# A New Method to Measure and Calculate Tri-axial Static Strain Change Based on Relative Displacements Between Points for Rock Mass and Structure

• Ruilin Yang
Technical Note

## Introduction

All rock excavations by means of natural gravity caving, mechanical excavation, or blasting cause redistribution of the static stress and strain in the remaining rock mass. It is critical to measure the change of the strain state. Such a measurement can serve as quantification of the effectiveness of an excavation method or a blast in terms of minimized damage to the remaining rock or structure.

Due to the discontinuous nature of the rock mass with local joints and cracks, small strain gauges placed at a spot in a rock surface cannot provide representative measurements for the strain in the rock mass on an engineering scale (e.g., a few meters) around the measurement point. Such measurement with strain gauges is not useful for the rock mass. At present, there is no method available to calculate strain changes over a large area of a rock slope or an underground structure. New measurement technologies developed over the last few decades, such as GPS, radar systems,...

## Keywords

Strain change Blast damage Piecewise average 3D strain Survey points Rock structures Displacement gradient Laser scanning

## List of Symbols

xA,yA, zA

Coordinates of the point A (m)

uA,vA, wA

Displacement components in x, y, and z directions of point A (m)

Δu, Δv, Δw

Relative displacement in x, y, and z directions between two points (m)

$$\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial u}{\partial z}$$

Gradients in x, y, and z directions of the displacement component in x direction

$$\varepsilon_{ij}$$

Strain component and where i, j = x, y, z (dimensionless)

$$\gamma_{ {\rm max} }$$

Maximum shear strain (dimensionless)

e

Extensional strain (dimensionless)

$$\varepsilon_{i}$$

Principal strain and where i = 1, 2, 3 (dimensionless)

$$\eta_{ij}$$

Finite strain component and where i, j = x, y, z (dimensionless)

α, β, γ

Directional cosines of a principal strain relative to x-, y-, and z-axes (dimensionless)

SVD

Singular value decomposition method

A

M × N Matrix M ≥ N

U

M × N Column-orthogonal matrix

W

N ×  N Diagonal matrix with positive or zero elements

V

Transpose of N × N orthogonal matrix

A

Unknown vector for the displacement gradients

B

Row vector established from M relative displacement components

Χ

Error residual for the least-square minimization

## Notes

### Acknowledgements

Orica Chile Colleagues Francis Fritis, Roberto Veragua, Natalia Ortega, Nuria Salvador, Rolando Fuentes, and Stephen Jeric supported the field setup and data collections. They should be greatly acknowledged.

## References

1. Bhandari AR, Powrie W, Harkness RM (2012) A digital image-based deformation measurement system for tri-axial tests. Geotech Test J 35(2):209–226Google Scholar
2. Bower AF (2011) Applied mechanics of solids. CRC Press, Taylor & Francis Group, Boca Raton, FLGoogle Scholar
3. Cai M, Peng H (2011) Advances of in situ stress measurement in China. Int J Rock Mech Min Sci Geomech Abstr 3(4):373–384Google Scholar
4. Cardozo N, Allmendinger RW (2009) SSPX: a program to compute strain from displacement/velocity data. Comput Geosci 35:1343–1357
5. Fischer DJ (1982) Near surface stress measurements in a candidate rock mass for superconductive magnetic energy storage, MS Thesis, University of Wisconsin-MadisonGoogle Scholar
6. Ghosh AK (2008) Rock stress measurements for underground excavations. In: The 12th international conference of international association for computer methods and advances in geomechanics (IACMAG), 1–6 October, 2008, Goa, IndiaGoogle Scholar
7. Hooker VE, Aggson JR, Bickel DL (1974) Improvements in the three-component borehole deformation gage and overcoring technique (with appendix by W.I. Duvall), USBR RI-7894Google Scholar
8. Hudson JA, Harrison JP (1997) Engineering Rock Mechanics, PergamonGoogle Scholar
9. Kim K, Franklin A (1987) Suggested methods for rock stress determination. Int J Rock Mech Min Sci Geomech Abstr 24(1):53–74
10. Lin W, Kwaśniewski M, Imamura T, Matsuki K (2006) Determination of three-dimensional in situ stresses from anelastic strain recovery measurement of cores at great depth. Tectonophysics 426(2006):221–238
11. McHugh EL, Dwyer J, Long DG, Sabine C (2006) Applications of ground-based radar to mine slope monitoring, national institute for occupational safety and health (NIOSH) Publication, Report of Investigations 9666, pp 1–32Google Scholar
12. Ness M, Schmelter S, Sharma R, Sullivan M (2012) Practical considerations for utilization of semi-automated slope monitoring data streams to mitigate geomechanical risk in open pit mining. SME, 2012, SeattleGoogle Scholar
13. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2005) Numerical recipes in C ++, the art of scientific computing, 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
14. Stacey TR (1981) A simple extension strain criterion for fracture of brittle rock. Int J Rock Mech Min Sci Geomech Abstr 18(6):469–474
15. Timoshenko SP, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill, New York, p 230Google Scholar
16. Widarsono B, Marsden JR, King (1998) In situ stress prediction using differential strain analysis and ultrasonic shear-wave splitting, Geological Society, vol 136. Special Publications 1998, London, pp 185–195Google Scholar
17. Wilkins R, Shwydiuk L (2012) Multi-sensor displacement monitoring at teck highland valley copper partnership, SME 2012, SeattleGoogle Scholar
18. Yang R (2010) Method to calculate dynamic strain from blast vibration monitoring. Orica mining services technical report, July 2010Google Scholar
19. Yang R, Kay DB (2011) Multiple seed blast vibration modeling for tunnel blasting in urban environments. Blast Fragmentation 5(2):109–122Google Scholar
20. Yang R, Lownds M (2011) Modeling effect of delay scatter on peak particle velocity of blast vibration using a multiple seed waveform vibration model. Blast Fragmentation 5(1):31–46Google Scholar
21. Yang R, Scovira DS (2008) A model of peak amplitude prediction for near field blast vibration based on non-linear charge weight superposition, time window broadening, and delay time modeling. Blast Fragm 2(2):91–116Google Scholar