground conditions at scales ranging from centimetres to tens of metres where less support is needed.

If the conditions appear to be uniformly good, it is likely that there are simply no areas present where the stresses are extreme. Overall conditions like these could exist on a mine-wide scale.

## Rock Stress and Its Measurement

Since we have insufficient evidence, this interpretation is debatable and will remain so until comprehensive and detailed stress databases from underground mines and civil projects have been compiled. The actual stress values have to be quantified for the local rock mass in question so that modelling can be used to quantify where failure may occur, and which stress combinations will cause failure.

This is done by finding the lithostatic stress line for the particular rock mass using the local density of the rock, or alternatively the generic value of 0. The Hoek-Brown stress limits are found using either local rock mass strength parameters or the generic parameters given in Table VII.

The divisions are then found by dividing the interval between the lithostatic line and the Hoek-Brown limits in tension and compression. This must be done at the appropriate depth, and the actual stresses so obtained for each division are relevant for modelling.

The directions of the horizontal stresses are not specified in the model, since there should be either local stress measurements, trends in geological structure, or other clues on the mine giving the direction of the maximum and minimum horizontal stresses. This stress model in Table VIII , combined with Equation [13] is a generic pre-mining stress model for southern Africa, given the stress measurements recorded over the last forty years.

It is incomplete, and should be modified as new data becomes available. It may change from one geological terrane to another, for example the model for the Bushveld Complex would be different to that for gold mines in the Witwatersrand Basin. As an approximation it assumes that the vertical and horizontal stresses are principal stresses Amadei and Stephansson, , pp. It needs to be adjusted for local mining conditions and for the local rock mass properties.

The most important local feature will be the geological trends, which will provide information on the directions and possibly even the relative magnitudes of the horizontal stresses. In some circumstances there may be geological data that shows the pre-mining stress tensor has inclined principal components. The generic stress model is therefore not the stress model that one should expect to find in every situation, but it is one that can be adapted to local conditions using the guidelines provided.

The proposed model will remain a very general model of pre-mining stress until more detail of the stress state in the Earth's crust becomes available. Mining has barely penetrated the Earth's crust, and our knowledge of crustal stresses is sketchy at best.

Pre-mining stress data remains relatively rare, and until much more data is available, the proposed stress model will remain general. The variety of stress values obtained and shown is an imprint of the various geological histories of the rock formations in which they were made. Thus the Southern African Stress Database represents a polyglot of stress states from rock formations of different ages, geologic histories, and structures. Despite this, there is a strong relationship between stress and depth for all the vertical and horizontal principal stress components. The variability of measured stresses, coupled with the simplicity of pre-mining stress models such as the rigid confinement model, confirms why none of the simple models show a good fit to the data.

The proposed model recognizes this variability, which seems to exist at all scales. It also recognizes the limits on stress variability imposed by the rock mass strength, and uses generic rock mass parameters and the Hoek-Brown Failure Criterion to define these limits. Any extensions to greater depth should be supported by further research and additional measurements.

Ideally, at each mine or construction project, rock mechanics practitioners should generate the range of stress states possible using the model described above, and combine it with locally measured rock stresses if these are available. From this, it will be possible to construct a probability-based range of stresses for the mine or construction project. The stress measurements should also be used to confirm whether the vertical-horizontal principal stress tensor approximation is valid, and if so, to define the directions and magnitudes of the horizontal principal stress components.

If the vertical-horizontal principal stress tensor approximation is not valid, then rock mass structure such as dip, strike, jointing, bedding, and other features should be used to deduce an overall orientation for the principal stress tensor. The following specific conclusions arise from this study:. Abramowitz, M. Handbook of Mathematical Functions. Amadei, B. Rock Stress and its Measurement. Chapman and Hall, London. Brady, B. Rock Mechanics for Underground Mining.

Springer, Dordrecht, Netherlands. Deacon, D. Discussion: The measurement of stress in rock, by E. Gay, N. In situ stress measurements in Southern Africa. Tectonophysics, vol. Handley, M. May A review of sequential grid mining method employed at Elandsrand Gold Mine. Rock Mechanics for Mining Practitioners. In preparation. Haxby, W. Stresses induced by the addition and removal of overburden and associated thermal effects.

Geology, vol. Heidbach, O. Commission for the Geological Map of the World, Paris, doi Underground Excavations in Rock. Institution of Mining and Metallurgy, London. Jaeger, J. Fundamentals of Rock Mechanics. Blackwell Publishing, pp. Leeman, E. The measurement of stress in rock. The determination of the complete state of stress in rock in a single borehole - laboratory and underground measurements.

Lowrie, W. Fundamentals of Geophysics. Cambridge University Press. McCarthy, T. Struik Nature, Cape Town. McGarr, A. State of stress in the Earth's crust. Annual Reviews of Earth and Planetary Sciences, vol. Mooney, W. CRUST 5. Journal of Geophysical Research, vol. Ortlepp, W. The elastic analysis of observed strata movement by means of an electrical analogue.

Pallister, G. The Measurement of Primitive Rock Stress. Price, N. Pergamon Press, oxford. Rummel, F. Stress and tectonics of the upper continental crust - a review.

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Ryder, J. Salamon, M. Elastic analysis of displacements and stresses induced by the mining of seam or reef deposits. An analogue solution for determining the elastic response of strata surrounding tabular mining excavations.

- Pre-mining stress model for subsurface excavations in southern Africa;
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Stacey, T. Department of Mineral Resources, Johannesburg. In situ stresses in mining areas in South Africa. Taylor, J. An Introduction to Error Analysis: The study of uncertainties in physical measurements.

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